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Learning Continuous Neural Representation of Stochastic Hybrid Systems
Sangli Teng, Hang Liu, Koushil Sreenath
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
Last extracted: 10/4/2026, 3:26:46 AM
Summary
This paper proposes a method to approximate Stochastic Hybrid Systems (SHS) with continuous Stochastic Differential Equations (SDEs) in a higher-dimensional latent space. By introducing auxiliary variables to encode stochastic reset branches, the authors transform the system into an Augmented SHS (A-SHS) with deterministic resets. Using topological gluing and the Whitney Embedding Theorem, the discontinuous hybrid flow is mapped to a continuous manifold, allowing the probability evolution to be modeled by a single latent SDE without explicit reset terms, mode labeling, or trajectory segmentation.
Entities (6)
Relation Signals (8)
Hybrid Fokker-Planck Equation → describesevolutionof → Stochastic Hybrid System
confidence 95% · Its probability evolution can be described by a hybrid Fokker-Planck (HFP) equation...
Stochastic Hybrid System → hascomponent → Markov Reset Kernel
confidence 95% · ...and a Markov reset kernel triggered on the guard surface.
Stochastic Hybrid System → hascomponent → Stochastic Differential Equation
confidence 95% · A stochastic hybrid system (SHS) is governed by a stochastic differential equation (SDE) describing the continuous dynamics...
Latent SDE → models → Stochastic Hybrid System
confidence 93% · enabling a single latent SDE to recover the probability evolution of the SHS...
Stochastic Hybrid System → isapproximatedby → Augmented Stochastic Hybrid System
confidence 92% · We present an augmented-state SHS (A-SHS) with a deterministic reset to approximate an SHS with a Markov reset kernel.
Topological Gluing → allowsembeddingvia → Whitney Embedding Theorem
confidence 90% · By the embedding theorem, the glued manifold can then be embedded into a higher-dimensional Euclidean space.
Augmented Stochastic Hybrid System → uses → Auxiliary Variables
confidence 90% · The key to this result is to encode different branches of the reset kernel using auxiliary variables...
Augmented Stochastic Hybrid System → enables → Topological Gluing
confidence 88% · ...transforming the resets into deterministic ones that enable topological gluing.
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Abstract
Abstract:A stochastic hybrid system (SHS) is governed by a stochastic differential equation (SDE) describing the continuous dynamics and a Markov reset kernel triggered on the guard surface. Its probability evolution can be described by a hybrid Fokker-Planck (HFP) equation with a partial differential term corresponding to the SDE and an integral term arising from the reset kernel. This work shows that such an SHS can be approximated by an SDE in a higher-dimensional latent space where the sample paths are continuous. The key to this result is to encode different branches of the reset kernel using auxiliary variables, transforming the resets into deterministic ones that enable topological gluing. By the embedding theorem, the glued manifold can then be embedded into a higher-dimensional Euclidean space. We show that the probability evolution on the embedded image no longer requires explicit reset terms in the HFP equation. Building on this theorem, we design a loss that matches the evolving state distributions, enabling a single latent SDE to recover the probability evolution of the SHS without mode labeling, trajectory segmentation, or event-based simulations.
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- Source: https://arxiv.org/abs/2609.38893v1
- Canonical: https://arxiv.org/abs/2609.38893v1
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Learning Continuous Neural Representation of Stochastic Hybrid Systems Sangli Teng Hang Liu Koushil Sreenath Abstract A stochastic hybrid system (SHS) is governed by a stochastic differential equation (SDE) describing the continuous dynamics and a Markov reset kernel triggered on the guard surface. Its probability evolution can be described by a hybrid Fokker–Planck (HFP) equation with a partial differential term corresponding to the SDE and an integral term arising from the reset kernel. This work shows that such an SHS can be approximated by an SDE in a higher-dimensional latent space where the sample paths are continuous. The key to this result is to encode different branches of the reset kernel using auxiliary variables, transforming the resets into deterministic ones that enable topological gluing. By the embedding theorem, the glued manifold can then be embedded into a higher-dimensional Euclidean space. We show that the probability evolution on the embedded image no longer requires explicit reset terms in the HFP equation. Building on this theorem, we design a loss that matches the evolving state distributions, enabling a single latent SDE to recover the probability evolution of the SHS without mode labeling, trajectory segmentation, or event-based simulations. 1 Introduction Figure 1: We approximate SHSs with a Markov reset kernel by an A-SHS with a deterministic reset function that allows the topological gluing to make the hybrid flow continuous. Hybrid systems are a powerful abstraction for dynamical systems such as robotics (Posa et al., 2014, Westervelt et al., 2003) and autonomous driving (Wang et al., 2024). To model the stochasticity, we can consider stochastic hybrid systems (SHSs) (Hu et al., 2000, Bujorianu and Lygeros, 2006). As SHSs are governed by compositions of continuous-time dynamics and discrete events, learning both components concurrently from time-series data is challenging. To address this challenge, the recent work (Teng et al., 2026b, Teng et al., 2026a) applies the geometric hybrid system theory (Simic et al., 2005, Hirsch, 1976) to represent a hybrid system by continuous dynamics on a manifold. In (Teng et al., 2026b, Teng et al., 2026a), the manifold representation admits an embedding in a higher-dimensional Euclidean space, which can be learned by a continuous latent ODE (Chen et al., 2018). However, the results in (Simic et al., 2005) assume deterministic resets, which cannot be applied to SHSs where the reset kernel is a probability distribution. In this work, we extend the geometric framework (Teng et al., 2026b, Teng et al., 2026a) to SHSs by representing different reset branches with auxiliary variables, which enables us to represent SHSs by a continuous SDE on a manifold (see Figure 1). The main contributions are summarized as: 1. We present an augmented-state SHS (A-SHS) with a deterministic reset to approximate an SHS with a Markov reset kernel. The A-SHS is then embedded into a higher-dimensional Euclidean space where the sample paths become continuous. 2. We derive the hybrid Fokker-Planck (HFP) equation of the A-SHS and show that the source and sink terms in the HFP equation vanish in the embedded image. This result enables us to predict the probability distribution of A-SHS by a continuous SDE without identifying the guard surface after the embedding. 3. Based on the theoretical result, we can accuretely predict the distribution evolution of SHS from time-series data vy a latent SDE without any trajectory segmentation, mode labeling, or event-based simulations. 2 Related Works Hybrid System Theory: Hybrid automata (Lygeros et al., 2003, Goebel et al., 2009) model continuous-time dynamics within modes and state-triggered transitions and resets between them. Stochastic hybrid systems (Hu et al., 2000) incorporate stochasticity into both continuous dynamics and discrete resets. Specifically, (Hu et al., 2000) models continuous evolution as a stochastic differential equation (SDE) and resets as Markov kernels triggered on lower-dimensional guard surfaces, while (Bujorianu and Lygeros, 2006) further allows spontaneous jumps within the state-space interior. Hybrid inclusions instead use set-valued flow and reset maps to represent multiple admissible reset branches (Aubin et al., 2002, Goebel and Teel, 2006). Whereas SHSs assign probability laws to reset outcomes, hybrid inclusions specify only the admissible outcomes. To consider the state distribution, the hybrid Fokker–Planck (HFP) equations include additional terms that account for reset-induced sources and sinks (Wang and Lee, 2020, Wang and Lee, 2022). Although difficult to solve, they provide a powerful tool for studying SHS distribution evolution. Learning Hybrid Systems: Learning hybrid systems from time-series data requires jointly identifying modes, guard surfaces, and continuous dynamics, making the problem inherently combinatorial. (Roll et al., 2004) uses mixed-integer programming (MIP) to identify piecewise-affine (PWA) systems and reformulates the problem as a linear complementarity problem to avoid MIP. However, PWA systems are a restricted class of hybrid automata (HA) with identity resets. To learn general SHSs, (Poli et al., 2021) uses normalizing flows for event transitions and Neural ODEs (Chen et al., 2018) for continuous dynamics. Similarly, (Liu et al., 2025b) learns discrete-time HA using a mode classifier and state-space model. Both methods require heuristic trajectory segmentation, which is sensitive to stochastic noise, while (Ochoa and Sanfelice, 2026) learns vector fields and reset functions assuming known segmentations. Alternatively, (Teng et al., 2026b, Teng et al., 2026a) builds on the theory in (Simic et al., 2005) to represent HA as Neural ODEs in higher-dimensional latent spaces. Learning Stochastic Differential Equations: Recent work learns stochastic dynamics at different statistical levels. (Kidger et al., 2021) trains a Neural SDE generator against a Neural CDE critic using a Wasserstein path-level loss. Most closely related, (Bartosh et al., 2025) parameterizes posterior marginals and trains latent SDEs without simulation using an ELBO with a path-space KL loss. (Neklyudov et al., 2023) recovers a canonical probability flow from unpaired temporal marginals through a variational action objective, while (Kiyohara et al., 2025) learns conditional normalizing-flow transition kernels using likelihood and Chapman–Kolmogorov consistency. (Seifner et al., 2025) pretrains a transformer through supervised drift–diffusion regression, whereas (Ilersich and Nair, 2025) learns coupled macro–microscale latent SDEs using an ELBO and Product-of-Experts likelihood. Unlike these methods, the proposed latent SDE framework represents stochastic resets without explicit event-based simulation. 3 Preliminaries & Problem Formulation 3.1 Stochastic Hybrid Systems We refer to (Hu et al., 2000, Wang and Lee, 2020) for a preliminary overview of SHSs. Consider a SHS with state-triggered Markov reset kernel as illustrated in Figure 1: dx dx =f(t,x)dt+g(t,x)dw,x∉S, =f(t,x)dt+g(t,x)dw,\ x∉ S, (SHS) x+ x^+ ∼K(x−,dx+),x−∈S, K(x^-,dx^+),\ x^-∈ S, where w is an n−dimn- Wiener process, x evolves in the domain M⊂ℝnM ^n with smooth boundary and S⊂∂MS⊂∂M is the (n−1)−dim(n-1)- guard surface. When x∉Sx∉ S, the system is governed by the drift f:ℝ×M→Mf:R× M→ M and the diffusion matrix g:ℝ×M→ℝn×ng:R× M ^n× n. The reset is described by a Markov kernel K:S×ℬ(M)→[0,1]K:S×B(M)→[0,1], where ℬ(M)B(M) is all Borel sets in MM. K(x−,A)K(x^-,A) denotes the probability that the post-reset state x+x^+ lies in a measurable set A⊂MA⊂M conditioned on x−∈Sx^-∈ S. Assume x admits a density p(t,x)p(t,x) on M. Let D(t,x)=12g(t,x)g(t,x)⊤D(t,x)= 12g(t,x)g(t,x) . The probability flux associated with the continuous diffusion is: J(t,x)=f(t,x)p(t,x)−∇⋅(D(t,x)p(t,x))J(t,x)=f(t,x)p(t,x)-∇·(D(t,x)p(t,x)) (1) Let nS(x)n_S(x) be the chosen normal direction on S, and define the guard flux density by α(t,x)=J(t,x)⋅nS(x),∀x∈S.α(t,x)=J(t,x)· n_S(x),\ ∀ x∈ S. (2) Consider the induced density of K(x−,dx)K(x^-,dx), i.e., κ(x−,x+)κ(x^-,x^+) as defined by K(x−,A)=:∫Aκ(x−,x+)dx+,A⊂M.K(x^-,A)=: _Aκ(x^-,x^+)dx^+,A⊂M. The hybrid Fokker–Planck (HFP) equation with reset is: ∂p(t,x)∂t= ∂ p(t,x)∂ t= −∇⋅J(t,x)+∫Sκ(ξ,x)α(t,ξ)dσS(ξ)−∫Sδ(x−ξ)α(t,ξ)dσS(ξ). -∇· J(t,x)+ _S [rgb]0,0,1κ(ξ,x)α(t,ξ)d _S(ξ)- _S [rgb]1,0,0δ(x-ξ)α(t,ξ)d _S(ξ). (HFP) Here dσSd _S is the surface measure on S. The second term shows that the probability mass leaving the guard point ξ∈Sξ∈ S is redistributed to post-reset states by x+∼κ(ξ,x+)x^+ κ(ξ,x^+). The last term suggests that the outgoing mass is removed from the pre-reset point ξ∈Sξ∈ S. 3.2 Geometric Representation of Hybrid Systems When the stochasticity of SHS vanishes, the Markov kernel K(x−,dx+)K(x^-,dx^+) becomes a classical reset function R(⋅)R(·) and SHS reduces to a hybrid automaton: x˙ x =f(t,x),x∉S, =f(t,x),\ \ \ x∉ S, (HA) x+ x^+ =R(x−),x−∈S. =R(x^-),\ x^-∈ S. Assume R is a diffeomorphism and S∈∂MS∈∂ M satisfies the regularity condition specified in (Simic et al., 2005), we have the following result to make HA continuous: Theorem 1. Let ∼ be the equivalence relation by x∼R(x),∀x∈Sx R(x),∀ x∈ S. We collapse the equivalence class by ∼ to a point to obtain the quotient manifold M/∼M/ . By Simic et al. (2005), M/∼M/ is a n−dimn- manifold with boundary, and both M/∼M/ and its boundary are piecewise smooth. Theorem 2 (Whitney Embedding Theorem (Hirsch, 1976)). Any CrC^r-manifold (r≥1)(r≥ 1) of dimension n can be embedded into ℝ2nR^2n. Remark 1. By Theorem 2, M/∼M/ admits an embedding that is globally piecewise smooth. Therefore, the hybrid flow of HA can be represented by ODEs with continuous flow (Teng et al., 2026b, Teng et al., 2026a). However, Theorem 1 cannot be applied to SHS as the reset kernel is not a one-to-one mapping. 3.3 Problem Formulation Problem 1. Given N trajectories :=(t0:T,x0:T(k))k=1NX:=\(t_0:T,x_0:T^(k))\_k=1^N sampled from (SHS), learn its conditional path distribution p(x1:T∣x0)p(x_1:T x_0) at prescribed observation times t0:Tt_0:T and initial state x0x_0. We note that this problem is extremely challenging as the flow of SHS is discontinuous and stochastic, the guard surface S is unknown, and the reset is stochastic and set-valued. 4 Continuous Representation of Stochastic Hybrid System In this section, we extend the geometric framework to SHS. 4.1 Augmented Stochastic Hybrid Systems We define an augmented-state stochastic hybrid system (A-SHS) where different reset of x+∼K(x−,dx+)x^+ K(x^-,dx^+) are represented by an auxiliary variable v∈V⊂ℝnv∈ V ^n. For simplification, let the augmented state be y:=[x⊤,v⊤]⊤∈M×V⊆ℝ2ny:= bmatrixx ,\ v bmatrix ∈ M× V ^2n and we have: dy dy =f¯(t,y)dt+g¯(t,y)dw¯,(x,v)∉S×V, = f(t,y)dt+ g(t,y)d w, (x,v)∉ S× V, (A-SHS) x+ x^+ =x−+v−,v+=v−,(x−,v−)∈S×V. =x^-+v^-, v^+=v^-, (x^-,v^-)∈ S× V. with f¯=[fx⊤fv⊤]⊤:ℝ×(M×V)→ℝ2n f= bmatrixf_x &f_v bmatrix :R×(M× V) ^2n the drift, g¯=[gx⊤gv⊤]:ℝ×(M×V)→ℝ2n×2n g= bmatrixg_x &g_v bmatrix:R×(M× V) ^2n× 2n the diffusion, and w¯ w the Wiener process of dimension 2n2n. The reset map in the augmented state thus can be represented by R(x−,v−)=(x−+v−,v−).R(x^-,v^-)=(x^-+v^-,v^-).11 1 It is possible to consider other diffeomorphism for the augmented systems. Under the Euclidean addition, R is injective and is a diffeomorphism onto its image, since (x+,v+)(x^+,v^+) uniquely determines (x−,v−)=(x+−v+,v+)(x^-,v^-)=(x^+-v^+,v^+). We denote the density of y in A-SHS as p¯(t,y)=p¯(t,x,v) p(t,y)= p(t,x,v). Then we have the D¯(t,y)=12g¯(t,y)g¯(t,y)⊤∈ℝ2n×2n D(t,y)= 12 g(t,y) g(t,y) ^2n× 2n and the current of y: J¯(t,y)=f¯(t,y)p¯(t,y)−∇⋅(D¯(t,x)p¯(t,y)) J(t,y)= f(t,y) p(t,y)-∇·( D(t,x) p(t,y)) (3) Given the normal vector of S×VS× V, i.e, n¯(x,v)=(nS(x),0n) n(x,v)=(n_S(x),0_n), we have the outgoing flux density as: α¯(t,x,v)=J¯(t,x,v)⋅n¯(x,v),(x,v)∈S×V. α(t,x,v)= J(t,x,v)· n(x,v),\ (x,v)∈ S× V. (4) Therefore, the augmented hybrid Fokker–Planck equation is the measure-valued equation ∂p¯(t,y)∂t=−∇⋅J¯(t,y)+∫S∫V[δ(y−[x′+v′v′])−δ(y−[x′v′])]α¯(t,x,v)dv′dσS(x′). ∂ p(t,y)∂ t=-∇· J(t,y)+ _S _V [δ(y- bmatrixx +v \\ v bmatrix)-δ(y- bmatrixx \\ v bmatrix) ] α(t,x,v)dv \ d _S(x ). (A-HFP) The last term indicates the mass injection and removal by the deterministic reset (x+,v+)=(x−+v−,v−)(x^+,v^+)=(x^-+v^-,v^-). Compared to (HFP), the source and sink terms only involve Dirac functions. Ideally, (A-SHS) will distinguish different reset branch by the auxiliary variable v, as shown in Figure 1. 4.2 Embedding A-SHS into Higher-Dimensional Space We now leverage the Whitney Embedding Theorem to eliminate the discontinuity of A-SHS. Define the equivalence relation on M×VM× V by (x,v)∼(x+v,v),(x,v)∈S×V.(x,v) (x+v,v),\ (x,v)∈ S× V. Let π:M×V→(M×V)/∼π:M× V→(M× V)/ be the quotient map22 2 The map that identifies all points in each equivalence class as a single point.. Then we have the following theorem: Lemma 1. There exists a piecewise smooth function E:M×V→ℝm≥4nE:M× V ^m≥ 4n, such that E(x,v)=E(x+v,v),(x,v)∈S×V,E(x,v)=E(x+v,v),\ (x,v)∈ S× V, and the map E¯ E defined by E¯∘π:=E E π:=E is an embedding on (M×V)/∼(M× V)/ . Proof. By (Simic et al., 2005), the quotient manifold (M×V)/∼(M× V)/ is an (n+n)−(n+n)-dimensional piecewise continuous manifold. By the Whitney Embedding theorem, (M×V)/∼(M× V)/ can be embedded into ℝmR^m whenever m≥2(n+n)=4nm≥ 2(n+n)=4n. Let the embedding be E¯ E. Since π identifies (x,v)∼(x+v,v)(x,v) (x+v,v), the composed map E=E¯∘πE= E π satisfies E(x,v)=E(x+v,v),(x,v)∈S×V.E(x,v)=E(x+v,v),\ (x,v)∈ S× V. ∎ By Lemma 1, we can eliminate the reset of A-SHS in the embedded image as shown in Figure 1: Theorem 3 (Embedding of A-SHS). Let z=E(x,v)∈Z:=ImgEz=E(x,v)∈ Z:=ImgE, with E the embedding defined in Lemma 1. We have the pushforward dynamics of z by a piecewise smooth SDE: dz dz =F(t,z)dt+G(t,z)dw,∀z∈Z. =F(t,z)dt+G(t,z)dw,\ ∀ z∈ Z. (5) Let Dz(t,z):=12G(t,z)G(t,z)⊤D_z(t,z):= 12G(t,z)G(t,z) , the density pz(t,z)p_z(t,z) is governed by the Fokker–Planck equation: ∂pz(t,z)∂t=−∇⋅(F(t,z)pz(t,z))+∇⋅∇⋅(Dz(t,z)pz(t,z)),z∈Z. ∂ p_z(t,z)∂ t=-∇· (F(t,z)p_z(t,z) )+∇·∇· (D_z(t,z)p_z(t,z) ),\ z∈ Z. (6) Proof. We first show the pushforward dynamics of A-SHS. By E(x,v)=E(x+v,v),(x,v)∈S×VE(x,v)=E(x+v,v),(x,v)∈ S× V, the discrete reset vanishes at the guard surface S×VS× V by: z+=E(x+,v+)=E(x−+v−,v−)=E(x−,v−)=z−,∀(x−,v−)∈S×V.z^+=E(x^+,v^+)=E(x^-+v^-,v^-)=E(x^-,v^-)=z^-,\ ∀(x^-,v^-)∈ S× V. (7) Then we have the pushforward dynamics by the Taylor expansion of Itô process (Øksendal, 2003): F(t,z):=∂E∂yf¯(t,x,v)+12∑i,j=12nD¯ij(t,x,v)∂2E∂yi∂yj,G(t,z):=∂E∂yg¯(t,x,v). F(t,z):= ∂ E∂ y f(t,x,v)+ 12Σ _i,j=1^2n D_ij(t,x,v) ∂^2E∂ y_i∂ y_j, \ G(t,z):= ∂ E∂ y g(t,x,v). (8) We note that z is an embedding of (x,v)(x,v) that is injective and has a unique inverse. Therefore, F and G can be uniquely represented by t and z. Thus we have (5) as a function of z. It remains to check the reset contribution. For the reset source term, integrating against a smooth test function on compact support, i.e., ϕ(z):Z→ℝ,ϕ(z)∈Cc∞(Z)φ(z):Z ,φ(z)∈ C_c^∞(Z) yields: ∫M×Vϕ(E(y′))(∫S∫Vδ(y′−(x+v,v))α¯(t,x,v)vdσS(x))dy′. _M× Vφ(E(y )) ( _S _Vδ (y -(x+v,v) ) α(t,x,v)\,dv\,d _S(x) )dy . (9) By the property of the Dirac measure, we move the test function to nested integration to reduce the above term to ∫S∫Vϕ(E(s+v,v))α¯(t,s,v)vdσS(s) _S _Vφ (E(s+v,v) ) α(t,s,v)\,dv\,d _S(s). For any smooth test function ϕ:Z→ℝφ:Z , the net reset source-sink contribution to the forward equation in the embedded image is IR[ϕ]=∫S∫V(ϕ(E(x+v,v))−ϕ(E(x,v)))α¯(t,x,v)vdσS(x).I_R[φ]= _S _V (φ (E(x+v,v) )-φ (E(x,v) ) ) α(t,x,v)dvd _S(x). As E(x+v,v)=E(x,v),∀(x,v)∈S×VE(x+v,v)=E(x,v),∀(x,v)∈ S× V, this integrand vanishes point-wise, yielding IR[ϕ]=0I_R[φ]=0. Therefore, the pushed-forward dynamics on z can be governed entirely by (6). ∎ 4.3 A Construction of A-SHS Figure 2: Target distribution. The dashed lines indicate the boundary of the partition of unity. A-SHS provides a pathway to replace the discrete reset with a continuous SDE. Now we construct such an A-SHS to approximate SHS. We define a residual reset kernel Q:S×ℬ(V)→[0,1]Q:S×B(V)→[0,1]: K(x−,A)=Q(x−,v∈V|x−+v∈A),A∈ℬ(M).K(x^-,A)=Q (x^-,\v∈ V\ |\ x^-+v∈ A\ ),\ A (M). (10) Equivalently, if v−∼Q(x−,dv)v^- Q(x^-,dv), then x+=x−+v−∼K(x−,dx+)x^+=x^-+v^- K(x^-,dx^+). Similar to κ(x−,x+)κ(x^-,x^+), we define the density of Q by κQ(x−,v) _Q(x^-,v). Now we show that we can construct a target distribution whose marginal density in the x−x-direction is equivalent to the solution of SHS. Construction 1 (Target Density). Define the pre-reset set Σ−:=S×V ^-:=S× V, its post-reset image Σ+ ^+, and the interior Σ∘:=(M×V)∖(Σ−∪Σ+)¯ :=(M× V) ( ^-∪ ^+). Let χ−(x)+χ+(x)+χ∘(x)=1χ^-(x)+χ^+(x)+χ (x)=1 be a smooth partition of unity33 3 Weight functions that allows local constructions to be blended smoothly into a global one. In Construction 1, v has the identical distribution as (10) on the guard surface and the pushforward on its image. associated with Σ− ^-, Σ+ ^+, and Σ∘ , and prescribe the conditional density of v: p⋆(v∣x)=χ−(x)κQ(x,v)+χ+(x)(R#κQ)(x,v)+χ∘(x)(v,μ,δ2I).p (v x)=χ^-(x) _Q(x,v)+χ^+(x)(R_\# _Q)(x,v)+χ (x)N(v;μ,δ^2I). (11) where R#κQR_\# _Q is the pushforward density of κQ _Q under the reset R. Let the solution of HFP be p⋆(t,x)p (t,x), we have the target density that preserves the marginal density of x for some T<∞T<∞: p¯⋆(t,x,v)=p⋆(t,x)p⋆(v|x),t∈[0,T], p (t,x,v)=p (t,x)p (v|x),\ t∈[0,T], (Target Density) and the target density J⋆:=[Jx⋆⊤,Jv⋆⊤]⊤J :=[J _x,\ J _v] given by (3). The target distribution is shown in Figure 2. Now we design A-SHS whose evolution of density will approximate p¯⋆(t,x,v) p (t,x,v). Construction 2 (Approximate A-SHS). Choose fx=ff_x=f and gx=gg_x=g to make the x-dynamics equivalent to SHS in the interior. Given the x-current Jx⋆=fxp¯⋆−∇x⋅(Dxxp¯⋆)−∇v⋅(Dxvp¯⋆)J_x =f_x p - _x·(D_x p )- _v·(D_xv p ), we solve for DxvD_xv. For any η>0η>0, assume D invertible, choose Dvv=Dxv⊤D−1Dxv+ηI,D_v=D_xv D^-1D_xv+η I, and solve ∂p⋆∂t=−∇⋅J⋆ ∂ p ∂ t=-∇· J for the v-dynamics. The resulting A-SHS dynamics are given by: f¯=[fJv⋆+∇x⋅(Dxv⊤p¯⋆)+∇v⋅(Dvvp¯⋆)p¯⋆],D¯=12g¯g¯⊤,g¯:=[g0gxvgv]. f= bmatrixf\\ J_v + _x· (D_xv p )+ _v· (D_v p ) p bmatrix, D= 12 g g , g:= bmatrixg&0\\ g_xv&g_v bmatrix. (12) Here, g¯ g is obtained from D¯ D by choosing gxv=Dxv⊤D−1g_xv=D_xv D^-1g and gv=2ηIg_v= 2ηI. Together with the deterministic reset R(x,v)=(x+v,v)R(x,v)=(x+v,v), these coefficients define an approximate A-SHS whose target density is p¯⋆(t,x,v) p (t,x,v). The construction of DxvD_xv is deferred to Appendix A. Remark 2 (Interior mixing and exponential forgetting). In the interior where χ∘=1χ =1, we have p⋆(v∣x)=(v,μ,δ2I)p (v x)=N(v;μ,δ^2I), Dxv=0D_xv=0, and J¯v⋆=0 J_v =0. Hence, we can choose fv=−λ(v−μ),Dvv=λδ2I,f_v=-λ(v-μ),\ D_v=λδ^2I, and the v−v-dynamics reduce to the Ornstein–Uhlenbeck (OU) process, i.e., dv=−λ(v−μ)dt+2λδdwv.dv=-λ(v-μ)\,dt+ 2λ\,δ\,dw_v. If the trajectory remains in the interior for τ, then v(t+τ)=μ+e−λτ(v(t)−μ)+δ1−e−2λτξ,ξ∼(0,I).v(t+τ)=μ+e^-λτ(v(t)-μ)+δ 1-e^-2λτ\,ξ,\ ξ (0,I). Consequently, the dependence of the receding reset v(t)v(t) is forgotten exponentially at rate λ and converges to Gaussian. As shown in Figure 2, Construction 2 first generates the desired density on Σ− ^- and then mixes it to a Gaussian in Σ∘ before the next reset. The details of both constructions are referred in Appendix A. 5 Main Algorithms Algorithm 1 Dataset X; sampler schedule βℓ=1L\ _ \_ =1^L; batch size B; rollouts per initial state M; training steps J; learning rate η. for j=1,…,Jj=1,…,J do Sample x0:T(k)k=1B\x_0:T^(k)\_k=1^B from X at t0:Tt_0:T zi(k,r)∼pθ(⋅∣xi(k))z_i^(k,r) p_θ(· x_i^(k)) ⊳ Encode z^0(k,r)←z0(k,r) z_0^(k,r)← z_0^(k,r) ⊳ Initialize z^0:T(k,r)←(13)z^0(k,r) z_0:T^(k,r) ( eq:SDE-EM) z_0^(k,r) ⊳ Rollout x^i(k,r)←Dθ(z^i(k,r)) x_i^(k,r)← D_θ( z_i^(k,r)) ⊳ Decode Compute ℒ(θ)L(θ) using (14) ⊳ Loss θ←θ−η∇θℒ(θ)θ←θ-η _θL(θ) ⊳ Update end for return pθ(z∣x)p_θ(z x), FθF_θ, GθG_θ, and DθD_θ Building on Theorem 3, we learn (SHS) using a stochastic encoder, a latent SDE, and a deterministic decoder. For clarity, we describe these components using a single observed trajectory x=x0:Tx=x_0:T at observation times t0:Tt_0:T and a single predicted trajectory x^=x^0:T x= x_0:T. Stochastic Encoder: The reset kernel x+∼κ(x−,x+)x^+ κ(x^-,x^+) assigns a distribution of post-reset states to each state x−x^-, inducing a distribution of reset displacements v=x+−x−v=x^+-x^-. By Theorem 3, the augmented state (x,v)(x,v) can be represented by a latent state z=E(x,v)z=E(x,v) by Lemma 1. We therefore model the encoder as z∼pθ(z∣x)z p_θ(z x) and sample zi∼pθ(z∣xi)z_i p_θ(z x_i) at tit_i. In practice, pθ(z∣x)p_θ(z x) is represented by a conditional diffusion sampler. Starting from ζL∼(0,I) _L (0,I), we apply L differentiable DDIM (Song et al., 2021) updates conditioned on x: ζℓ−1=α¯ℓ−1ζℓ−1−α¯ℓϵθ(ζℓ,ℓ,x)α¯ℓ+1−α¯ℓ−1ϵθ(ζℓ,ℓ,x),ℓ=L,…,1, _ -1= α_ -1 _ - 1- α_ \, _θ( _ , ,x) α_ + 1- α_ -1\, _θ( _ , ,x),\ =L,…,1, where α¯ℓ=∏s=1ℓ(1−βs) α_ = _s=1 (1- _s) and α¯0=1 α_0=1. The output is z=ζ0z= _0, and the noise predictor ϵθ _θ is trained end-to-end without a separate denoising loss. Latent SDE: Let dz=dimZd_z= Z. We represent the latent drift and diffusion by Fθ:ℝ×Z→ℝdzF_θ:R× Z ^d_z and Gθ:ℝ×Z→ℝdz×2nG_θ:R× Z ^d_z× 2n. Given z^0=z0 z_0=z_0, we generate z^=z^0:T z= z_0:T using the Euler–Maruyama scheme: z^i+1=z^i+Fθ(ti,z^i)Δti+Gθ(ti,z^i)Δwi,Δwi∼(0,ΔtiI2n), z_i+1= z_i+F_θ(t_i, z_i) t_i+G_θ(t_i, z_i) w_i,\ w_i (0, t_iI_2n), (13) where Δti=ti+1−ti t_i=t_i+1-t_i, i=0,…,T−1i=0,…,T-1, and the Brownian increments are independent. Deterministic Decoder: The inverse E−1E^-1 recovers an equivalence class in (M×V)/∼(M× V)/ . Choosing its post-reset representative at the glued seam defines a single-valued state decoder, which we approximate by Dθ:Z→MD_θ:Z→ M. The underlying decoder can be discontinuous at the seam: its one-sided limits recover x−x^- and x+x^+ while the latent trajectory remains continuous. We obtain the predicted trajectory x^=x^0:T x= x_0:T by applying x^i=Dθ(z^i) x_i=D_θ( z_i) at each observation time. Cost Function Design: At time tit_i, let pu,ip_u,i and p^u,i p_u,i denote the data and predicted distributions for u∈x,zu∈\x,z\, with pz,ip_z,i induced by encoding xix_i. Let pxp_x denote the data path distribution and p^x p_x the predicted path distribution when x0x_0 is drawn from its data distribution. For two distributions pAp_A and pBp_B with finite first moments, their squared energy distance is denoted by ℰ2(pA,pB)E^2(p_A,p_B)44 4 ℰ2(pA,pB)=2‖a−b‖2−‖a−a′‖2−‖b−b′‖2E^2(p_A,p_B)=2E\|a-b\|_2-E\|a-a \|_2-E\|b-b \|_2, where a,a′∼pAa,a p_A and b,b′∼pBb,b p_B are mutually independent.. Inspired by Theorem 3, we define the state distribution loss ℒu=1T+1∑i=0Tℰ2(pu,i,p^u,i)L_u= 1T+1 _i=0^TE^2(p_u,i, p_u,i), u∈x,zu∈\x,z\, to match the distribution evolution at each observation time. We also define the unconditional path distribution loss ℒxtraj=ℰ2(px,p^x)L_x^traj=E^2(p_x, p_x) to match the joint distribution across observation times. For the observed trajectory xx with initial state x0x_0, the conditional path distribution loss is ℒxtraj,c=12(T+1)dxℰ2(δx,p^x|x0)L_x^traj,c= 12 (T+1)d_xE^2( _x, p_x x_0), where dxd_x is the state dimension and p^x|x0 p_x x_0 is the conditional path distribution induced by the stochastic encoder, latent SDE, and decoder. With w denoting the weights and the loss ℒgL_g55 5 ℒg=1T+1∑i=0T‖Gθ(ti,z^i)‖F2L_g= 1T+1 _i=0^T\|G_θ(t_i, z_i)\|_F^2. to regularize diffusion, the overall objective is ℒ(θ)=wzℒz+wxℒx+wxtrajℒxtraj+wxtraj,cℒxtraj,c+wgℒg. (θ)=w_zL_z+w_xL_x+w_x^trajL_x^traj+w_x^traj,cL_x^traj,c+w_gL_g. (14) For multiple observed and independently generated trajectories, we estimate the distributional losses from empirical samples and average the conditional loss and diffusion regularizer over the corresponding trajectories. We summarize the training scheme in Algorithm 1, where we use the superscript (k,r)(k,r) to denote the r-th encoder sample corresponding to the k-th data trajectory, with batch size B and M encoder samples per state. 6 Numerical Experiments Figure 3: The density evolution of A-SHS for (15). We find that the marginal density of A-SHS in x matches that of (15). As shown in the x−vx-v plot, the density is mixed by the OU process first and then redistributed to the GMM to approximate the reset by Construction 1 and 2. Figure 4: The time evolution of the A-SHS and the embedded trajectory in 3D space. We can see that x is reset to a different state after it hits the guard at x=1x=1. The v replaces the role of the reset function with continuous paths. The augmented path can be embedded into a higher-dimensional space with variable z to mitigate the discontinuity. 6.1 Construction of A-SHS We illustrate the construction in Section 4.3 using a 1−dim1- hybrid system defined on (0,1](0,1] with constant drift and a Gaussian mixture reset with three peaks :=0,0.15,0.3C:=\0,0.15,0.3\: dx=dt+σdw,x<1,x+∼13∑xc∈TN(−∞,1)(xc,η2),x−=1. dx=dt+σ\,dw, x<1, x^+ 13 _x_c TN_(-∞,1)(x_c,η^2), x^-=1. (15) Here, TN(−∞,1)(xc,η2)TN_(-∞,1)(x_c,η^2) is the truncated normal distribution that denotes (xc,η2)N(x_c,η^2) conditioned on x<1x<1, ensuring that x+<1x^+<1 almost surely. In the extended state space X×VX× V, the guard surface and its reset image are given by S:=(x,v)∣x=1,v∈VS:=\(x,v) x=1,\ v∈ V\ and R(S)=(x,v)∣x=v+1R(S)=\(x,v) x=v+1\, respectively. We use Monte Carlo simulations to estimate the density evolution of (15) and use the rollout to construct a time-varying A-SHS. As shown in Figure 4, the partitioned OU process enables the A-SHS to closely reproduce the density evolution of (15). To verify Theorem 3, we consider the 3−dim3- embedding E(x,v)=(v,1−cosθ,sinθ),E(x,v)= (v,1- θ, θ ), with θ=2πx−1−v−vθ=2π x-1-v-v. As illustrated in Figure 4, (15) resets x−=1x^-=1 to different x+∈x^+ , while the A-SHS has deterministic resets on X×VX× V and can be embedded into a higher-dimensional space with continuous sample paths. 6.2 Learning Stochastic Hybrid Systems We now apply Algorithm 1 to learn SHS from time-series data. We consider some classic examples in hybrid system theory with different topologies, including the bouncing ball, the Klein bottle, and the torus systems. For a deterministic system, the Torus system can be embedded into 3D Euclidean space, while the Klein bottle system can only be embedded into 4D space. We consider these systems with different reset kernels, such as a GMM distribution or a uniform distribution. We also consider a switching system that randomly chooses the reset kernel of the deterministic torus system or the Klein bottle system. Then we consider the k-bouncing ball system in the x−yx-y plane with dimension equal to 4k4k (velocity and position in horizontal and vertical directions). We consider 0.32s0.32s training horizon. All comparisons and ablations are evaluated on the same normalized test data with horizon 1s1s, 3s3s or 5s5s. By default, we consider the latent dimension of our method to be dz=2(dx+dx)d_z=2(d_x+d_x) as inspired by Lemma 1. We present the result with horizon 1s1s, and the remaining cases are shown in Appendix B.3 and Appendix B.4. We note that the baselines exhibit worse results with a horizon longer than 1s1s, while our method has consistent accuracy. Comparison with baselines: We compare our method with (Teng et al., 2026b), a latent ODE framework for deterministic hybrid system learning; (Li et al., 2020) that learns latent SDEs through variational inference; (Bartosh et al., 2025) that solves a simulation-free variational objective; and (Kiyohara et al., 2025) that learns conditional transition kernels with Chapman–Kolmogorov consistency. We also evaluate (Li et al., 2020) augmented with additional losses, i.e., ℒxL_x and ℒxtrajL_x^traj. As shown in Table 1, our method achieves the lowest mean conditional and unconditional path distribution losses across all 11 benchmark settings. The relative performance of the competing approaches varies across datasets and metrics. Adding ℒxL_x and ℒxtrajL_x^traj to (Li et al., 2020) improves performance on some datasets, but does not consistently improve the results or close the gap to our method. (Kiyohara et al., 2025) and (Bartosh et al., 2025) exhibit larger errors or divergence under the evaluated configurations, with particularly large errors in the unconditional path distribution loss. As (Teng et al., 2026b) omits the stochastic effect, it triggers the divergence criterion on most cases. The predicted distribution evolution of the bouncing ball system is illustrated in Figure 5, showing that our method correctly captures the distribution evolution. As shown in Figure 6, our method correctly captures the path distribution by predicting bounces with different restitution coefficients. More visualizations and comparisons are illustrated in Appendix B.3 and B.5. Figure 5: The predicted probability evolution of bouncing ball systems under GMM reset. Our method correctly captures the long-term distribution with a training horizon of 0.32s0.32s. Figure 6: We plotted five predicted trajectories of the bouncing ball system under GMM reset conditioned on the same initial condition. The reset function independently chooses restitution by a GMM centered at 0.50.5 or 0.90.9 at each collision. Our method correctly captures this pattern. Table 1: Baseline comparison of conditional (unconditional) path distribution loss across 11 benchmark problems. Per-case means and sample standard deviations are reported in Tables 5 and 6. ⋆ denotes using ℒxL_x and ℒxtrajL_x^traj. Horizon: 1s1\,s. Ours (Teng et al., 2026b) (Li et al., 2020) (Li et al., 2020)⋆ (Bartosh et al., 2025) (Kiyohara et al., 2025) b-ball (GMM) 0.257(0.214)0.257\;(0.214) Diverged 0.369(1.12)0.369\;(1.12) 0.338(0.879)0.338\;(0.879) 0.620(4.14)0.620\;(4.14) 0.594(4.22)0.594\;(4.22) b-ball (Uniform) 0.289(0.249)0.289\;(0.249) Diverged 0.404(1.03)0.404\;(1.03) 0.380(0.873)0.380\;(0.873) 0.732(4.84)0.732\;(4.84) 0.542(3.23)0.542\;(3.23) Torus (GMM) 0.603(0.333)0.603\;(0.333) Diverged 1.84(6.17)1.84\;(6.17) 4.32(11.3)4.32\;(11.3) Diverged 0.909(6.27)0.909\;(6.27) Torus (Uniform) 0.560(0.304)0.560\;(0.304) Diverged 2.67(40.5)2.67\;(40.5) 1.51(10.3)1.51\;(10.3) Diverged 0.944(7.19)0.944\;(7.19) Klein (GMM) 0.612(0.341)0.612\;(0.341) Diverged Diverged Diverged 10.2(256)10.2\;(256) 0.903(6.05)0.903\;(6.05) Klein (Uniform) 0.560(0.298)0.560\;(0.298) Diverged 1.33(15.1)1.33\;(15.1) 1.96(29.2)1.96\;(29.2) 7.84(193)7.84\;(193) 0.919(6.66)0.919\;(6.66) Klein–Torus 0.455(0.257)0.455\;(0.257) Diverged 2.67(44.3)2.67\;(44.3) 2.97(52.3)2.97\;(52.3) Diverged 0.989(8.25)0.989\;(8.25) 1-ball (GMM) 0.569(0.570)0.569\;(0.570) Diverged 0.770(1.68)0.770\;(1.68) 0.978(4.43)0.978\;(4.43) 5.69(140)5.69\;(140) Diverged 1-ball (Uniform) 0.569(0.525)0.569\;(0.525) Diverged 0.781(1.46)0.781\;(1.46) 0.751(1.76)0.751\;(1.76) 6.83(192)6.83\;(192) Diverged 2-balls (GMM) 0.722(1.04)0.722\;(1.04) 1.03(2.07)1.03\;(2.07) 0.869(2.22)0.869\;(2.22) 0.883(2.70)0.883\;(2.70) Diverged Diverged 2-balls (Uniform) 0.760(1.07)0.760\;(1.07) 1.06(2.00)1.06\;(2.00) 0.985(3.39)0.985\;(3.39) 0.984(4.23)0.984\;(4.23) Diverged Diverged Table 2: Ablation summary across 11 benchmark settings on the conditional (unconditional) trajectory distribution loss. Mean change is the average of per-setting percentage changes relative to the default model; positive values indicate degradation. Worse cases count the settings with a higher mean test loss than the default model (out of 11), using the means over five training seeds. MLP replaces the diffusion sampler with a deterministic MLP encoder; loss labels identify the removed objectives. Detailed results are presented in Tables 11, 13, 12, 14 and 15. Setting dz=dxd_z=d_x dz=2dxd_z=2d_x dz=3dxd_z=3d_x MLP ℒxtraj,cL_x^traj,c ℒzL_z ℒxL_x ℒxtrajL_x^traj ℒxtraj,c,ℒxtrajL_x^traj,c,\ L_x^traj ℒx,ℒxtrajL_x,\ L_x^traj ℒxtraj,c,ℒxL_x^traj,c,\ L_x Mean change (%) +14.1(+71.4)+14.1\;(+71.4) +2.9(+12.2)+2.9\;(+12.2) +0.3(+4.7)+0.3\;(+4.7) +3.6(+14.7)+3.6\;(+14.7) +6.2(+6.6)+6.2\;(+6.6) +19.5(+246.6)+19.5\;(+246.6) +1.2(+8.8)+1.2\;(+8.8) +0.0(+2.1)+0.0\;(+2.1) +23.3(+54.7)+23.3\;(+54.7) −0.7(+3.8)-0.7\;(+3.8) +5.7(+4.4)+5.7\;(+4.4) Worse cases (/11) 11(11)11\;(11) 8(6)8\;(6) 5(5)5\;(5) 8(8)8\;(8) 7(7)7\;(7) 11(11)11\;(11) 7(9)7\;(9) 6(5)6\;(5) 11(11)11\;(11) 5(7)5\;(7) 6(5)6\;(5) Ablations: We study the effects of 1) latent dimension, 2) loss function design, and 3) encoder randomness. Table 2 summarizes the changes in conditional and unconditional path distribution losses relative to the default model. Reducing the latent dimension generally degrades performance, although dz=3dxd_z=3d_x performs comparably to the default dz=4dxd_z=4d_x at the short horizon. At longer horizons, smaller latent dimensions can lead to substantial degradation or divergence, including degradation for dz=3dxd_z=3d_x on some benchmarks, as shown in Tables 16 and 22. Thus, the longer-horizon results reinforce the benefit of a larger latent space and indicate that comparable short-horizon performance does not ensure comparable long-horizon behavior. For loss function design, removing either path distribution loss alone has a relatively modest effect on average, whereas removing both consistently degrades both metrics. Removing ℒzL_z also consistently degrades performance, with larger effects at longer horizons and divergence on one benchmark over the full trajectory. These trends are consistent across horizons, as reported in Tables 16 and 22. The benefit of latent-space supervision is consistent with the findings of Teng et al. (2026b). Finally, replacing the stochastic encoder with a deterministic MLP moderately degrades performance on average, particularly for the unconditional path distribution loss. This trend persists at longer horizons, as shown in Tables 13, 19 and 25. These results support the benefit of encoder randomness, consistent with the one-to-many mapping from state variables to latent variables. However, the moderate degradation suggests that a deterministic encoder remains viable, as the latent SDE (13) can also provide randomness. The details of the ablations and the cases on longer horizons are shown in Appendix B.4. 7 Discussions Latent Dimension: Whitney Embedding Theorem provides a general dimension bound to embed the 2n2n-dimensional glued manifold (X×V)/∼(X× V)/ into ℝ4nR^4n. The structure of the reset (x+,v+)=(x−+v−,v−)(x^+,v^+)=(x^-+v^-,v^-) may improve this bound: for fixed v, identifying x∼x+vx x+v on the guard S yields an n-dimensional space, embeddable into ℝ2nR^2n. If these embeddings vary smoothly with v and form a global embedding, retaining the n coordinates of v can give a 3n3n bound. Whether the v-wise manifolds vary smoothly remains open for future research. Loss Functions: As each sampled path is stochastic, we argue that there is no need to match a specific path. As discussed in Li et al. (2020), overfitting to a single path will result in the diffusion term converging to zero. In our method, we instead match the path distribution, considering the latent SDE as a sampler to generate the paths. As indicated in the comparative study, the loss function enables us to accurately recover the path distributions beyond the training horizon. Memory of Latent Variables: Latent variables can encode both reset events and past-state memory. When the closures of the guard and its images are disjoint, pure OU dynamics in an interior region allow arbitrarily fast exponential memory decay by increasing the mean-reversion parameter. However, this only approximately erases memory, without guaranteeing complete removal in finite time. Fully recovering the original system’s Markov property remains a direction for future research. Future Applications: The proposed methods can be applied to robotics control, such as (Teng et al., 2022c, Teng et al., 2024b, Jang et al., 2023, Teng et al., 2021a, Liu et al., 2025b, Teng et al., 2024a, Teng et al., 2023, Ghaffari et al., 2022, Teng et al., 2022b, Teng et al., 2022a, Teng et al., 2021b, Yu et al., 2023, He et al., 2024, Teng et al., 2024c, He et al., 2024, Chang et al., 2026, Teng et al., 2026c, Liu et al., 2025a, He et al., 2025, Iwasaki et al., 2025, Liu et al., 2026, Dong et al., 2025, Li et al., 2026) 8 Conclusion In this work, we show that a continuous latent SDE can accurately approximate a stochastic hybrid system with a Markov reset kernel. The key to this result is to represent different reset branches of the reset kernel by auxiliary variables, so that the augmented hybrid system admits a deterministic reset function that enables the topological gluing. In the embedded image, the glued system admits continuous sample paths, and its state distribution can be modeled by a Fokker-Planck equation without the source and sink terms due to the resets. Based on this theorem, we can accurately learn the flow of hybrid systems from time-series data. Reproducibility Statement We provide the assumptions and derivations underlying our theoretical results in the main text, with additional details and complete derivations in the appendix. The learning procedure and objective are described in Section 5, and the experimental settings, evaluation protocols, and benchmark construction are documented in Section 6 and the appendix. These details are intended to facilitate reproduction of both our theoretical and empirical results. We plan to release the implementation and experiment code upon publication. AI Disclosure In this work, we used generative AI tools to assist with mathematical proofs, revise and execute code for method implementation and experiments, and improve the clarity and readability of the manuscript. We did not use generative AI tools for other tasks requiring disclosure under the conference policy. The authors carefully reviewed and verified all AI-assisted proofs for mathematical correctness, reviewed and tested AI-assisted code, and checked the resulting experimental outputs. All AI-assisted writing was reviewed and revised by the authors. We take full responsibility for the final content of this work, including all text, mathematical claims, proofs, code, and experimental results produced with the assistance of generative AI. References Aubin et al. (2002) J. Aubin, J. Lygeros, M. Quincampoix, S. Sastry, and N. Seube Impulse differential inclusions: a viability approach to hybrid systems. IEEE Transactions on Automatic Control 47 (1), p. 2–20. Cited by: §2. 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Let p⋆(t,x)p (t,x) be the solution of HFP, and write the augmented drift and diffusion tensor as f¯(t,y)=[fx(t,x,v)fv(t,x,v)],D¯(t,y)=[Dxx(t,x,v)Dxv(t,x,v)Dxv(t,x,v)⊤Dvv(t,x,v)]. f(t,y)= bmatrixf_x(t,x,v)\\ f_v(t,x,v) bmatrix, D(t,y)= bmatrixD_x(t,x,v)&D_xv(t,x,v)\\ D_xv(t,x,v) &D_v(t,x,v) bmatrix. (16) Here, y=[x⊤,v⊤]⊤∈M×V⊆ℝ2ny=[x ,\ v ] ∈ M× V ^2n, each block of D¯ D is in ℝn×nR^n× n, and D¯=12g¯g¯⊤ D= 12 g g . We use J(t,x)J(t,x) and α(t,s)α(t,s) for the probability current and guard flux associated with p⋆(t,x)p (t,x): J(t,x)=f(t,x)p⋆(t,x)−∇x⋅(D(t,x)p⋆(t,x)),α(t,s)=J(t,s)⋅nS(s),s∈S.J(t,x)=f(t,x)p (t,x)- _x· (D(t,x)p (t,x) ), α(t,s)=J(t,s)· n_S(s),\ s∈ S. (17) We assume the densities and coefficients have the regularity required below, and that D(t,x)D(t,x) is positive definite on the region of construction. A.1 Construction of the Conditional Density We use the conditional density in Construction 1. Recall the pre-reset set Σ−:=S×V ^-:=S× V, its post-reset image Σ+:=R(Σ−) ^+:=R( ^-), and the interior Σ∘:=(M×V)∖(Σ−∪Σ+)¯ :=(M× V) ( ^-∪ ^+). With the smooth partition of unity χ−(x)+χ+(x)+χ∘(x)=1χ^-(x)+χ^+(x)+χ (x)=1, the target conditional density is p⋆(v∣x)=χ−(x)κQ(x,v)+χ+(x)(R#κQ)(x,v)+χ∘(x)(v,μ,δ2I).p (v x)=χ^-(x) _Q(x,v)+χ^+(x)(R_\# _Q)(x,v)+χ (x)N(v;μ,δ^2I). (18) The local densities are understood as smooth normalized extensions on their corresponding neighborhoods. We assume these extensions and the partition of unity are compatible with the prescribed guard trace. Since the weights are nonnegative, independent of v, and sum to one, we have ∫Vp⋆(v∣x)v=1 _Vp (v x)dv=1. In particular, p⋆(v∣s)=κQ(s,v),(s,v)∈Σ−.p (v s)= _Q(s,v), (s,v)∈ ^-. (19) In the interior where χ∘=1χ =1, the conditional density reduces to p⋆(v∣x)=(v,μ,δ2I)p (v x)=N(v;μ,δ^2I). The Gaussian is normalized on V=ℝnV=R^n; for a bounded auxiliary domain, it is replaced by its normalized restriction to V. A.2 Construction of the Target Augmented Density Define the target augmented density by p¯⋆(t,y)=p¯⋆(t,x,v)=p⋆(t,x)p⋆(v∣x),t∈[0,T]. p (t,y)= p (t,x,v)=p (t,x)p (v x), t∈[0,T]. (20) This factorization preserves the prescribed x-marginal: ∫Vp¯⋆(t,x,v)v=p⋆(t,x)∫Vp⋆(v∣x)v=p⋆(t,x). _V p (t,x,v)dv=p (t,x) _Vp (v x)dv=p (t,x). (21) We assume p¯⋆(t,x,v)>0 p (t,x,v)>0 on the region where the coefficients below are constructed, so division by p¯⋆ p is well-defined. A.3 Construction of the x-Direction Probability Current Following Construction 1, let J⋆=[Jx⋆⊤,Jv⋆⊤]⊤J =[J_x ,\ J_v ] denote the target augmented probability current. We prescribe its x-component by Jx⋆(t,x,v)=p⋆(v∣x)J(t,x).J_x (t,x,v)=p (v x)J(t,x). (22) Integrating over v gives ∫VJx⋆(t,x,v)v=J(t,x) _VJ_x (t,x,v)dv=J(t,x). On the augmented guard, (19) yields α¯(t,s,v) α(t,s,v) =Jx⋆(t,s,v)⋅nS(s) =J_x (t,s,v)· n_S(s) (23) =κQ(s,v)J(t,s)⋅nS(s)=α(t,s)κQ(s,v). = _Q(s,v)J(t,s)· n_S(s)=α(t,s) _Q(s,v). Since the augmented normal is n¯(s,v)=(nS(s),0n) n(s,v)=(n_S(s),0_n), this guard flux is independent of Jv⋆J_v . A.4 Construction of the x-Dynamics To preserve the continuous x-dynamics of SHS, choose fx(t,x,v)=f(t,x)f_x(t,x,v)=f(t,x) and Dxx(t,x,v)=D(t,x)D_x(t,x,v)=D(t,x). The cross-diffusion matrix Dxv(t,x,v)∈ℝn×nD_xv(t,x,v) ^n× n is chosen so that the resulting x-current equals (22): ∇v⋅(Dxv(t,x,v)p¯⋆(t,x,v))=f(t,x)p¯⋆(t,x,v)−∇x⋅(D(t,x)p¯⋆(t,x,v))−Jx⋆(t,x,v). _v· (D_xv(t,x,v) p (t,x,v) )=f(t,x) p (t,x,v)- _x· (D(t,x) p (t,x,v) )-J_x (t,x,v). (24) The right-hand side has zero integral over V. Indeed, ∫V[f(t,x)p¯⋆(t,x,v)−∇x⋅(D(t,x)p¯⋆(t,x,v))−Jx⋆(t,x,v)]v _V [f(t,x) p (t,x,v)- _x· (D(t,x) p (t,x,v) )-J_x (t,x,v) ]dv (25) =f(t,x)p⋆(t,x)−∇x⋅(D(t,x)p⋆(t,x))−J(t,x)=0. =f(t,x)p (t,x)- _x· (D(t,x)p (t,x) )-J(t,x)=0. For each i=1,…,ni=1,…,n, define the component-wise equation ri(t,x,v)=fi(t,x)p¯⋆(t,x,v)−∑j=1n∂xj(Dij(t,x)p¯⋆(t,x,v))−Jx,i⋆(t,x,v).r_i(t,x,v)=f_i(t,x) p (t,x,v)-Σ _j=1^n ∂ x_j (D_ij(t,x) p (t,x,v) )-J_x,i (t,x,v). (26) Then (24) is equivalent to ∑j=1n∂vj(Dxv,ij(t,x,v)p¯⋆(t,x,v))=ri(t,x,v),i=1,…,n.Σ _j=1^n ∂ v_j (D_xv,ij(t,x,v) p (t,x,v) )=r_i(t,x,v), i=1,…,n. (27) For a bounded connected domain V with smooth boundary, let ψi(t,x,⋅) _i(t,x,·) be the zero-mean solution of the Neumann–Poisson problem Δvψi(t,x,v) _v _i(t,x,v) =ri(t,x,v), =r_i(t,x,v), v∈V, v∈ V, (28) ∇vψi(t,x,v)⋅n∂V(v) _v _i(t,x,v)· n_∂ V(v) =0, =0, v∈∂V, v∈∂ V, ∫Vψi(t,x,v)v _V _i(t,x,v)dv =0. =0. Here, n∂Vn_∂ V is the outward unit normal to ∂V∂ V. The compatibility condition ∫Vri(t,x,v)v=0 _Vr_i(t,x,v)dv=0 follows from (25), and the zero-mean condition fixes the additive constant. Under the stated regularity assumptions, define Dxv,ij(t,x,v)=∂vjψi(t,x,v)p¯⋆(t,x,v),i,j=1,…,n.D_xv,ij(t,x,v)= _v_j _i(t,x,v) p (t,x,v), i,j=1,…,n. (29) Substituting into (27) gives ∑j=1n∂vj(Dxv,ijp¯⋆)=∑j=1n∂2ψi∂vj2=Δvψi=ri.Σ _j=1^n ∂ v_j (D_xv,ij p )=Σ _j=1^n ∂^2 _i∂ v_j^2= _v _i=r_i. (30) Thus, (24) holds. The Neumann condition also gives (Dxvp¯⋆)n∂V=0(D_xv p )n_∂ V=0 on ∂V∂ V. For V=ℝnV=R^n, replace the boundary and zero-mean conditions by suitable decay and an additive normalization, and assume the corresponding Poisson problems admit solutions with vanishing flux at infinity. To complete the augmented diffusion tensor, choose any η>0η>0 and set Dvv(t,x,v)=Dxv(t,x,v)⊤D(t,x)−1Dxv(t,x,v)+ηI.D_v(t,x,v)=D_xv(t,x,v) D(t,x)^-1D_xv(t,x,v)+η I. (31) The Schur complement satisfies Dvv−Dxv⊤D−1Dxv=ηI≻0D_v-D_xv D^-1D_xv=η I 0. Hence, D¯ D in (16) is positive definite. A.5 Construction of the v-Direction Probability Current In the interior Σ∘ , the target current must satisfy ∂p¯⋆(t,x,v)∂t=−∇x⋅Jx⋆(t,x,v)−∇v⋅Jv⋆(t,x,v),(x,v)∈Σ∘. ∂ p (t,x,v)∂ t=- _x· J_x (t,x,v)- _v· J_v (t,x,v), (x,v)∈ . (32) Define the residual rv(t,x,v)=−∂p¯⋆(t,x,v)∂t−∇x⋅Jx⋆(t,x,v).r^v(t,x,v)=- ∂ p (t,x,v)∂ t- _x· J_x (t,x,v). (33) Then Jv⋆J_v is required to solve ∇v⋅Jv⋆(t,x,v)=rv(t,x,v). _v· J_v (t,x,v)=r^v(t,x,v). (34) On a region where the entire v-fiber avoids the reset interfaces and the marginal equation has no reset contribution, we have ∫Vrv(t,x,v)v _Vr^v(t,x,v)dv =−∂t∫Vp¯⋆(t,x,v)dv−∇x⋅∫VJx⋆(t,x,v)dv =- ∂ t _V p (t,x,v)dv- _x· _VJ_x (t,x,v)dv (35) =−∂p⋆(t,x)∂t−∇x⋅J(t,x)=0. =- ∂ p (t,x)∂ t- _x· J(t,x)=0. Under this compatibility condition, one possible choice is a gradient current. For bounded V, let φ(t,x,⋅) (t,x,·) solve Δvφ(t,x,v) _v (t,x,v) =rv(t,x,v), =r^v(t,x,v), v∈V, v∈ V, (36) ∇vφ(t,x,v)⋅n∂V(v) _v (t,x,v)· n_∂ V(v) =0, =0, v∈∂V, v∈∂ V, ∫Vφ(t,x,v)v _V (t,x,v)dv =0. =0. Define Jv⋆(t,x,v)=∇vφ(t,x,v).J_v (t,x,v)= _v (t,x,v). (37) Then ∇v⋅Jv⋆=Δvφ=rv _v· J_v = _v =r^v and Jv⋆⋅n∂V=0J_v · n_∂ V=0. The same decay qualification as above applies when V=ℝnV=R^n. The gradient choice fixes one admissible current; the density evolution alone does not uniquely determine Jv⋆J_v . A.6 Recovery of the Augmented Drift and Diffusion Given Jv⋆J_v , define the v-component of the drift in Σ∘ by fv(t,x,v)=Jv⋆(t,x,v)+∇x⋅(Dxv(t,x,v)⊤p¯⋆(t,x,v))+∇v⋅(Dvv(t,x,v)p¯⋆(t,x,v))p¯⋆(t,x,v).f_v(t,x,v)= J_v (t,x,v)+ _x· (D_xv(t,x,v) p (t,x,v) )+ _v· (D_v(t,x,v) p (t,x,v) ) p (t,x,v). (38) All density factors in (38) are the joint density p¯⋆ p . As in Construction 2, choose g¯(t,x,v)=[g(t,x)0gxv(t,x,v)gv(t,x,v)],gxv=Dxv⊤D−1g,gv=2ηI. g(t,x,v)= bmatrixg(t,x)&0\\ g_xv(t,x,v)&g_v(t,x,v) bmatrix, g_xv=D_xv D^-1g, g_v= 2ηI. (39) Using gg⊤=2Dgg =2D, we obtain 12g¯g¯⊤=[DDxvDxv⊤Dxv⊤D−1Dxv+ηI]=D¯. 12 g g = bmatrixD&D_xv\\ D_xv &D_xv D^-1D_xv+η I bmatrix= D. (40) The resulting augmented dynamics are dy dy =f¯(t,y)dt+g¯(t,y)dw¯,(x,v)∉Σ−, = f(t,y)dt+ g(t,y)d w, (x,v)∉ ^-, (41) x+ x^+ =x−+v−,v+=v−,(x−,v−)∈Σ−. =x^-+v^-, v^+=v^-, (x^-,v^-)∈ ^-. Here, w¯=[wx⊤,wv⊤]⊤ w=[w_x ,\ w_v ] is a 2n2n-dimensional Wiener process. Evaluating the probability current at the target density gives f¯(t,y)p¯⋆(t,y)−∇y⋅(D¯(t,y)p¯⋆(t,y))=[Jx⋆(t,x,v)Jv⋆(t,x,v)]=J⋆(t,y),y∈Σ∘. f(t,y) p (t,y)- _y· ( D(t,y) p (t,y) )= bmatrixJ_x (t,x,v)\\ J_v (t,x,v) bmatrix=J (t,y), y∈ . (42) In a region where χ∘=1χ =1 and the marginal equation has no reset contribution, p⋆(v∣x)=(v,μ,δ2I)p (v x)=N(v;μ,δ^2I) is independent of x. Hence, ri=0r_i=0, and we may choose Dxv=0D_xv=0 and Jv⋆=0J_v =0. With η=λδ2η=λδ^2, (38) reduces to fv=−λ(v−μ)f_v=-λ(v-μ), giving the interior OU dynamics in Section 4.3: dv=−λ(v−μ)dt+2λδdwv.dv=-λ(v-μ)dt+ 2λ\,δ\,dw_v. (43) This expression applies on V=ℝnV=R^n; a bounded auxiliary domain additionally requires a boundary mechanism compatible with the imposed no-flux condition. A.7 Numerical Realization of the Two Constructions We describe the one-dimensional realization of Construction 1 and 2 used in the analytical example. The continuous state satisfies dx=dt+σdwdx=dt+σ\,dw, with σ=0.005σ=0.005 and guard x=1x=1. The auxiliary state represents the reset displacement, v=x+−1v=x^+-1, so that the augmented reset is deterministic: R(1,v)=(1+v,v).R(1,v)=(1+v,v). (44) For numerical simulation, we restrict the auxiliary variable to V=[vmin,vmax]=[−1.15,−0.55]V=[v_ ,v_ ]=[-1.15,-0.55] and use reflecting boundaries. Consequently, the augmented reset image lies on x=1+vx=1+v, with x∈[−0.15,0.45]x∈[-0.15,0.45]. Construction 1: conditional and joint target densities. Let κ(v)κ(v) denote the reset-displacement density and let ρ(v)ρ(v) denote the Gaussian interior profile. Before restriction to V, the profiles used in the numerical example are κ0(v) _0(v) =13∑c∈0,0.15,0.3(v,c−1,0.032), = 13 _c∈\0,0.15,0.3\N(v;c-1,0.03^2), (45) ρ0(v) _0(v) =(v,−0.85,0.22). =N(v;-0.85,0.2^2). For the normalized target construction on a bounded auxiliary domain, these profiles are replaced by κ(v)=κ0(v)∫Vκ0(u)u,ρ(v)=ρ0(v)∫Vρ0(u)u,v∈V.κ(v)= _0(v) _V _0(u)\,du, ρ(v)= _0(v) _V _0(u)\,du, v∈ V. (46) Since R leaves v unchanged, the outgoing and incoming reset fluxes have the same auxiliary-coordinate law. The implementation therefore uses the reset-displacement profile as a local extension near both the guard and the projected reset band. This extension is a numerical choice; it does not by itself establish the joint-density trace on the slanted reset image. We blend the reset and interior profiles using a twice continuously differentiable weight. Define s(u)=0,u≤0,6u5−15u4+10u3,0<u<1,1,u≥1,s(u)= cases0,&u≤ 0,\\ 6u^5-15u^4+10u^3,&0<u<1,\\ 1,&u≥ 1, cases (47) and set a+(x) a_+(x) =1−s(x−0.450.20), =1-s ( x-0.450.20 ), (48) a−(x) a_-(x) =s(x−0.700.30), =s ( x-0.700.30 ), a(x) a(x) =maxa+(x),a−(x). = \a_+(x),a_-(x)\. The two transition regions are disjoint. Thus, a=1a=1 on the projected reset band and at the guard, while a=0a=0 for x∈[0.65,0.70]x∈[0.65,0.70]. The conditional target and the augmented target are q(v∣x)=a(x)κ(v)+(1−a(x))ρ(v),p¯⋆(t,x,v)=p⋆(t,x)q(v∣x).q(v x)=a(x)κ(v)+(1-a(x))ρ(v), p (t,x,v)=p (t,x)q(v x). (49) Normalization of the local profiles gives ∫Vq(v∣x)v=1 _Vq(v x)\,dv=1, and hence ∫Vp¯⋆(t,x,v)v=p⋆(t,x) _V p (t,x,v)\,dv=p (t,x). In the numerical implementation, the time-dependent marginal used to evaluate the coefficients is obtained from a conservative finite-volume approximation of the original hybrid Fokker–Planck equation, ∂tp+∂xJ=α(t)κ(x−1),J=p−D0∂xp,D0=σ22. _tp+ _xJ=α(t)κ(x-1), J=p-D_0 _xp, D_0= σ^22. (50) The source is supported on the reset band. The computational interval is [−0.15,1][-0.15,1], with an absorbing right boundary and a no-flux left boundary. We use 600 spatial cells and ten finite-volume substeps per particle-integration step. The tabulated density and its spatial derivative provide the time-dependent current J(t,x)J(t,x). Separate Monte Carlo ensembles are used to compare the SHS and A-SHS marginal distributions. Construction 2: cross-diffusion and auxiliary drift. For normalized profiles, the one-dimensional Neumann problem reduces to a single cumulative integral, H(v)=∫vminv(κ(u)−ρ(u))u.H(v)= _v_ ^v (κ(u)-ρ(u) )\,du. (51) It satisfies H(vmin)=H(vmax)=0H(v_ )=H(v_ )=0 and H′(v)=κ(v)−ρ(v)H (v)=κ(v)-ρ(v). Prescribing Jx⋆=qJ_x =qJ gives ∂v(Dxvp¯⋆)=−D0p⋆∂xq. _v (D_xv p )=-D_0p _xq. (52) Therefore, a scalar cross-diffusion coefficient is Dxv(x,v)=−D0a′(x)H(v)q(v∣x).D_xv(x,v)=- D_0a (x)H(v)q(v x). (53) On a source-free region where ∂tp⋆+∂xJ=0 _tp + _xJ=0, the auxiliary current can be chosen as Jv⋆(t,x,v)=−a′(x)J(t,x)H(v).J_v (t,x,v)=-a (x)J(t,x)H(v). (54) Indeed, ∂vJv⋆=−J∂xq=−∂tp¯⋆−∂x(qJ) _vJ_v =-J _xq=- _t p - _x(qJ) on such a region. Choose an independent auxiliary-noise amplitude σv=0.5 _v=0.5 and write ηv=σv2/2 _v= _v^2/2. The remaining diffusion coefficient and the auxiliary drift are Dvv D_v =Dxv2D0+ηv, = D_xv^2D_0+ _v, (55) hv h_v =Jv⋆+∂x(Dxvp¯⋆)+∂v(Dvvp¯⋆)p¯⋆. = J_v + _x(D_xv p )+ _v(D_v p ) p . The corresponding continuous dynamics are dx dx =dt+σdw, =dt+σ\,dw, (56) dv dv =hv(t,x,v)dt+2Dxv(x,v)σdw+σvdu, =h_v(t,x,v)\,dt+ 2D_xv(x,v)σ\,dw+ _v\,du, where w and u are independent Wiener processes. Sharing w between the two equations realizes the prescribed cross-diffusion. At x=1x=1, we apply (x+,v+)=(1+v−,v−)(x^+,v^+)=(1+v^-,v^-) without resampling the auxiliary state. In the Gaussian interior, a=a′=0a=a =0, so that Dxv=Jv⋆=0D_xv=J_v =0. Away from numerical regularization and auxiliary boundaries, the v-dynamics reduce to dv=−λ(v−μ)dt+σvdu,μ=−0.85,λ=σv22δ2=3.125,δ=0.2.dv=-λ(v-μ)\,dt+ _v\,du, μ=-0.85, λ= _v^22δ^2=3.125, δ=0.2. (57) Thus, the OU process is the interior limit of the construction, rather than the complete augmented dynamics. In the transition regions, the shared noise and the time-dependent current contribute additional terms. Numerical approximation and scope. The displayed simulations use 5000 particles per system, step size Δt=5×10−4 t=5× 10^-4, and final time T=3T=3. Both particle ensembles start at x(0)=0.25x(0)=0.25. The coefficient table uses a regularized initial density with standard deviation 0.050.05. The auxiliary integral is tabulated on 4000 grid points, density denominators are floored at 10−310^-3, and the auxiliary drift is clipped to [−200,200][-200,200]. Appendix B Details of the Numerical Experiments In this appendix, we provide the details of the numerical experiments. Unless stated otherwise, values are the mean ± sample standard deviation over five training seeds. Bold entries indicate the lowest mean among complete numeric entries for each metric within each row; rows with only one available configuration do not provide a comparative ranking. B.1 System Setup In this section, we consider eleven benchmark settings constructed from six stochastic hybrid systems. The Bouncing Ball, Torus, Klein Bottle, and Klein–Torus systems have two-dimensional observed states. We additionally consider a single planar bouncing ball with a four-dimensional state and two interacting planar bouncing balls with an eight-dimensional state. Except for Klein–Torus, each system is evaluated with both Gaussian-mixture and uniform reset distributions. The continuous dynamics are deterministic in these experiments; randomness enters through the reset mechanism. For the Torus and Klein Bottle examples, the dynamics evolve on the fundamental domain M=[0,1]2M=[0,1]^2 with the constant vector field x˙=c,c=[2,22]⊤. x=c, c=[2,2 2] . (58) The guard consists of two neighboring boundaries, S=S1∪S2S=S_1∪ S_2, where S1=x∈M:x1=1,S2=x∈M:x2=1.S_1=\x∈ M:x_1=1\, S_2=\x∈ M:x_2=1\. (59) Let [a]1=a−⌊a⌋[a]_1=a- a denote wrapping into [0,1)[0,1). The stochastic reset maps are rt(x,ξ) r_t(x,ξ) =(0,[x2+ξ]1),x∈S1,([x1+ξ]1,0),x∈S2, = cases(0,[x_2+ξ]_1),&x∈ S_1,\\ ([x_1+ξ]_1,0),&x∈ S_2, cases (60) rk(x,ξ) r_k(x,ξ) =(0,[x2+ξ]1),x∈S1,([1−x1+ξ]1,0),x∈S2. = cases(0,[x_2+ξ]_1),&x∈ S_1,\\ ([1-x_1+ξ]_1,0),&x∈ S_2. cases When ξ=0ξ=0, these maps recover the canonical boundary identifications of the torus and Klein bottle, respectively. For the stochastic experiments, the reset perturbation is sampled independently at each event from ξ∼13∑μ∈−0.2,0,0.2(μ,0.012),GMM,(−0.2,0.2),Uniform.ξ cases 13 _μ∈\-0.2,0,0.2\N(μ,0.01^2),&GMM,\\[4.0pt] U(-0.2,0.2),&Uniform. cases (61) Initial states are sampled uniformly from [0,1]2[0,1]^2. The Klein–Torus example has the same continuous dynamics and guard, but randomly selects between the two canonical reset rules. The reset on S1S_1 is (1,x2)↦(0,x2)(1,x_2) (0,x_2). On S2S_2, we independently draw B∼Bernoulli(1/2)B (1/2) and apply rkt(x,B)=(x1,0),B=0,(1−x1,0),B=1.r_kt(x,B)= cases(x_1,0),&B=0,\\ (1-x_1,0),&B=1. cases (62) There is no additive reset perturbation in this example: the randomness is entirely due to the choice of reset rule. The Bouncing Ball system has state x=[p,v]⊤x=[p,v] , with height p and vertical velocity v. Its continuous dynamics are p˙=v,v˙=−g,g=9.81. p=v, v=-g, g=9.81. (63) The guard Sb=(p,v):p=0,v<0S_b=\(p,v):p=0,\ v<0\ represents contact with the ground. At each impact, the reset is p+=0,v+=−αv−+ε,ε∼(0,0.052),p^+=0, v^+=-α v^-+ , (0,0.05^2), (64) where the restitution coefficient is sampled independently from α∼12(0.5,0.012)+12(0.9,0.012),GMM,(0.25,0.90),Uniform.α cases 12N(0.5,0.01^2)+ 12N(0.9,0.01^2),&GMM,\\[4.0pt] U(0.25,0.90),&Uniform. cases (65) The Gaussian components are sampled without truncation. We initialize p(0)∼(0.2,1.5)p(0) (0.2,1.5) and v(0)=suv(0)=su, where u∼(5,7)u (5,7) and s∈−1,1s∈\-1,1\ is equiprobable. For the Planar Bouncing Ball examples, we consider k∈1,2k∈\1,2\ equal-mass balls with radius r=0.05r=0.05. Each ball has state xi=[px,i,py,i,vx,i,vy,i]⊤x_i=[p_x,i,p_y,i,v_x,i,v_y,i] , and the full observed state is x=[x1⊤,…,xk⊤]⊤∈ℝ4kx=[x_1 ,…,x_k ] ^4k. The continuous dynamics are p˙x,i=vx,i,p˙y,i=vy,i,v˙x,i=0,v˙y,i=−g. p_x,i=v_x,i, p_y,i=v_y,i, v_x,i=0, v_y,i=-g. (66) The physical side walls are located at px=±0.5p_x=± 0.5, the floor is located at py=0p_y=0, and there is no ceiling. Consequently, the admissible ball-center positions satisfy −0.45≤px,i≤0.45,py,i≥0.05.-0.45≤ p_x,i≤ 0.45, p_y,i≥ 0.05. (67) A wall or floor collision is triggered when a ball reaches the corresponding contact boundary while moving into the surface. The normal velocity component is reset according to vn,i+=−αvn,i−,v_n,i^+=-α v_n,i^-, (68) while the tangential component remains unchanged. The restitution coefficient follows (65). Unlike the one-dimensional bouncing ball, these planar systems do not include additive velocity noise at impact. For two balls, contact additionally occurs when ‖pi−pj‖=2r\|p_i-p_j\|=2r and the balls are approaching each other. Let nij=pi−pj‖pi−pj‖,sij=(vi−vj−)⊤nij<0.n_ij= p_i-p_j\|p_i-p_j\|, s_ij=(v_i^--v_j^-) n_ij<0. (69) The equal-mass collision reset is vi+ v_i^+ =vi−1+α2sijnij, =v_i^-- 1+α2s_ijn_ij, (70) vj+ v_j^+ =vj−+1+α2sijnij, =v_j^-+ 1+α2s_ijn_ij, with a newly sampled restitution coefficient. Initial positions are non-overlapping, and initial velocities are rescaled to satisfy a total mechanical-energy budget of at most kgkg, for unit masses. The implementation corrects numerical overlap and uses a small inward position margin of 10−610^-6 after contact. For each benchmark setting, we generate 4096 training trajectories and 512 independent test trajectories with a nominal duration of 5s5\,s and integration step Δt=0.01s t=0.01\,s. Collision and boundary-crossing events are handled during simulation, so the stored observation timestamps need not form a uniform grid. We normalize each coordinate using the training-set mean and standard deviation and reuse those statistics for testing. All qualitative trajectory plots are transformed back to the original physical or fundamental-domain coordinates. B.2 Model Structure We compare the proposed model with CHyLL (Teng et al., 2026b), Latent SDE (Li et al., 2020), SDE Matching (Bartosh et al., 2025), and Neural Stochastic Flows (Kiyohara et al., 2025). We additionally evaluate Latent SDE augmented with ℒxL_x and ℒxtrajL_x^traj. This variant uses the same architecture as Latent SDE and is denoted by Latent SDE + x losses in the figures. The stochastic baselines use the authors’ implementation cores, with conditional initial-state adapters where needed for forecasting from a given observation. Let n=dxn=d_x denote the observed-state dimension. All latent-state models use dz=4nd_z=4n in the default comparison. NSF instead models transitions directly in the observed state space and does not use a latent SDE of dimension 4n4n. We use the standard network widths for b-ball, Torus, Klein, Klein–Torus, and the single planar ball. For the two-ball examples, we double the hidden widths within each model family. Table 3: Hidden-width settings. The large configuration is used only for the two planar balls. H is the hidden width for Proposed and CHyLL; L and C are the hidden and context widths for Latent SDE; M is the hidden width for SDE Matching; F is the hidden width for NSF. Configuration H L C M F Standard 256 128 64 100 64 Large 512 256 128 200 128 The Proposed model uses a conditional diffusion sampler to generate latent initial states from the observed initial condition. Its score network has two hidden layers of width H with SiLU activations. We use 20 DDIM sampling steps and a scalar diffusion-time input. The discrete noise schedule is linearly spaced between 0.1/200.1/20 and 4/204/20. The latent drift and diffusion networks each contain two hidden layers of width H, and the decoder contains three hidden layers of width H. These networks use ReLU hidden activations and linear outputs. The CHyLL baseline (Teng et al., 2026b) uses a deterministic MLP encoder, a latent ODE, and an MLP decoder. Their hidden-layer widths are [H]×2[H]× 2, [H]×2[H]× 2, and [H]×3[H]× 3, respectively. The encoder, vector field, and decoder use ReLU hidden activations. This baseline is trained with its deterministic reconstruction and dynamics objectives, rather than with the stochastic distribution-matching objectives. For a fixed initial observation, repeated predictions coincide. The Latent SDE baseline (Li et al., 2020) uses a GRU with hidden width L, followed by a linear projection to a context of dimension C. The initial posterior parameters are obtained from this context through a linear layer. Both the posterior and prior drift networks contain two Softplus hidden layers of width L. The diffusion is diagonal: each latent coordinate has a scalar-input network with one Softplus hidden layer of width L and a sigmoid output. A linear projection maps latent states to observations. For conditional forecasting, an additional MLP with two Softplus hidden layers of width L maps the initial observation to the mean and log standard deviation of a diagonal Gaussian initial prior. The observation likelihood has fixed standard deviation 0.010.01 in normalized coordinates. The KL weight is annealed to one over the first 1000 updates. The Latent SDE + x losses variant adds ℒxL_x and ℒxtrajL_x^traj with unit weights, without changing these networks. The SDE Matching baseline (Bartosh et al., 2025) uses a GRU of width M and a time-conditioned MLP with two SiLU hidden layers of width M to parameterize its Gaussian posterior process. The prior drift has one Softplus hidden layer of width M. Its diagonal diffusion uses one scalar network per latent coordinate, each with one Softplus hidden layer of width M and a sigmoid output. The observation map is linear, with fixed likelihood standard deviation 0.010.01. We condition the initial Gaussian prior on the initial observation using an MLP with one Softplus hidden layer of width M. Training uses the official simulation-free objective. The implementation uses the fixed time conversion t~=t/0.31 t=t/0.31 and returns forecasts in the original time units. At evaluation, the observation mean is used without adding likelihood noise. The NSF baseline (Kiyohara et al., 2025) uses the official autonomous affine-coupling stochastic-flow and bridge models. The Gaussian-parameter and coupling-conditioner networks each have two hidden layers of width F with SiLU activations, and each flow uses four affine-coupling layers. Training alternates five bridge updates with one flow update, using the official consistency objective. Predictive trajectories are generated recursively from the conditional transition kernels, rather than by independently sampling each time marginal. Table 4: Network components used in the baseline comparison. [h]×ℓ[h]× denotes ℓ hidden layers of width h; output layers are not counted. Widths are specified in Table 3. Latent SDE with additional losses has the same architecture as Latent SDE. Component Proposed CHyLL Latent SDE SDE Matching NSF Initial-state network Score: [H]×2[H]× 2 [H]×2[H]× 2 [L]×2[L]× 2 [M]×1[M]× 1 Conditional flow Posterior context N/A N/A GRU(L), context C GRU(M), [M]×2[M]× 2 Bridge model Prior drift [H]×2[H]× 2 [H]×2[H]× 2 [L]×2[L]× 2 [M]×1[M]× 1 N/A Diffusion [H]×2[H]× 2, matrix N/A [L]×1[L]× 1, per coordinate [M]×1[M]× 1, per coordinate N/A Observation decoder [H]×3[H]× 3 [H]×3[H]× 3 Linear Linear Observed-state flow Flow conditioner N/A N/A N/A N/A [F]×2[F]× 2 We use a learning rate of 10−310^-3, batch size 256, and training windows of 32 stored observations. Each run uses 10,000 training steps; for NSF, these are 10,000 flow updates with the additional alternating bridge updates. The proposed model uses ten conditional samples during training. All methods are evaluated over five training seeds, 1101,2202,3303,4404,5505\1101,2202,3303,4404,5505\. The latent ODE/SDE forecasts use Euler or Euler–Maruyama integration on the supplied prediction grid. For a 32-point grid, this gives 31 integration steps. Predictions are interpolated to the recorded observation times when needed. NSF instead recursively samples its transition kernels. Forecasting conditions only on the initial observation; posterior access to a training window is not used to provide future observations during testing. For the horizon-based comparisons, we evaluate the saved checkpoints on all 512 test trajectories using ten predictions per initial condition. We report results at 1s1\,s, 3s3\,s, and the full stored horizon. For a fixed physical horizon, we retain observations up to the time cutoff and pad shorter retained sequences by repeating their final retained state and timestamp. The same protocol is applied across methods. Metrics are computed in normalized coordinates, while the qualitative figures display unnormalized trajectories. B.3 Comparative Studies B.3.1 Horizon 1s Tables 5 and 6 report the conditional and unconditional path distribution losses, respectively, both based on energy distance. Evaluation uses 512 test trajectories in normalized state space, 32-point windows, and ten predictive samples per condition. Latent SDE (Li et al., 2020) and SDE Matching (Bartosh et al., 2025) are our task-adapted reimplementations; the ⋆ variant of Latent SDE additionally uses the state distribution loss ℒxL_x and the unconditional path distribution loss ℒxtrajL_x^traj. NSF (Kiyohara et al., 2025) uses the official implementation and recursively samples trajectories from conditional transition kernels. Diverged denotes a numerical failure or either test-set path distribution loss exceeding 10310^3 in any seed. The same criterion applies to CHyLL; surviving seeds are not averaged separately. Table 5: Baseline comparison measured by test-set conditional path distribution loss ℒxtraj,cL_x^traj,c (↓ ). Horizon: 1s1\,s. Case Ours (Teng et al., 2026b) (Li et al., 2020) (Li et al., 2020)⋆ (Bartosh et al., 2025) (Kiyohara et al., 2025) b-ball (GMM) 0.2570.257±\,±\, 0.01580.0158 Diverged 0.3690.369±\,±\, 0.04640.0464 0.3380.338±\,±\, 0.02690.0269 0.6200.620±\,±\, 0.2650.265 0.5940.594±\,±\, 0.007400.00740 b-ball (Uniform) 0.2890.289±\,±\, 0.03170.0317 Diverged 0.4040.404±\,±\, 0.08560.0856 0.3800.380±\,±\, 0.09840.0984 0.7320.732±\,±\, 0.2950.295 0.5420.542±\,±\, 0.006760.00676 Torus (GMM) 0.6030.603±\,±\, 0.003470.00347 Diverged 1.841.84±\,±\, 2.482.48 4.324.32±\,±\, 7.817.81 Diverged 0.9090.909±\,±\, 0.02140.0214 Torus (Uniform) 0.5600.560±\,±\, 0.007560.00756 Diverged 2.672.67±\,±\, 3.933.93 1.511.51±\,±\, 1.601.60 Diverged 0.9440.944±\,±\, 0.01840.0184 Klein (GMM) 0.6120.612±\,±\, 0.01240.0124 Diverged Diverged Diverged 10.210.2±\,±\, 8.158.15 0.9030.903±\,±\, 0.01200.0120 Klein (Uniform) 0.5600.560±\,±\, 0.01780.0178 Diverged 1.331.33±\,±\, 0.8350.835 1.961.96±\,±\, 2.272.27 7.847.84±\,±\, 6.176.17 0.9190.919±\,±\, 0.01750.0175 Klein–Torus 0.4550.455±\,±\, 0.01490.0149 Diverged 2.672.67±\,±\, 1.861.86 2.972.97±\,±\, 2.152.15 Diverged 0.9890.989±\,±\, 0.03860.0386 1-ball (GMM) 0.5690.569±\,±\, 0.005360.00536 Diverged 0.7700.770±\,±\, 0.06250.0625 0.9780.978±\,±\, 0.2500.250 5.695.69±\,±\, 3.343.34 Diverged 1-ball (Uniform) 0.5690.569±\,±\, 0.01880.0188 Diverged 0.7810.781±\,±\, 0.07680.0768 0.7510.751±\,±\, 0.07520.0752 6.836.83±\,±\, 2.732.73 Diverged 2-balls (GMM) 0.7220.722±\,±\, 0.01850.0185 1.031.03±\,±\, 0.01550.0155 0.8690.869±\,±\, 0.01770.0177 0.8830.883±\,±\, 0.02510.0251 Diverged Diverged 2-balls (Uniform) 0.7600.760±\,±\, 0.01830.0183 1.061.06±\,±\, 0.009930.00993 0.9850.985±\,±\, 0.05900.0590 0.9840.984±\,±\, 0.04900.0490 Diverged Diverged Table 6: Baseline comparison measured by test-set unconditional path distribution loss ℒxtrajL_x^traj (↓ ). Horizon: 1s1\,s. Case Ours (Teng et al., 2026b) (Li et al., 2020) (Li et al., 2020)⋆ (Bartosh et al., 2025) (Kiyohara et al., 2025) b-ball (GMM) 0.2140.214±\,±\, 0.05580.0558 Diverged 1.121.12±\,±\, 0.5710.571 0.8790.879±\,±\, 0.2930.293 4.144.14±\,±\, 3.753.75 4.224.22±\,±\, 0.08880.0888 b-ball (Uniform) 0.2490.249±\,±\, 0.08190.0819 Diverged 1.031.03±\,±\, 0.7080.708 0.8730.873±\,±\, 0.6730.673 4.844.84±\,±\, 4.544.54 3.233.23±\,±\, 0.05270.0527 Torus (GMM) 0.3330.333±\,±\, 0.01330.0133 Diverged 6.176.17±\,±\, 9.079.07 11.311.3±\,±\, 16.316.3 Diverged 6.276.27±\,±\, 0.5510.551 Torus (Uniform) 0.3040.304±\,±\, 0.03190.0319 Diverged 40.540.5±\,±\, 76.176.1 10.310.3±\,±\, 15.115.1 Diverged 7.197.19±\,±\, 0.4920.492 Klein (GMM) 0.3410.341±\,±\, 0.02010.0201 Diverged Diverged Diverged 256256±\,±\, 231231 6.056.05±\,±\, 0.3300.330 Klein (Uniform) 0.2980.298±\,±\, 0.03170.0317 Diverged 15.115.1±\,±\, 17.517.5 29.229.2±\,±\, 49.449.4 193193±\,±\, 171171 6.666.66±\,±\, 0.4510.451 Klein–Torus 0.2570.257±\,±\, 0.02060.0206 Diverged 44.344.3±\,±\, 41.841.8 52.352.3±\,±\, 48.748.7 Diverged 8.258.25±\,±\, 1.081.08 1-ball (GMM) 0.5700.570±\,±\, 0.05180.0518 Diverged 1.681.68±\,±\, 0.4410.441 4.434.43±\,±\, 2.872.87 140140±\,±\, 125125 Diverged 1-ball (Uniform) 0.5250.525±\,±\, 0.05790.0579 Diverged 1.461.46±\,±\, 0.2770.277 1.761.76±\,±\, 0.8940.894 192192±\,±\, 96.496.4 Diverged 2-balls (GMM) 1.041.04±\,±\, 0.1620.162 2.072.07±\,±\, 0.06470.0647 2.222.22±\,±\, 0.3140.314 2.702.70±\,±\, 0.8340.834 Diverged Diverged 2-balls (Uniform) 1.071.07±\,±\, 0.08520.0852 2.002.00±\,±\, 0.1150.115 3.393.39±\,±\, 1.471.47 4.234.23±\,±\, 1.421.42 Diverged Diverged B.3.2 Horizon 3s Tables 7 and 8 report the conditional and unconditional path distribution losses over 3s3\,s from the first stored observation. Our method yields complete numeric results and the lowest mean losses for both metrics across all 11 benchmark settings. SDE Matching triggers the divergence criterion in all settings, while both Latent SDE variants diverge on all five topological benchmarks. Table 7: Baseline comparison measured by test-set conditional path distribution loss ℒxtraj,cL_x^traj,c (↓ ). Horizon: 3s3\,s. Case Ours (Teng et al., 2026b) (Li et al., 2020) (Li et al., 2020)⋆ (Bartosh et al., 2025) (Kiyohara et al., 2025) b-ball (GMM) 0.4870.487±\,±\, 0.01530.0153 Diverged 1.601.60±\,±\, 0.6470.647 1.181.18±\,±\, 0.3900.390 Diverged 0.8970.897±\,±\, 0.01160.0116 b-ball (Uniform) 0.4330.433±\,±\, 0.01520.0152 Diverged 1.421.42±\,±\, 1.041.04 1.451.45±\,±\, 1.701.70 Diverged 0.6930.693±\,±\, 0.007920.00792 Torus (GMM) 0.6710.671±\,±\, 0.002390.00239 Diverged Diverged Diverged Diverged 0.9130.913±\,±\, 0.02270.0227 Torus (Uniform) 0.6540.654±\,±\, 0.006910.00691 Diverged Diverged Diverged Diverged 0.9540.954±\,±\, 0.02010.0201 Klein (GMM) 0.6770.677±\,±\, 0.005300.00530 Diverged Diverged Diverged Diverged 0.9050.905±\,±\, 0.01500.0150 Klein (Uniform) 0.6530.653±\,±\, 0.01080.0108 Diverged Diverged Diverged Diverged 0.9220.922±\,±\, 0.01890.0189 Klein–Torus 0.5450.545±\,±\, 0.03250.0325 Diverged Diverged Diverged Diverged 0.9940.994±\,±\, 0.04330.0433 1-ball (GMM) 0.5230.523±\,±\, 0.007830.00783 Diverged 2.982.98±\,±\, 2.092.09 Diverged Diverged Diverged 1-ball (Uniform) 0.4700.470±\,±\, 0.02650.0265 Diverged Diverged 3.333.33±\,±\, 1.841.84 Diverged Diverged 2-balls (GMM) 0.5960.596±\,±\, 0.01140.0114 0.9080.908±\,±\, 0.007940.00794 1.191.19±\,±\, 0.1390.139 1.281.28±\,±\, 0.3090.309 Diverged Diverged 2-balls (Uniform) 0.5730.573±\,±\, 0.01950.0195 0.8590.859±\,±\, 0.009950.00995 1.911.91±\,±\, 0.8060.806 1.831.83±\,±\, 0.3170.317 Diverged Diverged Table 8: Baseline comparison measured by test-set unconditional path distribution loss ℒxtrajL_x^traj (↓ ). Horizon: 3s3\,s. Case Ours (Teng et al., 2026b) (Li et al., 2020) (Li et al., 2020)⋆ (Bartosh et al., 2025) (Kiyohara et al., 2025) b-ball (GMM) 0.7320.732±\,±\, 0.1370.137 Diverged 33.033.0±\,±\, 21.321.3 19.319.3±\,±\, 12.812.8 Diverged 9.629.62±\,±\, 0.2810.281 b-ball (Uniform) 0.6900.690±\,±\, 0.1560.156 Diverged 33.433.4±\,±\, 39.339.3 36.036.0±\,±\, 65.965.9 Diverged 5.505.50±\,±\, 0.09970.0997 Torus (GMM) 0.6150.615±\,±\, 0.02930.0293 Diverged Diverged Diverged Diverged 11.311.3±\,±\, 1.011.01 Torus (Uniform) 0.5950.595±\,±\, 0.04620.0462 Diverged Diverged Diverged Diverged 12.912.9±\,±\, 0.9770.977 Klein (GMM) 0.6260.626±\,±\, 0.02800.0280 Diverged Diverged Diverged Diverged 10.510.5±\,±\, 0.7290.729 Klein (Uniform) 0.6440.644±\,±\, 0.06240.0624 Diverged Diverged Diverged Diverged 11.611.6±\,±\, 0.8900.890 Klein–Torus 0.5550.555±\,±\, 0.06950.0695 Diverged Diverged Diverged Diverged 15.015.0±\,±\, 2.142.14 1-ball (GMM) 0.9170.917±\,±\, 0.1150.115 Diverged 101101±\,±\, 77.577.5 Diverged Diverged Diverged 1-ball (Uniform) 0.7760.776±\,±\, 0.1280.128 Diverged Diverged 147147±\,±\, 106106 Diverged Diverged 2-balls (GMM) 1.531.53±\,±\, 0.3030.303 3.203.20±\,±\, 0.2800.280 30.630.6±\,±\, 12.712.7 44.244.2±\,±\, 28.428.4 Diverged Diverged 2-balls (Uniform) 1.651.65±\,±\, 0.3980.398 2.692.69±\,±\, 0.1820.182 101101±\,±\, 78.778.7 95.795.7±\,±\, 30.830.8 Diverged Diverged B.3.3 Horizon around 5s Tables 9 and 10 report the losses over the complete stored trajectory, approximately 5s5\,s. Our method yields complete numeric results in all 11 settings and achieves the lowest mean losses wherever a competing method also has complete results. On Torus (GMM), 1-ball (GMM), and 1-ball (Uniform), all competing methods trigger the divergence criterion. Table 9: Baseline comparison measured by test-set conditional path distribution loss ℒxtraj,cL_x^traj,c (↓ ). Horizon: 5s5\,s. Case Ours (Teng et al., 2026b) (Li et al., 2020) (Li et al., 2020)⋆ (Bartosh et al., 2025) (Kiyohara et al., 2025) b-ball (GMM) 0.4320.432±\,±\, 0.009090.00909 Diverged Diverged 3.693.69±\,±\, 3.663.66 Diverged 0.7300.730±\,±\, 0.009930.00993 b-ball (Uniform) 0.3380.338±\,±\, 0.01190.0119 Diverged Diverged Diverged Diverged 0.5340.534±\,±\, 0.006270.00627 Torus (GMM) 0.6870.687±\,±\, 0.001090.00109 Diverged Diverged Diverged Diverged Diverged Torus (Uniform) 0.6770.677±\,±\, 0.005170.00517 Diverged Diverged Diverged Diverged 0.9550.955±\,±\, 0.02070.0207 Klein (GMM) 0.6880.688±\,±\, 0.003390.00339 Diverged Diverged Diverged Diverged 0.9030.903±\,±\, 0.01640.0164 Klein (Uniform) 0.6760.676±\,±\, 0.008320.00832 Diverged Diverged Diverged Diverged 0.9220.922±\,±\, 0.02050.0205 Klein–Torus 0.5810.581±\,±\, 0.05460.0546 Diverged Diverged Diverged Diverged 0.9950.995±\,±\, 0.04440.0444 1-ball (GMM) 0.4820.482±\,±\, 0.005780.00578 Diverged Diverged Diverged Diverged Diverged 1-ball (Uniform) 0.4390.439±\,±\, 0.02090.0209 Diverged Diverged Diverged Diverged Diverged 2-balls (GMM) 0.5250.525±\,±\, 0.006890.00689 0.8390.839±\,±\, 0.006770.00677 1.641.64±\,±\, 0.1850.185 1.841.84±\,±\, 0.6720.672 Diverged Diverged 2-balls (Uniform) 0.5090.509±\,±\, 0.01910.0191 0.8090.809±\,±\, 0.02330.0233 Diverged 3.013.01±\,±\, 0.8660.866 Diverged Diverged Table 10: Baseline comparison measured by test-set unconditional path distribution loss ℒxtrajL_x^traj (↓ ). Horizon: 5s5\,s. Case Ours (Teng et al., 2026b) (Li et al., 2020) (Li et al., 2020)⋆ (Bartosh et al., 2025) (Kiyohara et al., 2025) b-ball (GMM) 0.8230.823±\,±\, 0.1440.144 Diverged Diverged 154154±\,±\, 187187 Diverged 9.539.53±\,±\, 0.3410.341 b-ball (Uniform) 0.7090.709±\,±\, 0.1590.159 Diverged Diverged Diverged Diverged 5.405.40±\,±\, 0.1070.107 Torus (GMM) 0.7950.795±\,±\, 0.03150.0315 Diverged Diverged Diverged Diverged Diverged Torus (Uniform) 0.7940.794±\,±\, 0.06220.0622 Diverged Diverged Diverged Diverged 16.716.7±\,±\, 1.321.32 Klein (GMM) 0.8330.833±\,±\, 0.03240.0324 Diverged Diverged Diverged Diverged 13.413.4±\,±\, 1.041.04 Klein (Uniform) 0.8260.826±\,±\, 0.06960.0696 Diverged Diverged Diverged Diverged 14.714.7±\,±\, 1.271.27 Klein–Torus 0.7630.763±\,±\, 0.1180.118 Diverged Diverged Diverged Diverged 19.419.4±\,±\, 2.852.85 1-ball (GMM) 1.051.05±\,±\, 0.1410.141 Diverged Diverged Diverged Diverged Diverged 1-ball (Uniform) 0.9560.956±\,±\, 0.1850.185 Diverged Diverged Diverged Diverged Diverged 2-balls (GMM) 1.891.89±\,±\, 0.3740.374 4.024.02±\,±\, 0.9810.981 93.493.4±\,±\, 23.623.6 132132±\,±\, 83.583.5 Diverged Diverged 2-balls (Uniform) 2.142.14±\,±\, 0.6930.693 4.074.07±\,±\, 1.511.51 Diverged 277277±\,±\, 96.296.2 Diverged Diverged B.4 Ablations Percentage changes in the ablation summaries are computed per benchmark relative to the default model and then averaged over the 11 settings; unconditional results are given in parentheses. In the loss-function ablations, the indicated objectives are removed by setting their weights to zero; other settings are unchanged. B.4.1 Horizon 1s Latent Dimension: Tables 11 and 12 report the conditional and unconditional path distribution losses for dz∈dx,2dx,3dx,4dxd_z∈\d_x,2d_x,3d_x,4d_x\. The diffusion noise dimension follows dzd_z for the 1D ball and topological systems and is fixed at dx/2d_x/2 for the ball systems; other settings are unchanged. Stochastic Encoder: Table 13 compares the conditional diffusion sampler with a deterministic MLP encoder using both path distribution losses. Both variants use ten SDE trajectories per condition; other settings are unchanged. Loss Function Design: Tables 14 and 15 report the conditional and unconditional path distribution losses for each benchmark under the loss-function ablations. As summarized in Table 2, removing ℒzL_z increases the mean conditional (unconditional) loss by 19.5%19.5\% (246.6%246.6\%), while removing both path distribution losses increases them by 23.3%23.3\% (54.7%54.7\%). Both modifications worsen both metrics in all 11 settings. Table 11: Latent-dimension ablation measured by test-set conditional path distribution loss ℒxtraj,cL_x^traj,c (↓ ). Horizon: 1s1\,s. dz=dxd_z=d_x dz=2dxd_z=2d_x dz=3dxd_z=3d_x dz=4dxd_z=4d_x (default) b-ball (GMM) 0.2590.259±\,±\, 0.01720.0172 0.2750.275±\,±\, 0.02470.0247 0.2700.270±\,±\, 0.02270.0227 0.2570.257±\,±\, 0.01580.0158 b-ball (Uniform) 0.3400.340±\,±\, 0.07120.0712 0.3380.338±\,±\, 0.06560.0656 0.3000.300±\,±\, 0.02120.0212 0.2890.289±\,±\, 0.03170.0317 Torus (GMM) 0.6960.696±\,±\, 0.04400.0440 0.6140.614±\,±\, 0.01430.0143 0.6080.608±\,±\, 0.005350.00535 0.6030.603±\,±\, 0.003470.00347 Torus (Uniform) 0.6850.685±\,±\, 0.01710.0171 0.5630.563±\,±\, 0.01210.0121 0.5530.553±\,±\, 0.004060.00406 0.5600.560±\,±\, 0.007560.00756 Klein (GMM) 0.6890.689±\,±\, 0.02060.0206 0.6300.630±\,±\, 0.01890.0189 0.6110.611±\,±\, 0.003360.00336 0.6120.612±\,±\, 0.01240.0124 Klein (Uniform) 0.6400.640±\,±\, 0.03540.0354 0.5620.562±\,±\, 0.003710.00371 0.5590.559±\,±\, 0.002860.00286 0.5600.560±\,±\, 0.01780.0178 Klein–Torus 0.6700.670±\,±\, 0.05680.0568 0.4790.479±\,±\, 0.03840.0384 0.4410.441±\,±\, 0.01550.0155 0.4550.455±\,±\, 0.01490.0149 1-ball (GMM) 0.6280.628±\,±\, 0.01630.0163 0.5710.571±\,±\, 0.01560.0156 0.5710.571±\,±\, 0.007610.00761 0.5690.569±\,±\, 0.005360.00536 1-ball (Uniform) 0.6040.604±\,±\, 0.02100.0210 0.5610.561±\,±\, 0.01580.0158 0.5600.560±\,±\, 0.007960.00796 0.5690.569±\,±\, 0.01880.0188 2-balls (GMM) 0.7530.753±\,±\, 0.01930.0193 0.7100.710±\,±\, 0.01050.0105 0.7130.713±\,±\, 0.007210.00721 0.7220.722±\,±\, 0.01850.0185 2-balls (Uniform) 0.7960.796±\,±\, 0.01820.0182 0.7590.759±\,±\, 0.02090.0209 0.7660.766±\,±\, 0.02170.0217 0.7600.760±\,±\, 0.01830.0183 Table 12: Latent-dimension ablation measured by test-set unconditional path distribution loss ℒxtrajL_x^traj (↓ ). Horizon: 1s1\,s. dz=dxd_z=d_x dz=2dxd_z=2d_x dz=3dxd_z=3d_x dz=4dxd_z=4d_x (default) b-ball (GMM) 0.2740.274±\,±\, 0.04520.0452 0.3110.311±\,±\, 0.1070.107 0.3360.336±\,±\, 0.1470.147 0.2140.214±\,±\, 0.05580.0558 b-ball (Uniform) 0.6800.680±\,±\, 0.5240.524 0.4600.460±\,±\, 0.2900.290 0.2740.274±\,±\, 0.07620.0762 0.2490.249±\,±\, 0.08190.0819 Torus (GMM) 0.8770.877±\,±\, 0.9020.902 0.3420.342±\,±\, 0.02560.0256 0.3500.350±\,±\, 0.01650.0165 0.3330.333±\,±\, 0.01330.0133 Torus (Uniform) 0.5090.509±\,±\, 0.06290.0629 0.3020.302±\,±\, 0.02560.0256 0.2860.286±\,±\, 0.01550.0155 0.3040.304±\,±\, 0.03190.0319 Klein (GMM) 0.5530.553±\,±\, 0.1190.119 0.3810.381±\,±\, 0.04730.0473 0.3390.339±\,±\, 0.02720.0272 0.3410.341±\,±\, 0.02010.0201 Klein (Uniform) 0.4450.445±\,±\, 0.06010.0601 0.2960.296±\,±\, 0.01440.0144 0.2960.296±\,±\, 0.02230.0223 0.2980.298±\,±\, 0.03170.0317 Klein–Torus 0.5510.551±\,±\, 0.1350.135 0.2740.274±\,±\, 0.02500.0250 0.2400.240±\,±\, 0.009110.00911 0.2570.257±\,±\, 0.02060.0206 1-ball (GMM) 0.8410.841±\,±\, 0.1120.112 0.5530.553±\,±\, 0.06250.0625 0.5720.572±\,±\, 0.03740.0374 0.5700.570±\,±\, 0.05180.0518 1-ball (Uniform) 0.6750.675±\,±\, 0.1160.116 0.5150.515±\,±\, 0.06420.0642 0.4970.497±\,±\, 0.02630.0263 0.5250.525±\,±\, 0.05790.0579 2-balls (GMM) 1.281.28±\,±\, 0.1070.107 0.9170.917±\,±\, 0.05740.0574 0.9570.957±\,±\, 0.08080.0808 1.041.04±\,±\, 0.1620.162 2-balls (Uniform) 1.361.36±\,±\, 0.1750.175 1.081.08±\,±\, 0.1380.138 1.141.14±\,±\, 0.05850.0585 1.071.07±\,±\, 0.08520.0852 Table 13: Sampler ablation measured by test-set conditional and unconditional path distribution losses (↓ ). Horizon: 1s1\,s. DDIM MLP Case Conditional Unconditional Conditional Unconditional b-ball (GMM) 0.2570.257±\,±\, 0.01580.0158 0.2140.214±\,±\, 0.05580.0558 0.2850.285±\,±\, 0.01620.0162 0.3440.344±\,±\, 0.09830.0983 b-ball (Uniform) 0.2890.289±\,±\, 0.03170.0317 0.2490.249±\,±\, 0.08190.0819 0.3320.332±\,±\, 0.03030.0303 0.4570.457±\,±\, 0.1220.122 Torus (GMM) 0.6030.603±\,±\, 0.003470.00347 0.3330.333±\,±\, 0.01330.0133 0.6100.610±\,±\, 0.007990.00799 0.3420.342±\,±\, 0.02140.0214 Torus (Uniform) 0.5600.560±\,±\, 0.007560.00756 0.3040.304±\,±\, 0.03190.0319 0.5600.560±\,±\, 0.003770.00377 0.2990.299±\,±\, 0.01360.0136 Klein (GMM) 0.6120.612±\,±\, 0.01240.0124 0.3410.341±\,±\, 0.02010.0201 0.6130.613±\,±\, 0.005920.00592 0.3450.345±\,±\, 0.008730.00873 Klein (Uniform) 0.5600.560±\,±\, 0.01780.0178 0.2980.298±\,±\, 0.03170.0317 0.5650.565±\,±\, 0.004730.00473 0.3170.317±\,±\, 0.01100.0110 Klein–Torus 0.4550.455±\,±\, 0.01490.0149 0.2570.257±\,±\, 0.02060.0206 0.4890.489±\,±\, 0.02640.0264 0.2810.281±\,±\, 0.02650.0265 1-ball (GMM) 0.5690.569±\,±\, 0.005360.00536 0.5700.570±\,±\, 0.05180.0518 0.5880.588±\,±\, 0.005130.00513 0.5910.591±\,±\, 0.05510.0551 1-ball (Uniform) 0.5690.569±\,±\, 0.01880.0188 0.5250.525±\,±\, 0.05790.0579 0.5910.591±\,±\, 0.01510.0151 0.5780.578±\,±\, 0.08350.0835 2-balls (GMM) 0.7220.722±\,±\, 0.01850.0185 1.041.04±\,±\, 0.1620.162 0.7070.707±\,±\, 0.006400.00640 0.9290.929±\,±\, 0.06470.0647 2-balls (Uniform) 0.7600.760±\,±\, 0.01830.0183 1.071.07±\,±\, 0.08520.0852 0.7510.751±\,±\, 0.01780.0178 1.021.02±\,±\, 0.08150.0815 Table 14: Loss-function ablation measured by test-set conditional path distribution loss ℒxtraj,cL_x^traj,c (↓ ). Column labels other than Default model identify the removed objectives (their weights are set to zero). Values are the mean ± sample standard deviation over five training seeds. The lowest mean among finite entries in each row is bold. Diverged denotes a configuration with nonfinite results or either test metric exceeding 10310^3 in any seed. Horizon: 1s1\,s. Case Default model ℒxtraj,cL_x^traj,c ℒzL_z ℒxL_x ℒxtrajL_x^traj ℒxtraj,c,ℒxtrajL_x^traj,c,\ L_x^traj ℒx,ℒxtrajL_x,\ L_x^traj ℒxtraj,c,ℒxL_x^traj,c,\ L_x b-ball (GMM) 0.2570.257±\,±\, 0.01580.0158 0.2550.255±\,±\, 0.007460.00746 0.2880.288±\,±\, 0.03480.0348 0.2800.280±\,±\, 0.03670.0367 0.2610.261±\,±\, 0.02080.0208 0.2700.270±\,±\, 0.02300.0230 0.2580.258±\,±\, 0.006310.00631 0.2540.254±\,±\, 0.009870.00987 b-ball (Uniform) 0.2890.289±\,±\, 0.03170.0317 0.2920.292±\,±\, 0.02020.0202 0.3350.335±\,±\, 0.04890.0489 0.2970.297±\,±\, 0.03530.0353 0.2980.298±\,±\, 0.03460.0346 0.3170.317±\,±\, 0.01630.0163 0.2880.288±\,±\, 0.01500.0150 0.2830.283±\,±\, 0.005730.00573 Torus (GMM) 0.6030.603±\,±\, 0.003470.00347 0.6070.607±\,±\, 0.003510.00351 0.6490.649±\,±\, 0.03840.0384 0.6060.606±\,±\, 0.004630.00463 0.6170.617±\,±\, 0.004110.00411 0.7090.709±\,±\, 0.001380.00138 0.6160.616±\,±\, 0.004030.00403 0.6060.606±\,±\, 0.001670.00167 Torus (Uniform) 0.5600.560±\,±\, 0.007560.00756 0.5580.558±\,±\, 0.003340.00334 0.6470.647±\,±\, 0.06820.0682 0.5530.553±\,±\, 0.004320.00432 0.5550.555±\,±\, 0.002960.00296 0.7090.709±\,±\, 0.001290.00129 0.5590.559±\,±\, 0.003890.00389 0.5540.554±\,±\, 0.002160.00216 Klein (GMM) 0.6120.612±\,±\, 0.01240.0124 0.6100.610±\,±\, 0.008080.00808 0.7090.709±\,±\, 0.05950.0595 0.6070.607±\,±\, 0.007780.00778 0.6140.614±\,±\, 0.01080.0108 0.7080.708±\,±\, 0.001720.00172 0.6160.616±\,±\, 0.006060.00606 0.6050.605±\,±\, 0.002170.00217 Klein (Uniform) 0.5600.560±\,±\, 0.01780.0178 0.5590.559±\,±\, 0.004680.00468 0.7180.718±\,±\, 0.01750.0175 0.5620.562±\,±\, 0.02250.0225 0.5640.564±\,±\, 0.01760.0176 0.7040.704±\,±\, 0.0008410.000841 0.5640.564±\,±\, 0.01080.0108 0.5540.554±\,±\, 0.002590.00259 Klein–Torus 0.4550.455±\,±\, 0.01490.0149 0.4730.473±\,±\, 0.01750.0175 0.7230.723±\,±\, 0.04780.0478 0.4550.455±\,±\, 0.02840.0284 0.4500.450±\,±\, 0.009090.00909 0.7130.713±\,±\, 0.002570.00257 0.4460.446±\,±\, 0.01520.0152 0.4710.471±\,±\, 0.01180.0118 1-ball (GMM) 0.5690.569±\,±\, 0.005360.00536 0.6350.635±\,±\, 0.01380.0138 0.6500.650±\,±\, 0.02630.0263 0.5780.578±\,±\, 0.01350.0135 0.5830.583±\,±\, 0.01890.0189 0.6950.695±\,±\, 0.02050.0205 0.5850.585±\,±\, 0.01910.0191 0.6370.637±\,±\, 0.02180.0218 1-ball (Uniform) 0.5690.569±\,±\, 0.01880.0188 0.6500.650±\,±\, 0.03350.0335 0.6650.665±\,±\, 0.03410.0341 0.5790.579±\,±\, 0.01910.0191 0.5630.563±\,±\, 0.02330.0233 0.7180.718±\,±\, 0.01960.0196 0.5550.555±\,±\, 0.01200.0120 0.6550.655±\,±\, 0.03560.0356 2-balls (GMM) 0.7220.722±\,±\, 0.01850.0185 0.8500.850±\,±\, 0.007650.00765 0.8250.825±\,±\, 0.07750.0775 0.7240.724±\,±\, 0.01560.0156 0.7060.706±\,±\, 0.009630.00963 0.9040.904±\,±\, 0.005280.00528 0.6970.697±\,±\, 0.008160.00816 0.8480.848±\,±\, 0.005630.00563 2-balls (Uniform) 0.7600.760±\,±\, 0.01830.0183 0.9160.916±\,±\, 0.01190.0119 0.8760.876±\,±\, 0.06350.0635 0.7550.755±\,±\, 0.02130.0213 0.7240.724±\,±\, 0.01680.0168 0.9570.957±\,±\, 0.01570.0157 0.7170.717±\,±\, 0.007310.00731 0.9130.913±\,±\, 0.01090.0109 Table 15: Loss-function ablation measured by test-set unconditional path distribution loss ℒxtrajL_x^traj (↓ ). Column labels other than Default model identify the removed objectives (their weights are set to zero). Values are the mean ± sample standard deviation over five training seeds. The lowest mean among finite entries in each row is bold. Diverged denotes a configuration with nonfinite results or either test metric exceeding 10310^3 in any seed. Horizon: 1s1\,s. Case Default model ℒxtraj,cL_x^traj,c ℒzL_z ℒxL_x ℒxtrajL_x^traj ℒxtraj,c,ℒxtrajL_x^traj,c,\ L_x^traj ℒx,ℒxtrajL_x,\ L_x^traj ℒxtraj,c,ℒxL_x^traj,c,\ L_x b-ball (GMM) 0.2140.214±\,±\, 0.05580.0558 0.2150.215±\,±\, 0.03950.0395 0.4110.411±\,±\, 0.2730.273 0.3790.379±\,±\, 0.2470.247 0.1940.194±\,±\, 0.02680.0268 0.3080.308±\,±\, 0.1570.157 0.2160.216±\,±\, 0.04090.0409 0.2050.205±\,±\, 0.04570.0457 b-ball (Uniform) 0.2490.249±\,±\, 0.08190.0819 0.2790.279±\,±\, 0.1100.110 0.4080.408±\,±\, 0.1690.169 0.2600.260±\,±\, 0.1130.113 0.2680.268±\,±\, 0.1070.107 0.3350.335±\,±\, 0.06750.0675 0.2180.218±\,±\, 0.05770.0577 0.2330.233±\,±\, 0.03970.0397 Torus (GMM) 0.3330.333±\,±\, 0.01330.0133 0.3230.323±\,±\, 0.005860.00586 0.8420.842±\,±\, 0.4930.493 0.3480.348±\,±\, 0.02040.0204 0.3860.386±\,±\, 0.03450.0345 0.5070.507±\,±\, 0.01340.0134 0.3810.381±\,±\, 0.03610.0361 0.3220.322±\,±\, 0.01320.0132 Torus (Uniform) 0.3040.304±\,±\, 0.03190.0319 0.2850.285±\,±\, 0.004010.00401 1.011.01±\,±\, 0.7090.709 0.2990.299±\,±\, 0.02050.0205 0.3020.302±\,±\, 0.01360.0136 0.5450.545±\,±\, 0.01120.0112 0.3220.322±\,±\, 0.02110.0211 0.2820.282±\,±\, 0.003760.00376 Klein (GMM) 0.3410.341±\,±\, 0.02010.0201 0.3140.314±\,±\, 0.01130.0113 1.671.67±\,±\, 1.021.02 0.3390.339±\,±\, 0.01240.0124 0.3440.344±\,±\, 0.03340.0334 0.5150.515±\,±\, 0.008330.00833 0.3920.392±\,±\, 0.03680.0368 0.3130.313±\,±\, 0.01250.0125 Klein (Uniform) 0.2980.298±\,±\, 0.03170.0317 0.2840.284±\,±\, 0.01400.0140 1.601.60±\,±\, 0.4890.489 0.3000.300±\,±\, 0.03730.0373 0.3300.330±\,±\, 0.04120.0412 0.5440.544±\,±\, 0.009190.00919 0.3280.328±\,±\, 0.02680.0268 0.2820.282±\,±\, 0.007490.00749 Klein–Torus 0.2570.257±\,±\, 0.02060.0206 0.2630.263±\,±\, 0.01190.0119 1.981.98±\,±\, 0.4790.479 0.2630.263±\,±\, 0.03100.0310 0.2480.248±\,±\, 0.01950.0195 0.5900.590±\,±\, 0.007760.00776 0.2460.246±\,±\, 0.01080.0108 0.2620.262±\,±\, 0.01120.0112 1-ball (GMM) 0.5700.570±\,±\, 0.05180.0518 0.6080.608±\,±\, 0.07430.0743 1.171.17±\,±\, 0.4960.496 0.5850.585±\,±\, 0.06950.0695 0.6210.621±\,±\, 0.1370.137 0.6890.689±\,±\, 0.06940.0694 0.6660.666±\,±\, 0.1430.143 0.6000.600±\,±\, 0.06260.0626 1-ball (Uniform) 0.5250.525±\,±\, 0.05790.0579 0.6340.634±\,±\, 0.1270.127 1.411.41±\,±\, 0.4170.417 0.5430.543±\,±\, 0.05510.0551 0.5100.510±\,±\, 0.03920.0392 0.6740.674±\,±\, 0.05160.0516 0.5520.552±\,±\, 0.09380.0938 0.6350.635±\,±\, 0.1090.109 2-balls (GMM) 1.041.04±\,±\, 0.1620.162 1.181.18±\,±\, 0.1160.116 3.133.13±\,±\, 2.582.58 1.051.05±\,±\, 0.07400.0740 1.041.04±\,±\, 0.1400.140 1.361.36±\,±\, 0.06720.0672 1.001.00±\,±\, 0.03040.0304 1.201.20±\,±\, 0.08310.0831 2-balls (Uniform) 1.071.07±\,±\, 0.08520.0852 1.481.48±\,±\, 0.1470.147 3.183.18±\,±\, 1.751.75 1.091.09±\,±\, 0.1470.147 1.011.01±\,±\, 0.07320.0732 1.581.58±\,±\, 0.2090.209 0.9960.996±\,±\, 0.08530.0853 1.491.49±\,±\, 0.1450.145 B.4.2 Horizon 3s Latent Dimension: Tables 17 and 18 report the latent-dimension ablation over 3s3\,s. As summarized in Table 16, using dz=dxd_z=d_x increases the mean conditional (unconditional) loss by 25.9%25.9\% (417.1%417.1\%), with both metrics worsening in all 11 settings. Using dz=2dxd_z=2d_x or 3dx3d_x gives smaller mean increases of 1.5%1.5\% (15.0%15.0\%) or 0.7%0.7\% (14.4%14.4\%), respectively. Stochastic Encoder: Table 19 reports the sampler ablation over 3s3\,s. Replacing the stochastic encoder with a deterministic MLP increases the mean losses by 2.9%2.9\% (13.9%13.9\%), with degradation in 7/117/11 (9/119/11) settings. Loss Function Design: Tables 20 and 21 report the conditional and unconditional path distribution losses for each benchmark under the loss-function ablations. As summarized in Table 16, removing ℒzL_z increases the mean losses by 63.8%63.8\% (1611.9%1611.9\%), while removing both path distribution losses increases them by 12.5%12.5\% (24.6%24.6\%). Both modifications worsen both metrics in all 11 settings. Table 16: Ablation summary across 11 benchmark settings on the conditional (unconditional) trajectory distribution loss. Mean change is the average of per-setting percentage changes relative to the default model; positive values indicate degradation. Worse cases count the settings with a higher mean test loss than the default model (out of 11), using the means over five training seeds. MLP replaces the diffusion sampler with a deterministic MLP encoder; loss labels identify the removed objectives. Per-case latent-dimension and loss-ablation results are reported in Tables 17, 18, 20 and 21. Per-case conditional and unconditional results for the MLP encoder ablation are reported in Table 19. Horizon: 3s3\,s. Setting dz=dxd_z=d_x dz=2dxd_z=2d_x dz=3dxd_z=3d_x MLP ℒxtraj,cL_x^traj,c ℒzL_z ℒxL_x ℒxtrajL_x^traj ℒxtraj,c,ℒxtrajL_x^traj,c,\ L_x^traj ℒx,ℒxtrajL_x,\ L_x^traj ℒxtraj,c,ℒxL_x^traj,c,\ L_x Mean change (%) +25.9(+417.1)+25.9\;(+417.1) +1.5(+15.0)+1.5\;(+15.0) +0.7(+14.4)+0.7\;(+14.4) +2.9(+13.9)+2.9\;(+13.9) +4.4(+2.6)+4.4\;(+2.6) +63.8(+1611.9)+63.8\;(+1611.9) +0.5(+6.4)+0.5\;(+6.4) −0.7(+2.7)-0.7\;(+2.7) +12.5(+24.6)+12.5\;(+24.6) −1.1(+5.1)-1.1\;(+5.1) +3.9(−0.0)+3.9\;(-0.0) Worse cases (/11) 11(11)11\;(11) 5(7)5\;(7) 5(6)5\;(6) 7(9)7\;(9) 8(5)8\;(5) 11(11)11\;(11) 6(7)6\;(7) 6(7)6\;(7) 11(11)11\;(11) 6(7)6\;(7) 7(4)7\;(4) Table 17: Latent-dimension ablation measured by test-set conditional path distribution loss ℒxtraj,cL_x^traj,c (↓ ). Horizon: 3s3\,s. dz=dxd_z=d_x dz=2dxd_z=2d_x dz=3dxd_z=3d_x dz=4dxd_z=4d_x (default) b-ball (GMM) 0.5320.532±\,±\, 0.02940.0294 0.5050.505±\,±\, 0.03450.0345 0.5290.529±\,±\, 0.05230.0523 0.4870.487±\,±\, 0.01530.0153 b-ball (Uniform) 1.351.35±\,±\, 1.971.97 0.4860.486±\,±\, 0.07460.0746 0.4380.438±\,±\, 0.02670.0267 0.4330.433±\,±\, 0.01520.0152 Torus (GMM) 0.7220.722±\,±\, 0.05590.0559 0.6750.675±\,±\, 0.004970.00497 0.6730.673±\,±\, 0.002190.00219 0.6710.671±\,±\, 0.002390.00239 Torus (Uniform) 0.7010.701±\,±\, 0.005760.00576 0.6520.652±\,±\, 0.005650.00565 0.6450.645±\,±\, 0.003200.00320 0.6540.654±\,±\, 0.006910.00691 Klein (GMM) 0.7050.705±\,±\, 0.008100.00810 0.6830.683±\,±\, 0.007860.00786 0.6750.675±\,±\, 0.001660.00166 0.6770.677±\,±\, 0.005300.00530 Klein (Uniform) 0.6860.686±\,±\, 0.01360.0136 0.6510.651±\,±\, 0.003530.00353 0.6530.653±\,±\, 0.003160.00316 0.6530.653±\,±\, 0.01080.0108 Klein–Torus 0.6990.699±\,±\, 0.01490.0149 0.5780.578±\,±\, 0.05430.0543 0.5340.534±\,±\, 0.02460.0246 0.5450.545±\,±\, 0.03250.0325 1-ball (GMM) 0.5490.549±\,±\, 0.01490.0149 0.5150.515±\,±\, 0.006560.00656 0.5220.522±\,±\, 0.003000.00300 0.5230.523±\,±\, 0.007830.00783 1-ball (Uniform) 0.4760.476±\,±\, 0.01210.0121 0.4500.450±\,±\, 0.007600.00760 0.4580.458±\,±\, 0.006770.00677 0.4700.470±\,±\, 0.02650.0265 2-balls (GMM) 0.6110.611±\,±\, 0.01050.0105 0.5960.596±\,±\, 0.02450.0245 0.6200.620±\,±\, 0.02930.0293 0.5960.596±\,±\, 0.01140.0114 2-balls (Uniform) 0.5830.583±\,±\, 0.01400.0140 0.5640.564±\,±\, 0.02260.0226 0.5700.570±\,±\, 0.01640.0164 0.5730.573±\,±\, 0.01950.0195 Table 18: Latent-dimension ablation measured by test-set unconditional path distribution loss ℒxtrajL_x^traj (↓ ). Horizon: 3s3\,s. dz=dxd_z=d_x dz=2dxd_z=2d_x dz=3dxd_z=3d_x dz=4dxd_z=4d_x (default) b-ball (GMM) 1.391.39±\,±\, 0.3750.375 1.031.03±\,±\, 0.3770.377 1.371.37±\,±\, 0.7090.709 0.7320.732±\,±\, 0.1370.137 b-ball (Uniform) 28.428.4±\,±\, 60.360.3 1.181.18±\,±\, 0.8380.838 0.6440.644±\,±\, 0.2650.265 0.6900.690±\,±\, 0.1560.156 Torus (GMM) 2.172.17±\,±\, 3.183.18 0.6230.623±\,±\, 0.03240.0324 0.6430.643±\,±\, 0.02750.0275 0.6150.615±\,±\, 0.02930.0293 Torus (Uniform) 0.7850.785±\,±\, 0.07960.0796 0.5800.580±\,±\, 0.05550.0555 0.5600.560±\,±\, 0.02180.0218 0.5950.595±\,±\, 0.04620.0462 Klein (GMM) 0.8970.897±\,±\, 0.1670.167 0.6890.689±\,±\, 0.06780.0678 0.6380.638±\,±\, 0.04070.0407 0.6260.626±\,±\, 0.02800.0280 Klein (Uniform) 0.7860.786±\,±\, 0.06010.0601 0.6230.623±\,±\, 0.02580.0258 0.6310.631±\,±\, 0.02140.0214 0.6440.644±\,±\, 0.06240.0624 Klein–Torus 0.9760.976±\,±\, 0.1700.170 0.5840.584±\,±\, 0.06390.0639 0.5230.523±\,±\, 0.02820.0282 0.5550.555±\,±\, 0.06950.0695 1-ball (GMM) 1.141.14±\,±\, 0.1470.147 0.8230.823±\,±\, 0.09010.0901 0.9510.951±\,±\, 0.05410.0541 0.9170.917±\,±\, 0.1150.115 1-ball (Uniform) 0.8530.853±\,±\, 0.1340.134 0.6870.687±\,±\, 0.06270.0627 0.7310.731±\,±\, 0.02420.0242 0.7760.776±\,±\, 0.1280.128 2-balls (GMM) 1.821.82±\,±\, 0.1350.135 2.412.41±\,±\, 2.232.23 2.842.84±\,±\, 1.401.40 1.531.53±\,±\, 0.3030.303 2-balls (Uniform) 1.671.67±\,±\, 0.1010.101 1.771.77±\,±\, 0.8000.800 1.681.68±\,±\, 0.3150.315 1.651.65±\,±\, 0.3980.398 Table 19: Sampler ablation measured by test-set conditional and unconditional path distribution losses (↓ ). Horizon: 3s3\,s. DDIM MLP Case Conditional Unconditional Conditional Unconditional b-ball (GMM) 0.4870.487±\,±\, 0.01530.0153 0.7320.732±\,±\, 0.1370.137 0.5440.544±\,±\, 0.03050.0305 1.431.43±\,±\, 0.5700.570 b-ball (Uniform) 0.4330.433±\,±\, 0.01520.0152 0.6900.690±\,±\, 0.1560.156 0.4740.474±\,±\, 0.01660.0166 0.9710.971±\,±\, 0.2290.229 Torus (GMM) 0.6710.671±\,±\, 0.002390.00239 0.6150.615±\,±\, 0.02930.0293 0.6740.674±\,±\, 0.002700.00270 0.6230.623±\,±\, 0.03380.0338 Torus (Uniform) 0.6540.654±\,±\, 0.006910.00691 0.5950.595±\,±\, 0.04620.0462 0.6490.649±\,±\, 0.002620.00262 0.6070.607±\,±\, 0.03860.0386 Klein (GMM) 0.6770.677±\,±\, 0.005300.00530 0.6260.626±\,±\, 0.02800.0280 0.6770.677±\,±\, 0.003370.00337 0.6440.644±\,±\, 0.03200.0320 Klein (Uniform) 0.6530.653±\,±\, 0.01080.0108 0.6440.644±\,±\, 0.06240.0624 0.6520.652±\,±\, 0.003580.00358 0.6740.674±\,±\, 0.01210.0121 Klein–Torus 0.5450.545±\,±\, 0.03250.0325 0.5550.555±\,±\, 0.06950.0695 0.5890.589±\,±\, 0.04350.0435 0.6190.619±\,±\, 0.08200.0820 1-ball (GMM) 0.5230.523±\,±\, 0.007830.00783 0.9170.917±\,±\, 0.1150.115 0.5430.543±\,±\, 0.007170.00717 0.9460.946±\,±\, 0.06010.0601 1-ball (Uniform) 0.4700.470±\,±\, 0.02650.0265 0.7760.776±\,±\, 0.1280.128 0.4790.479±\,±\, 0.008000.00800 0.7950.795±\,±\, 0.07700.0770 2-balls (GMM) 0.5960.596±\,±\, 0.01140.0114 1.531.53±\,±\, 0.3030.303 0.5910.591±\,±\, 0.007800.00780 1.501.50±\,±\, 0.1220.122 2-balls (Uniform) 0.5730.573±\,±\, 0.01950.0195 1.651.65±\,±\, 0.3980.398 0.5620.562±\,±\, 0.01100.0110 1.511.51±\,±\, 0.1440.144 Table 20: Loss-function ablation measured by test-set conditional path distribution loss ℒxtraj,cL_x^traj,c (↓ ). Column labels other than Default model identify the removed objectives (their weights are set to zero). Values are the mean ± sample standard deviation over five training seeds. The lowest mean among finite entries in each row is bold. Diverged denotes a configuration with nonfinite results or either test metric exceeding 10310^3 in any seed. Horizon: 3s3\,s. Case Default model ℒxtraj,cL_x^traj,c ℒzL_z ℒxL_x ℒxtrajL_x^traj ℒxtraj,c,ℒxtrajL_x^traj,c,\ L_x^traj ℒx,ℒxtrajL_x,\ L_x^traj ℒxtraj,c,ℒxL_x^traj,c,\ L_x b-ball (GMM) 0.4870.487±\,±\, 0.01530.0153 0.5080.508±\,±\, 0.01030.0103 0.5880.588±\,±\, 0.05340.0534 0.5180.518±\,±\, 0.03430.0343 0.4980.498±\,±\, 0.01160.0116 0.5180.518±\,±\, 0.02740.0274 0.4970.497±\,±\, 0.01850.0185 0.5050.505±\,±\, 0.006210.00621 b-ball (Uniform) 0.4330.433±\,±\, 0.01520.0152 0.4460.446±\,±\, 0.02680.0268 0.5080.508±\,±\, 0.05560.0556 0.4360.436±\,±\, 0.02270.0227 0.4340.434±\,±\, 0.01400.0140 0.4560.456±\,±\, 0.01310.0131 0.4380.438±\,±\, 0.01370.0137 0.4320.432±\,±\, 0.01400.0140 Torus (GMM) 0.6710.671±\,±\, 0.002390.00239 0.6730.673±\,±\, 0.001590.00159 0.7310.731±\,±\, 0.04640.0464 0.6730.673±\,±\, 0.002240.00224 0.6780.678±\,±\, 0.001800.00180 0.7070.707±\,±\, 0.0009570.000957 0.6770.677±\,±\, 0.003000.00300 0.6720.672±\,±\, 0.001460.00146 Torus (Uniform) 0.6540.654±\,±\, 0.006910.00691 0.6480.648±\,±\, 0.002290.00229 0.7300.730±\,±\, 0.04860.0486 0.6470.647±\,±\, 0.002850.00285 0.6470.647±\,±\, 0.002340.00234 0.7080.708±\,±\, 0.0009330.000933 0.6520.652±\,±\, 0.002840.00284 0.6450.645±\,±\, 0.001700.00170 Klein (GMM) 0.6770.677±\,±\, 0.005300.00530 0.6740.674±\,±\, 0.004780.00478 0.8130.813±\,±\, 0.07080.0708 0.6750.675±\,±\, 0.003680.00368 0.6780.678±\,±\, 0.003940.00394 0.7090.709±\,±\, 0.001200.00120 0.6790.679±\,±\, 0.0007660.000766 0.6730.673±\,±\, 0.001370.00137 Klein (Uniform) 0.6530.653±\,±\, 0.01080.0108 0.6510.651±\,±\, 0.003300.00330 0.7790.779±\,±\, 0.05040.0504 0.6520.652±\,±\, 0.01120.0112 0.6540.654±\,±\, 0.01040.0104 0.7060.706±\,±\, 0.0005030.000503 0.6550.655±\,±\, 0.006930.00693 0.6470.647±\,±\, 0.001510.00151 Klein–Torus 0.5450.545±\,±\, 0.03250.0325 0.5550.555±\,±\, 0.02340.0234 0.8320.832±\,±\, 0.01640.0164 0.5420.542±\,±\, 0.04060.0406 0.5410.541±\,±\, 0.01830.0183 0.7090.709±\,±\, 0.0005190.000519 0.5250.525±\,±\, 0.006400.00640 0.5550.555±\,±\, 0.03090.0309 1-ball (GMM) 0.5230.523±\,±\, 0.007830.00783 0.5640.564±\,±\, 0.01770.0177 0.7400.740±\,±\, 0.09650.0965 0.5240.524±\,±\, 0.009400.00940 0.5260.526±\,±\, 0.01180.0118 0.6040.604±\,±\, 0.01390.0139 0.5310.531±\,±\, 0.01340.0134 0.5630.563±\,±\, 0.01730.0173 1-ball (Uniform) 0.4700.470±\,±\, 0.02650.0265 0.5130.513±\,±\, 0.01830.0183 0.7680.768±\,±\, 0.1710.171 0.4760.476±\,±\, 0.01890.0189 0.4560.456±\,±\, 0.01190.0119 0.5640.564±\,±\, 0.01600.0160 0.4510.451±\,±\, 0.007610.00761 0.5180.518±\,±\, 0.02040.0204 2-balls (GMM) 0.5960.596±\,±\, 0.01140.0114 0.6550.655±\,±\, 0.004790.00479 2.282.28±\,±\, 1.551.55 0.5970.597±\,±\, 0.01190.0119 0.5860.586±\,±\, 0.005680.00568 0.6870.687±\,±\, 0.004680.00468 0.5760.576±\,±\, 0.007170.00717 0.6530.653±\,±\, 0.004470.00447 2-balls (Uniform) 0.5730.573±\,±\, 0.01950.0195 0.6490.649±\,±\, 0.005290.00529 1.511.51±\,±\, 0.7580.758 0.5650.565±\,±\, 0.01700.0170 0.5440.544±\,±\, 0.01480.0148 0.6810.681±\,±\, 0.008100.00810 0.5320.532±\,±\, 0.004650.00465 0.6460.646±\,±\, 0.004710.00471 Table 21: Loss-function ablation measured by test-set unconditional path distribution loss ℒxtrajL_x^traj (↓ ). Column labels other than Default model identify the removed objectives (their weights are set to zero). Values are the mean ± sample standard deviation over five training seeds. The lowest mean among finite entries in each row is bold. Diverged denotes a configuration with nonfinite results or either test metric exceeding 10310^3 in any seed. Horizon: 3s3\,s. Case Default model ℒxtraj,cL_x^traj,c ℒzL_z ℒxL_x ℒxtrajL_x^traj ℒxtraj,c,ℒxtrajL_x^traj,c,\ L_x^traj ℒx,ℒxtrajL_x,\ L_x^traj ℒxtraj,c,ℒxL_x^traj,c,\ L_x b-ball (GMM) 0.7320.732±\,±\, 0.1370.137 0.9030.903±\,±\, 0.1870.187 2.252.25±\,±\, 0.9080.908 1.181.18±\,±\, 0.2680.268 0.8670.867±\,±\, 0.1340.134 1.071.07±\,±\, 0.5680.568 0.9540.954±\,±\, 0.2650.265 0.8420.842±\,±\, 0.1510.151 b-ball (Uniform) 0.6900.690±\,±\, 0.1560.156 0.6460.646±\,±\, 0.2160.216 1.621.62±\,±\, 1.191.19 0.6960.696±\,±\, 0.1760.176 0.7170.717±\,±\, 0.09340.0934 0.7100.710±\,±\, 0.1210.121 0.7830.783±\,±\, 0.2480.248 0.5720.572±\,±\, 0.1500.150 Torus (GMM) 0.6150.615±\,±\, 0.02930.0293 0.6030.603±\,±\, 0.01680.0168 2.812.81±\,±\, 1.701.70 0.6510.651±\,±\, 0.03480.0348 0.6950.695±\,±\, 0.05170.0517 0.7410.741±\,±\, 0.01520.0152 0.6800.680±\,±\, 0.06730.0673 0.6040.604±\,±\, 0.02110.0211 Torus (Uniform) 0.5950.595±\,±\, 0.04620.0462 0.5510.551±\,±\, 0.009960.00996 2.812.81±\,±\, 1.531.53 0.5890.589±\,±\, 0.03810.0381 0.5860.586±\,±\, 0.02390.0239 0.7760.776±\,±\, 0.02030.0203 0.6330.633±\,±\, 0.04390.0439 0.5430.543±\,±\, 0.008640.00864 Klein (GMM) 0.6260.626±\,±\, 0.02800.0280 0.5860.586±\,±\, 0.01430.0143 5.895.89±\,±\, 3.803.80 0.6400.640±\,±\, 0.03940.0394 0.6540.654±\,±\, 0.05030.0503 0.7330.733±\,±\, 0.01080.0108 0.7030.703±\,±\, 0.06170.0617 0.5900.590±\,±\, 0.01670.0167 Klein (Uniform) 0.6440.644±\,±\, 0.06240.0624 0.6180.618±\,±\, 0.02220.0222 3.903.90±\,±\, 1.991.99 0.6430.643±\,±\, 0.06780.0678 0.6840.684±\,±\, 0.05290.0529 0.8410.841±\,±\, 0.01980.0198 0.6890.689±\,±\, 0.03460.0346 0.6080.608±\,±\, 0.01660.0166 Klein–Torus 0.5550.555±\,±\, 0.06950.0695 0.5480.548±\,±\, 0.02090.0209 6.086.08±\,±\, 0.9260.926 0.5480.548±\,±\, 0.06360.0636 0.5330.533±\,±\, 0.01900.0190 0.9310.931±\,±\, 0.01850.0185 0.5190.519±\,±\, 0.02240.0224 0.5500.550±\,±\, 0.05110.0511 1-ball (GMM) 0.9170.917±\,±\, 0.1150.115 0.9380.938±\,±\, 0.08900.0890 7.097.09±\,±\, 3.993.99 0.9380.938±\,±\, 0.08300.0830 0.9730.973±\,±\, 0.1590.159 0.9510.951±\,±\, 0.06720.0672 1.011.01±\,±\, 0.1910.191 0.9070.907±\,±\, 0.06930.0693 1-ball (Uniform) 0.7760.776±\,±\, 0.1280.128 0.8930.893±\,±\, 0.1250.125 9.669.66±\,±\, 6.556.55 0.8080.808±\,±\, 0.1470.147 0.7110.711±\,±\, 0.05740.0574 0.8820.882±\,±\, 0.05610.0561 0.7490.749±\,±\, 0.1080.108 0.8710.871±\,±\, 0.08590.0859 2-balls (GMM) 1.531.53±\,±\, 0.3030.303 1.551.55±\,±\, 0.07710.0771 130130±\,±\, 119119 1.541.54±\,±\, 0.08640.0864 1.531.53±\,±\, 0.1240.124 1.721.72±\,±\, 0.1410.141 1.461.46±\,±\, 0.05300.0530 1.561.56±\,±\, 0.08220.0822 2-balls (Uniform) 1.651.65±\,±\, 0.3980.398 1.891.89±\,±\, 0.3600.360 69.569.5±\,±\, 60.560.5 1.591.59±\,±\, 0.2450.245 1.511.51±\,±\, 0.2850.285 2.062.06±\,±\, 0.4870.487 1.341.34±\,±\, 0.1210.121 1.851.85±\,±\, 0.3120.312 B.4.3 Horizon around 5s Latent Dimension: Tables 23 and 24 report the latent-dimension ablation over the complete stored trajectory. Using dz=dxd_z=d_x triggers the divergence criterion on b-ball (Uniform), while dz=2dxd_z=2d_x triggers it on 2-balls (GMM). Using dz=3dxd_z=3d_x increases the mean losses by 20.9%20.9\% (396.2%396.2\%), largely driven by 2-balls (GMM), where the unconditional loss reaches 82.682.6, compared with 1.891.89 for the default dz=4dxd_z=4d_x. These results show that comparable short-horizon performance does not necessarily imply comparable performance over the full trajectory. Stochastic Encoder: Table 25 reports the sampler ablation over the complete stored trajectory. Replacing the stochastic encoder with a deterministic MLP increases the mean losses by 3.1%3.1\% (14.2%14.2\%), with degradation in 7/117/11 (10/1110/11) settings. Loss Function Design: Tables 26 and 27 report the conditional and unconditional path distribution losses for each benchmark under the loss-function ablations. Removing ℒzL_z triggers the divergence criterion on 2-balls (GMM). As summarized in Table 22, removing both path distribution losses increases the mean losses by 9.9%9.9\% (17.4%17.4\%), with both metrics worsening in all 11 settings. Table 22: Ablation summary across 11 benchmark settings on the conditional (unconditional) trajectory distribution loss. Mean change is the average of per-setting percentage changes relative to the default model; positive values indicate degradation. Worse cases count the settings with a higher mean test loss than the default model (out of 11), using the means over five training seeds. MLP replaces the diffusion sampler with a deterministic MLP encoder; loss labels identify the removed objectives. Per-case latent-dimension and loss-ablation results are reported in Tables 23, 24, 26 and 27. Per-case conditional and unconditional results for the MLP encoder ablation are reported in Table 25. Horizon: 5s5\,s. Setting dz=dxd_z=d_x dz=2dxd_z=2d_x dz=3dxd_z=3d_x MLP ℒxtraj,cL_x^traj,c ℒzL_z ℒxL_x ℒxtrajL_x^traj ℒxtraj,c,ℒxtrajL_x^traj,c,\ L_x^traj ℒx,ℒxtrajL_x,\ L_x^traj ℒxtraj,c,ℒxL_x^traj,c,\ L_x Mean change (%) Diverged(Diverged)Diverged\;(Diverged) Diverged(Diverged)Diverged\;(Diverged) +20.9(+396.2)+20.9\;(+396.2) +3.1(+14.2)+3.1\;(+14.2) +3.7(+1.6)+3.7\;(+1.6) Diverged(Diverged)Diverged\;(Diverged) +0.8(+11.6)+0.8\;(+11.6) −0.4(+4.0)-0.4\;(+4.0) +9.9(+17.4)+9.9\;(+17.4) −1.0(+4.8)-1.0\;(+4.8) +3.2(−1.1)+3.2\;(-1.1) Worse cases (/11) Diverged(Diverged)Diverged\;(Diverged) Diverged(Diverged)Diverged\;(Diverged) 7(5)7\;(5) 7(10)7\;(10) 8(4)8\;(4) Diverged(Diverged)Diverged\;(Diverged) 7(7)7\;(7) 6(6)6\;(6) 11(11)11\;(11) 6(8)6\;(8) 8(4)8\;(4) Table 23: Latent-dimension ablation measured by test-set conditional path distribution loss ℒxtraj,cL_x^traj,c (↓ ). Horizon: 5s5\,s. dz=dxd_z=d_x dz=2dxd_z=2d_x dz=3dxd_z=3d_x dz=4dxd_z=4d_x (default) b-ball (GMM) 0.4690.469±\,±\, 0.01700.0170 0.4440.444±\,±\, 0.02390.0239 0.4650.465±\,±\, 0.03840.0384 0.4320.432±\,±\, 0.009090.00909 b-ball (Uniform) Diverged 0.3810.381±\,±\, 0.05760.0576 0.3430.343±\,±\, 0.01920.0192 0.3380.338±\,±\, 0.01190.0119 Torus (GMM) 0.7710.771±\,±\, 0.1530.153 0.6890.689±\,±\, 0.002930.00293 0.6890.689±\,±\, 0.001800.00180 0.6870.687±\,±\, 0.001090.00109 Torus (Uniform) 0.7050.705±\,±\, 0.004150.00415 0.6740.674±\,±\, 0.004800.00480 0.6710.671±\,±\, 0.002850.00285 0.6770.677±\,±\, 0.005170.00517 Klein (GMM) 0.7070.707±\,±\, 0.005790.00579 0.6930.693±\,±\, 0.005650.00565 0.6880.688±\,±\, 0.0007170.000717 0.6880.688±\,±\, 0.003390.00339 Klein (Uniform) 0.6940.694±\,±\, 0.008230.00823 0.6750.675±\,±\, 0.003880.00388 0.6770.677±\,±\, 0.002250.00225 0.6760.676±\,±\, 0.008320.00832 Klein–Torus 0.7050.705±\,±\, 0.008160.00816 0.6280.628±\,±\, 0.07430.0743 0.5740.574±\,±\, 0.04010.0401 0.5810.581±\,±\, 0.05460.0546 1-ball (GMM) 0.5010.501±\,±\, 0.01150.0115 0.4790.479±\,±\, 0.005660.00566 0.4820.482±\,±\, 0.003280.00328 0.4820.482±\,±\, 0.005780.00578 1-ball (Uniform) 0.4410.441±\,±\, 0.01050.0105 0.4230.423±\,±\, 0.007670.00767 0.4270.427±\,±\, 0.005710.00571 0.4390.439±\,±\, 0.02090.0209 2-balls (GMM) 0.5320.532±\,±\, 0.005390.00539 Diverged 1.711.71±\,±\, 1.711.71 0.5250.525±\,±\, 0.006890.00689 2-balls (Uniform) 0.5150.515±\,±\, 0.01130.0113 0.5160.516±\,±\, 0.03630.0363 0.5050.505±\,±\, 0.01560.0156 0.5090.509±\,±\, 0.01910.0191 Table 24: Latent-dimension ablation measured by test-set unconditional path distribution loss ℒxtrajL_x^traj (↓ ). Horizon: 5s5\,s. dz=dxd_z=d_x dz=2dxd_z=2d_x dz=3dxd_z=3d_x dz=4dxd_z=4d_x (default) b-ball (GMM) 1.631.63±\,±\, 0.4650.465 1.161.16±\,±\, 0.4010.401 1.531.53±\,±\, 0.7970.797 0.8230.823±\,±\, 0.1440.144 b-ball (Uniform) Diverged 1.171.17±\,±\, 0.8160.816 0.6540.654±\,±\, 0.2540.254 0.7090.709±\,±\, 0.1590.159 Torus (GMM) 6.816.81±\,±\, 13.213.2 0.8010.801±\,±\, 0.03200.0320 0.8250.825±\,±\, 0.03280.0328 0.7950.795±\,±\, 0.03150.0315 Torus (Uniform) 0.9620.962±\,±\, 0.09220.0922 0.7690.769±\,±\, 0.06170.0617 0.7610.761±\,±\, 0.03010.0301 0.7940.794±\,±\, 0.06220.0622 Klein (GMM) 1.131.13±\,±\, 0.1990.199 0.8900.890±\,±\, 0.08360.0836 0.8300.830±\,±\, 0.05520.0552 0.8330.833±\,±\, 0.03240.0324 Klein (Uniform) 0.9590.959±\,±\, 0.05240.0524 0.7920.792±\,±\, 0.01850.0185 0.8070.807±\,±\, 0.02450.0245 0.8260.826±\,±\, 0.06960.0696 Klein–Torus 1.241.24±\,±\, 0.1990.199 0.8400.840±\,±\, 0.1200.120 0.7250.725±\,±\, 0.05940.0594 0.7630.763±\,±\, 0.1180.118 1-ball (GMM) 1.271.27±\,±\, 0.1240.124 0.9770.977±\,±\, 0.08150.0815 1.101.10±\,±\, 0.07870.0787 1.051.05±\,±\, 0.1410.141 1-ball (Uniform) 1.011.01±\,±\, 0.1350.135 0.8530.853±\,±\, 0.08730.0873 0.9030.903±\,±\, 0.05320.0532 0.9560.956±\,±\, 0.1850.185 2-balls (GMM) 2.082.08±\,±\, 0.2570.257 Diverged 82.682.6±\,±\, 112112 1.891.89±\,±\, 0.3740.374 2-balls (Uniform) 1.991.99±\,±\, 0.2320.232 4.684.68±\,±\, 5.535.53 2.392.39±\,±\, 0.6580.658 2.142.14±\,±\, 0.6930.693 Table 25: Sampler ablation measured by test-set conditional and unconditional path distribution losses (↓ ). Horizon: 5s5\,s. DDIM MLP Case Conditional Unconditional Conditional Unconditional b-ball (GMM) 0.4320.432±\,±\, 0.009090.00909 0.8230.823±\,±\, 0.1440.144 0.4690.469±\,±\, 0.02100.0210 1.481.48±\,±\, 0.5670.567 b-ball (Uniform) 0.3380.338±\,±\, 0.01190.0119 0.7090.709±\,±\, 0.1590.159 0.3700.370±\,±\, 0.01390.0139 0.9780.978±\,±\, 0.2220.222 Torus (GMM) 0.6870.687±\,±\, 0.001090.00109 0.7950.795±\,±\, 0.03150.0315 0.6890.689±\,±\, 0.001470.00147 0.8150.815±\,±\, 0.03440.0344 Torus (Uniform) 0.6770.677±\,±\, 0.005170.00517 0.7940.794±\,±\, 0.06220.0622 0.6730.673±\,±\, 0.003160.00316 0.8150.815±\,±\, 0.05500.0550 Klein (GMM) 0.6880.688±\,±\, 0.003390.00339 0.8330.833±\,±\, 0.03240.0324 0.6900.690±\,±\, 0.001860.00186 0.8400.840±\,±\, 0.03430.0343 Klein (Uniform) 0.6760.676±\,±\, 0.008320.00832 0.8260.826±\,±\, 0.06960.0696 0.6750.675±\,±\, 0.002350.00235 0.8610.861±\,±\, 0.02460.0246 Klein–Torus 0.5810.581±\,±\, 0.05460.0546 0.7630.763±\,±\, 0.1180.118 0.6420.642±\,±\, 0.06070.0607 0.8860.886±\,±\, 0.1460.146 1-ball (GMM) 0.4820.482±\,±\, 0.005780.00578 1.051.05±\,±\, 0.1410.141 0.5060.506±\,±\, 0.01110.0111 1.181.18±\,±\, 0.09860.0986 1-ball (Uniform) 0.4390.439±\,±\, 0.02090.0209 0.9560.956±\,±\, 0.1850.185 0.4530.453±\,±\, 0.009490.00949 1.011.01±\,±\, 0.08570.0857 2-balls (GMM) 0.5250.525±\,±\, 0.006890.00689 1.891.89±\,±\, 0.3740.374 0.5240.524±\,±\, 0.009220.00922 2.002.00±\,±\, 0.4320.432 2-balls (Uniform) 0.5090.509±\,±\, 0.01910.0191 2.142.14±\,±\, 0.6930.693 0.5000.500±\,±\, 0.009900.00990 1.901.90±\,±\, 0.2600.260 Table 26: Loss-function ablation measured by test-set conditional path distribution loss ℒxtraj,cL_x^traj,c (↓ ). Column labels other than Default model identify the removed objectives (their weights are set to zero). Values are the mean ± sample standard deviation over five training seeds. The lowest mean among finite entries in each row is bold. Diverged denotes a configuration with nonfinite results or either test metric exceeding 10310^3 in any seed. Horizon: 5s5\,s. Case Default model ℒxtraj,cL_x^traj,c ℒzL_z ℒxL_x ℒxtrajL_x^traj ℒxtraj,c,ℒxtrajL_x^traj,c,\ L_x^traj ℒx,ℒxtrajL_x,\ L_x^traj ℒxtraj,c,ℒxL_x^traj,c,\ L_x b-ball (GMM) 0.4320.432±\,±\, 0.009090.00909 0.4460.446±\,±\, 0.007690.00769 0.5120.512±\,±\, 0.04050.0405 0.4560.456±\,±\, 0.01930.0193 0.4420.442±\,±\, 0.01180.0118 0.4540.454±\,±\, 0.01900.0190 0.4440.444±\,±\, 0.01440.0144 0.4450.445±\,±\, 0.006190.00619 b-ball (Uniform) 0.3380.338±\,±\, 0.01190.0119 0.3480.348±\,±\, 0.02040.0204 0.4030.403±\,±\, 0.05240.0524 0.3420.342±\,±\, 0.01670.0167 0.3420.342±\,±\, 0.01130.0113 0.3560.356±\,±\, 0.009490.00949 0.3420.342±\,±\, 0.009110.00911 0.3390.339±\,±\, 0.008320.00832 Torus (GMM) 0.6870.687±\,±\, 0.001090.00109 0.6880.688±\,±\, 0.001110.00111 0.7550.755±\,±\, 0.05500.0550 0.6890.689±\,±\, 0.001230.00123 0.6910.691±\,±\, 0.0009860.000986 0.7090.709±\,±\, 0.0004820.000482 0.6900.690±\,±\, 0.001780.00178 0.6880.688±\,±\, 0.0006960.000696 Torus (Uniform) 0.6770.677±\,±\, 0.005170.00517 0.6730.673±\,±\, 0.001300.00130 0.7480.748±\,±\, 0.04340.0434 0.6700.670±\,±\, 0.002010.00201 0.6710.671±\,±\, 0.002210.00221 0.7080.708±\,±\, 0.0009510.000951 0.6750.675±\,±\, 0.002480.00248 0.6700.670±\,±\, 0.001070.00107 Klein (GMM) 0.6880.688±\,±\, 0.003390.00339 0.6870.687±\,±\, 0.002810.00281 0.8970.897±\,±\, 0.2080.208 0.6890.689±\,±\, 0.002490.00249 0.6900.690±\,±\, 0.003260.00326 0.7070.707±\,±\, 0.0008180.000818 0.6900.690±\,±\, 0.0007680.000768 0.6850.685±\,±\, 0.0007210.000721 Klein (Uniform) 0.6760.676±\,±\, 0.008320.00832 0.6740.674±\,±\, 0.001200.00120 0.7950.795±\,±\, 0.06440.0644 0.6750.675±\,±\, 0.007720.00772 0.6770.677±\,±\, 0.006100.00610 0.7070.707±\,±\, 0.001220.00122 0.6780.678±\,±\, 0.006510.00651 0.6710.671±\,±\, 0.0007040.000704 Klein–Torus 0.5810.581±\,±\, 0.05460.0546 0.5920.592±\,±\, 0.03900.0390 0.8680.868±\,±\, 0.01320.0132 0.5750.575±\,±\, 0.05320.0532 0.5810.581±\,±\, 0.02750.0275 0.7080.708±\,±\, 0.0005330.000533 0.5490.549±\,±\, 0.009640.00964 0.5900.590±\,±\, 0.05150.0515 1-ball (GMM) 0.4820.482±\,±\, 0.005780.00578 0.5130.513±\,±\, 0.01080.0108 0.9870.987±\,±\, 0.6740.674 0.4840.484±\,±\, 0.005400.00540 0.4870.487±\,±\, 0.007300.00730 0.5460.546±\,±\, 0.01380.0138 0.4940.494±\,±\, 0.01160.0116 0.5130.513±\,±\, 0.01120.0112 1-ball (Uniform) 0.4390.439±\,±\, 0.02090.0209 0.4740.474±\,±\, 0.01370.0137 1.081.08±\,±\, 0.7240.724 0.4460.446±\,±\, 0.01560.0156 0.4290.429±\,±\, 0.01130.0113 0.5120.512±\,±\, 0.01260.0126 0.4250.425±\,±\, 0.006600.00660 0.4730.473±\,±\, 0.01650.0165 2-balls (GMM) 0.5250.525±\,±\, 0.006890.00689 0.5680.568±\,±\, 0.003300.00330 Diverged 0.5360.536±\,±\, 0.02430.0243 0.5160.516±\,±\, 0.003920.00392 0.5950.595±\,±\, 0.005030.00503 0.5110.511±\,±\, 0.006100.00610 0.5670.567±\,±\, 0.003370.00337 2-balls (Uniform) 0.5090.509±\,±\, 0.01910.0191 0.5680.568±\,±\, 0.007770.00777 4.304.30±\,±\, 3.283.28 0.5070.507±\,±\, 0.02020.0202 0.4860.486±\,±\, 0.01600.0160 0.6010.601±\,±\, 0.006370.00637 0.4740.474±\,±\, 0.004190.00419 0.5630.563±\,±\, 0.006860.00686 Table 27: Loss-function ablation measured by test-set unconditional path distribution loss ℒxtrajL_x^traj (↓ ). Column labels other than Default model identify the removed objectives (their weights are set to zero). Values are the mean ± sample standard deviation over five training seeds. The lowest mean among finite entries in each row is bold. Diverged denotes a configuration with nonfinite results or either test metric exceeding 10310^3 in any seed. Horizon: 5s5\,s. Case Default model ℒxtraj,cL_x^traj,c ℒzL_z ℒxL_x ℒxtrajL_x^traj ℒxtraj,c,ℒxtrajL_x^traj,c,\ L_x^traj ℒx,ℒxtrajL_x,\ L_x^traj ℒxtraj,c,ℒxL_x^traj,c,\ L_x b-ball (GMM) 0.8230.823±\,±\, 0.1440.144 0.9900.990±\,±\, 0.2190.219 2.592.59±\,±\, 1.061.06 1.291.29±\,±\, 0.1880.188 1.091.09±\,±\, 0.2950.295 1.191.19±\,±\, 0.5380.538 1.181.18±\,±\, 0.3470.347 0.9310.931±\,±\, 0.1910.191 b-ball (Uniform) 0.7090.709±\,±\, 0.1590.159 0.6530.653±\,±\, 0.2070.207 2.052.05±\,±\, 1.901.90 0.6990.699±\,±\, 0.1770.177 0.7380.738±\,±\, 0.1070.107 0.7110.711±\,±\, 0.1170.117 0.7930.793±\,±\, 0.2320.232 0.5930.593±\,±\, 0.1550.155 Torus (GMM) 0.7950.795±\,±\, 0.03150.0315 0.7700.770±\,±\, 0.02160.0216 4.454.45±\,±\, 2.872.87 0.8350.835±\,±\, 0.04480.0448 0.8830.883±\,±\, 0.06340.0634 0.8990.899±\,±\, 0.01770.0177 0.8690.869±\,±\, 0.07690.0769 0.7820.782±\,±\, 0.02210.0221 Torus (Uniform) 0.7940.794±\,±\, 0.06220.0622 0.7320.732±\,±\, 0.006320.00632 4.084.08±\,±\, 2.072.07 0.7890.789±\,±\, 0.06120.0612 0.7780.778±\,±\, 0.03210.0321 0.9240.924±\,±\, 0.02330.0233 0.8260.826±\,±\, 0.05680.0568 0.7260.726±\,±\, 0.005250.00525 Klein (GMM) 0.8330.833±\,±\, 0.03240.0324 0.7710.771±\,±\, 0.01420.0142 12.712.7±\,±\, 13.813.8 0.8280.828±\,±\, 0.05360.0536 0.8440.844±\,±\, 0.06740.0674 0.9000.900±\,±\, 0.02080.0208 0.8980.898±\,±\, 0.07890.0789 0.7800.780±\,±\, 0.01400.0140 Klein (Uniform) 0.8260.826±\,±\, 0.06960.0696 0.7890.789±\,±\, 0.02500.0250 5.785.78±\,±\, 3.493.49 0.8290.829±\,±\, 0.08660.0866 0.8770.877±\,±\, 0.06600.0660 0.9910.991±\,±\, 0.01860.0186 0.8720.872±\,±\, 0.03740.0374 0.7860.786±\,±\, 0.01250.0125 Klein–Torus 0.7630.763±\,±\, 0.1180.118 0.7410.741±\,±\, 0.05990.0599 9.669.66±\,±\, 0.9450.945 0.7470.747±\,±\, 0.1110.111 0.7380.738±\,±\, 0.03860.0386 1.151.15±\,±\, 0.02930.0293 0.6960.696±\,±\, 0.04800.0480 0.7560.756±\,±\, 0.09330.0933 1-ball (GMM) 1.051.05±\,±\, 0.1410.141 1.111.11±\,±\, 0.1220.122 27.927.9±\,±\, 39.239.2 1.091.09±\,±\, 0.05760.0576 1.151.15±\,±\, 0.1690.169 1.091.09±\,±\, 0.1080.108 1.151.15±\,±\, 0.1680.168 1.091.09±\,±\, 0.1350.135 1-ball (Uniform) 0.9560.956±\,±\, 0.1850.185 1.051.05±\,±\, 0.1470.147 35.435.4±\,±\, 43.443.4 0.9760.976±\,±\, 0.1100.110 0.9000.900±\,±\, 0.08760.0876 1.011.01±\,±\, 0.06670.0667 0.9570.957±\,±\, 0.1090.109 0.9860.986±\,±\, 0.09990.0999 2-balls (GMM) 1.891.89±\,±\, 0.3740.374 1.791.79±\,±\, 0.06110.0611 Diverged 2.792.79±\,±\, 1.581.58 1.831.83±\,±\, 0.1290.129 1.961.96±\,±\, 0.2040.204 1.741.74±\,±\, 0.1570.157 1.791.79±\,±\, 0.09940.0994 2-balls (Uniform) 2.142.14±\,±\, 0.6930.693 2.572.57±\,±\, 0.8520.852 415415±\,±\, 370370 2.502.50±\,±\, 1.171.17 2.042.04±\,±\, 0.6930.693 2.652.65±\,±\, 0.8610.861 1.681.68±\,±\, 0.2970.297 2.422.42±\,±\, 0.7730.773 B.5 Visualizations We provide qualitative comparisons on all eleven datasets in Figures 7–17. The rows follow the method names shown in the legends: Proposed denotes our method; CHyLL denotes the deterministic latent ODE framework of Teng et al. (2026b); Latent SDE denotes the variational latent SDE baseline (Li et al., 2020); Latent SDE + x losses denotes the same baseline augmented with ℒxL_x and ℒxtrajL_x^traj; SDE Matching denotes the simulation-free variational baseline (Bartosh et al., 2025); and NSF denotes the conditional stochastic-flow baseline (Kiyohara et al., 2025). For each dataset, all methods are conditioned on the same initial observation from a fixed test trajectory. Blue curves show the ground-truth (GT) trajectory, and orange curves show five predictive rollouts. One rollout is emphasized for readability without selection by prediction error. The five CHyLL predictions coincide because this baseline is deterministic. We use training seed 1101 and test trajectory index 0 throughout. Predictions are generated over the full recorded horizon (approximately five seconds), without subsequent conditioning on GT. All signals are displayed in their original, unnormalized coordinates. These plots assess full-horizon forecasting behavior rather than directly visualizing the short-window errors reported in the tables. Since the systems are stochastic, individual predictive samples are not expected to reproduce the particular GT realization exactly. For two-coordinate systems, the two columns show the state coordinates as functions of time. For the planar bouncing-ball systems, each row additionally includes spatial trajectories: the four time-series panels show (x,y,vx,vy)(x,y,v_x,v_y) for the single ball or the first ball, while the right-hand panels show the spatial paths of all balls. The boundary lines indicate ball-center contact locations, x=±0.45x=± 0.45 and y=0.05y=0.05, accounting for the ball radius r=0.05r=0.05. Time-series limits are initially determined by the GT range with 20% padding on each side; spatial panels initially show the collision region with additional margins. These limits are retained whenever at least one predicted point is visible in the corresponding panel. Only when no predicted point is visible are the limits expanded to include the actual signal range. Thus, predictions may leave the displayed region without being numerically non-finite. The label Non-finite instead indicates that a complete finite rollout could not be obtained; only GT is shown in that row. Figure 7: b-ball (GMM): one-dimensional bouncing ball with GMM-distributed restitution coefficients. The two columns show position and velocity over time. Figure 8: b-ball (Uniform): one-dimensional bouncing ball with uniformly distributed restitution coefficients. The two columns show position and velocity over time. Figure 9: Torus (GMM): coordinate trajectories in the fundamental domain with GMM-distributed reset perturbations. Figure 10: Torus (Uniform): coordinate trajectories in the fundamental domain with uniformly distributed reset perturbations. Figure 11: Klein (GMM): coordinate trajectories in the fundamental domain with GMM-distributed reset perturbations and Klein-bottle boundary gluing. Figure 12: Klein (Uniform): coordinate trajectories in the fundamental domain with uniformly distributed reset perturbations and Klein-bottle boundary gluing. Figure 13: Klein–Torus: coordinate trajectories with random selection between canonical Klein-bottle and torus resets at the top boundary, each with probability 1/21/2. Figure 14: 1-ball (GMM): a single planar bouncing ball with GMM-distributed wall and floor restitution coefficients. The left 2×22× 2 panels show (x,y)(x,y) above (vx,vy)(v_x,v_y); the right panel shows the spatial path with radius-adjusted ball-center contact boundaries. Figure 15: 1-ball (Uniform): a single planar bouncing ball with uniformly distributed wall and floor restitution coefficients. The left panels show the four state components over time; the right panel shows the spatial path and ball-center contact boundaries. Figure 16: 2-balls (GMM): two interacting planar bouncing balls with GMM-distributed wall and floor restitution coefficients. The left panels show the first ball’s four state components; the two right panels show the spatial paths of both balls and their ball-center contact boundaries. Figure 17: 2-balls (Uniform): two interacting planar bouncing balls with uniformly distributed wall and floor restitution coefficients. The left panels show the first ball’s four state components; the two right panels show the spatial paths of both balls and their ball-center contact boundaries.