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D-JEPA: Design-Recoverable JEPA Representation with Swappable Physics Decoders
Nitin Nagesh Kulkarni, Aashwin Anand Mishra, Yin Yu, Peter Lyu
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Abstract
Abstract:Joint-Embedding Predictive Architectures (JEPAs) provide a framework for learning compact representations without directly reconstructing high-dimensional observations. However, in parameterized physical systems, learned representations can entangle geometry with operating conditions and task-specific physical responses, limiting their reuse across prediction tasks. We introduce D-JEPA (Design-recoverable JEPA), a geometry-centric JEPA that computes a compact representation from geometry alone and reuses it across operating conditions and physical response spaces through lightweight physics-specific decoders. An explicit design-recoverability objective encourages the geometry latent to preserve information about the underlying design variables, enabling the representation to support design analysis and optimization. We further identify a case-level collapse failure mode in which target representations become nearly invariant across distinct geometries despite low reconstruction error, and mitigate it using case-level variation constraints and auxiliary target reconstruction. Across four 3D aerodynamic, hydrodynamic, and structural benchmarks, D-JEPA maintains or improves full-field prediction accuracy while achieving near-perfect linear recoverability of design parameters. The frozen geometry representation can be reused at held-out operating conditions and transferred to a structural response task with fewer trainable parameters. Finally, the representation supports differentiable design optimization, with designs validated using high-fidelity CFD, preserving the predicted ranking of candidate designs. These results demonstrate that separating a reusable geometry representation from physics-specific prediction provides a practical representation for scientific surrogate modeling and design.
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- Source: https://arxiv.org/abs/2609.33110v1
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D-JEPA: Design-Recoverable JEPA Representation with Swappable Physics Decoders Nitin Nagesh Kulkarni, Aashwin Anand Mishra, Yin Yu, Peter Lyu Luminary San Mateo, CA, USA Abstract Joint-Embedding Predictive Architectures (JEPAs) provide a framework for learning compact representations without directly reconstructing high-dimensional observations. However, in parameterized physical systems, learned representations can entangle geometry with operating conditions and task-specific physical responses, limiting their reuse across prediction tasks. We introduce D-JEPA (Design-recoverable JEPA), a geometry-centric JEPA that computes a compact representation from geometry alone and reuses it across operating conditions and physical response spaces through lightweight physics-specific decoders. An explicit design-recoverability objective encourages the geometry latent to preserve information about the underlying design variables, enabling the representation to support design analysis and optimization. We further identify a case-level collapse failure mode in which target representations become nearly invariant across distinct geometries despite low reconstruction error, and mitigate it using case-level variation constraints and auxiliary target reconstruction. Across four 3D aerodynamic, hydrodynamic, and structural benchmarks, D-JEPA maintains or improves full-field prediction accuracy while achieving near-perfect linear recoverability of design parameters. The frozen geometry representation can be reused at held-out operating conditions and transferred to a structural response task with fewer trainable parameters. Finally, the representation supports differentiable design optimization, with designs validated using high-fidelity CFD, preserving the predicted ranking of candidate designs. These results demonstrate that separating a reusable geometry representation from physics-specific prediction provides a practical representation for scientific surrogate modeling and design. 1 Introduction Many scientific and engineering problems require repeatedly solving parameterized partial differential equations (PDEs) as geometry, operating conditions, or boundary conditions vary. In aerodynamic design, Aerodynamic Databases (AeroDBs) map flight conditions and vehicle geometries to quantities of interest (QoIs) such as lift, drag, and structural loads, supporting applications from preliminary aircraft design and aeroelasticity to Certification by Analysis (CbA). However, populating an AeroDB requires evaluating a combinatorial space of geometries and operating conditions, making high-fidelity computational fluid dynamics (CFD) prohibitively expensive. The scale of this cost is evident in NASA’s Space Launch System databases, which required thousands of CFD simulations and tens of millions of CPU-core hours (NASA Advanced Supercomputing Division, 2017; NASA Advanced Supercomputing Division, 2018), and the NASA X-57 aerodynamic database, which involved thousands of simulations with meshes containing hundreds of millions of cells (NASA Advanced Supercomputing Division, 2021; Frederick and others, 2025). These computational demands have motivated neural surrogates that amortize simulation cost by learning the underlying input–output mapping (Brunton et al., 2020). While recent surrogates can substantially accelerate CFD prediction, many downstream aerodynamic applications require only a small set of QoIs rather than complete high-dimensional flow fields, motivating reusable geometry representations that directly support physical prediction and downstream design tasks. A large body of recent work has developed neural surrogates for predicting high-dimensional physical fields on complex engineering geometries. Transformer-, mesh- and geometry-aware architectures such as Transolver, DoMINO, GeoTransolver and AB-UPT have demonstrated accurate prediction of physical responses on general engineering domains (Wu et al., 2024a; Ranade et al., 2025; Adams et al., 2025; Alkin et al., 2025). These approaches are particularly valuable when the complete physical field is required. However, aerodynamic design workflows also require repeated evaluation of integrated quantities such as lift, drag and moments across large design spaces. More fundamentally, a surrogate optimized for a particular physical response does not necessarily provide a representation that can be reused when operating conditions or the physical quantity of interest changes. This motivates learning a reusable geometry representation that can support multiple downstream physical prediction and design tasks beyond a single geometry-to-field mapping. Joint-Embedding Predictive Architectures (JEPAs) provide a natural framework for learning such representations. Rather than reconstructing high-dimensional observations directly, JEPA learns predictive relationships between latent representations (Assran et al., 2023; Bardes et al., 2024). Related predictive representation methods have likewise demonstrated that useful information can be retained in compact latent spaces without explicit reconstruction (Grill et al., 2020; Baevski et al., 2022). For engineering surrogate modeling, however, the representation must remain useful as geometry, operating conditions and physical response spaces change. A representation that entangles these factors may be effective for a particular prediction task but difficult to reuse when the operating condition or physical quantity of interest changes. More broadly, self-supervised representation learning has identified representation collapse as a potential failure mode, motivating mechanisms such as redundancy reduction and variance preservation (Zbontar et al., 2021; Bardes et al., 2022; aron et al., 2021; Balestriero and LeCun, 2025). Recent work has begun to explore predictive latent representations for aerodynamic surrogate modeling. AeroJEPA, for example, predicts aerodynamic latent representations from geometry and operating conditions and demonstrates that such latent spaces can support linear probing and design-oriented operations (Giral et al., 2026). However, its representation remains tied to the aerodynamic prediction setting and operating conditions, leaving open a more general question: can geometry itself be represented independently of operating conditions and physical response, while retaining information that is explicitly recoverable for downstream design? We propose D-JEPA (Design-recoverable JEPA), a joint-embedding architecture that separates geometry representation from physics prediction by encoding each geometry into a compact latent space and conditioning lightweight physics decoders on operating conditions. The geometry encoder receives neither operating conditions nor target physics, while an explicit design-recoverability objective encourages the latent to preserve geometric design information. This factorization enables the learned representation to be evaluated for predictive accuracy, design-variable recoverability and reuse across operating conditions and physical response spaces. We demonstrate these capabilities through full-field prediction, condition and cross-physics transfer, as well as differentiable design optimization validated with high-fidelity CFD. Our contributions are as follows: 1. A geometry-centered JEPA formulation for scientific surrogate modeling that explicitly factorizes a reusable geometry representation from operating-condition- and physics-specific prediction. 2. An explicit design-recoverability objective that encourages the shared geometry latent to preserve information about the underlying geometric design variables, making this information directly recoverable through a shallow probe. 3. A reusable geometry–physics factorization in which the geometry encoder is shared across operating conditions and physical response spaces, while lightweight physics-specific decoders can be swapped without retraining the geometry encoder. 4. Empirical evaluation of the resulting representation across parameterized aerodynamic benchmarks, operating-condition reuse, cross-physics transfer and differentiable design optimization with high-fidelity CFD validation. 2 Method In this section, we present the mathematical formulation of D-JEPA, including the geometry representation, latent prediction, physics-specific decoding and design-recoverability objective. 2.1 Problem Formulation We consider the problem of predicting steady physical fields on a set of parameterized geometries under varying operating conditions and multiple sets of governing equations. Let a design be defined by its boundary surface ∂Ω∂ embedded in Ω⊂ℝ3 ^3, represented in a canonical reference coordinate system. The surface is discretized as a point cloud P=xii=1NP=\x_i\_i=1^N, with corresponding surface normals N=nii=1N=\n_i\_i=1^N. Let c∈ℝncondc ^n_cond denote the operating environment, such as Reynolds number, angle of attack, Mach number or structural load cases. We index different physical regimes by s∈s , where each regime corresponds to a particular set of governing equations, such as fluid or structural physics. The corresponding ground-truth field is denoted by Fs:Ω→ℝdsF_s: ^d_s. Each geometry is associated with a design vector θ∈ℝDθ ^D. The design parameters are never provided as inputs to the physics decoder; instead, they are used during training only through an auxiliary design-recoverability objective. The primary objective is to learn a geometry encoder EϑE_ that maps the surface representation to a compact latent representation g=Eϑ(P,N),g∈ℝdg,g=E_ (P,N), g ^d_g, (1) where dg=128d_g=128, together with lightweight physics-specific decoders Dϕs\D_ _s\. Field prediction is given by F^s(q)=Dϕs(g,q,c),g=Eϑ(P,N), F_s(q)=D_ _s(g,q,c), g=E_ (P,N), (2) where q denotes a query location. By structural construction, g does not receive c or s as inputs. Thus, operating-condition and physics-specific information can enter the prediction only through the downstream decoder, while the geometry encoder is shared across conditions and physics regimes. This provides the factorization underlying D-JEPA: geometry is represented once in g, while operating conditions and physical response spaces are handled by the corresponding decoder. 2.2 D-JEPA Framework The D-JEPA framework separates geometry g, operating conditions c and governing physics through the decoder ϕs _s, enabling design recoverability, reuse across operating conditions and transfer across physical response spaces (Figure 1). During training, the geometry encoder, target encoder, latent predictor, physics decoder and design probe are jointly optimized. The target encoder EtargetE_target receives an independently sampled ground-truth field point cloud and produces target tokens ZtZ_t, while the latent predictor fpred,θ(g,c)f_pred,θ(g,c) produces predicted tokens Z^t Z_t. Gradients from ℒlatL_lat propagate through both the predictor and target encoder; no stop-gradient or EMA teacher is used. The target encoder is used only during training and is discarded at inference. The JEPA predictor provides an auxiliary latent prediction signal that shapes the geometry representation, whereas the physics decoder Dϕs(g,q,c)D_ _s(g,q,c) is responsible for direct physical-field prediction. The two pathways therefore serve distinct roles. The JEPA pathway provides representation-level supervision by requiring the geometry-dependent prediction Z^t Z_t to match target tokens derived from the physical field, while the physics decoder maps the shared geometry representation directly to the physical field at arbitrary query locations. Thus, ℒlatL_lat shapes the representation, whereas ℒrecL_rec evaluates its utility for physical-field prediction. Figure 1: Overview of D-JEPA: (a) D-JEPA architecture and training pathways. (b) Design recoverability, condition reuse and cross-physics transfer. 2.3 Coordinate-Aware Geometry Encoding A local point-attention mechanism can capture geometric features such as local curvature but does not by itself provide an explicit representation of global proportions and orientation. We augment the learned point-transformer representation (Wu et al., 2024b) with two parameter-free geometric descriptor blocks computed in the canonical reference frame. The first descriptor is a global descriptor sglobs_glob with 15 dimensions. It contains the per-axis mean and standard deviation, bounding-box extent and the eigenvalues and off-diagonal entries of the 3×33× 3 position covariance matrix. These features provide compact information about the global scale, extent and spatial distribution of the geometry. The second descriptor is a regional descriptor sregs_reg obtained from a 5×5×55× 5× 5 voxel grid. Each voxel contains 13 statistics summarizing the mean and standard deviation of positions, normal statistics and occupancy, resulting in a 16251625-dimensional descriptor. In parallel, a learned point-transformer branch tokenizes the point cloud into M=512M=512 tokens with width d=128d=128, following the hierarchical point-set construction of Qi et al. (2017). Mean and max pooling produce sloc∈ℝ256s_loc ^256. Layer sizes and neighbourhood parameters are given in Appendix B. The local, global and regional descriptors are concatenated, [sloc;sglob;sreg]∈ℝ1896[s_loc;s_glob;s_reg] ^1896, and passed through an MLP to obtain the geometry latent g∈ℝ128g ^128. The resulting latent receives neither operating conditions c nor physics labels s, and the same geometry representation is used by every downstream physics decoder and operating point. 2.4 FiLM-Modulated Implicit Neural Field Decoder For each physics regime s, the decoder maps a query location q to the corresponding local physical field and is modulated by the geometry latent g through Feature-wise Linear Modulation (FiLM) (Perez et al., 2018). This follows the implicit neural representation formulation used for continuous fields (Sitzmann et al., 2020; Mildenhall et al., 2021). The query location is encoded using a Fourier feature embedding with 16 octaves (Tancik et al., 2020), producing a 9999-dimensional representation. The decoder additionally receives the local surface normal n(q)n(q) and operating condition c. The decoder consists of four residual FiLM blocks. For each block l, the latent representation generates the modulation parameters (γl,βl)=split(Alg),( _l, _l)=split(A_lg), (3) and the hidden representation is updated as hl+1=hl+GELU(γl⊙WlLayerNorm(hl)+βl).h_l+1=h_l+GELU ( _l W_lLayerNorm(h_l)+ _l ). (4) The predicted field is obtained from the final hidden representation as F^s(q)=WouthL. F_s(q)=W_outh_L. (5) This formulation allows the same geometry representation to condition different physics-specific decoders while operating conditions enter explicitly through the decoder. Automatic differentiation provides surrogate gradients with respect to the latent and design variables, without requiring a separate PDE-adjoint computation. 2.5 Training Objectives and Design Recoverability During Stage 1, the geometry encoder, latent predictor, primary physics decoder and design probe are jointly optimized. A target encoder EtargetE_target receives an independently sampled ground-truth field point cloud and produces target tokens Zt∈ℝM×d.Z_t ^M× d. (6) The predictor maps (g,c)(g,c) to predicted target tokens Z^t Z_t. An auxiliary target decoder DtD_t reconstructs the physical field from ZtZ_t, while the primary physics decoder DϕsD_ _s predicts the field from the geometry representation g. The composite training objective is ℒ=ℒlat+λrℒrec+λdℒdis+λsℒsig+λcℒsig-case+λvℒvar+λtℒrec-t.L=L_lat+ _rL_rec+ _dL_dis+ _sL_sig+ _cL_sig -case+ _vL_var+ _tL_rec -t. (7) The latent prediction loss is ℒlat=‖Z^t−Zt‖22L_lat= \| Z_t-Z_t \|_2^2 and the primary field reconstruction loss is ℒrec=q[‖Dϕs(g,q,c)−Fs(q)‖22]L_rec=E_q [ \|D_ _s(g,q,c)-F_s(q) \|_2^2 ]. To explicitly encourage the geometry representation to preserve information about the underlying design, we introduce a design-recoverability loss ℒdis=‖probe(g)−θ‖22.L_dis= \|probe(g)-θ \|_2^2. (8) The design parameters are never supplied to the physics decoder. Instead, they supervise the representation through ℒdisL_dis, explicitly encouraging g to retain design-relevant information. Unlike a purely post-hoc probing analysis (Alain and Bengio, 2017), design recoverability is therefore directly optimized during representation learning. 2.6 Preventing Representation Collapse As design recoverability requires the representation to distinguish different geometries, we additionally constrain variation across design cases. Standard SIGReg (Balestriero and LeCun, 2025) operates on the flattened token ensemble and thus does not guarantee diversity across cases. In our setting, representations can exhibit a near-constant token pattern across designs, with between-case standard deviation of approximately 0.00320.0032, while satisfying the token-level SIGReg constraint. We address this failure mode with three complementary terms. First, case-level SIGReg is applied to the mean target representation ℒsig-case=SIGReg(1M∑mZt,m).L_sig -case=SIGReg ( 1M _mZ_t,m ). (9) Second, we explicitly constrain the variance of the representation across design cases: ℒvar=1d∑jReLU(γ−varcase(Z¯t)j+ϵ).L_var= 1d _jReLU (γ- var_case( Z_t)_j+ε ). (10) Finally, we require the target representation to retain sufficient information to reconstruct the physical field: ℒrec-t=q[‖Dt(Zt,q)−Fs(q)‖22].L_rec -t=E_q [ \|D_t(Z_t,q)-F_s(q) \|_2^2 ]. (11) We set λc=λv=λt=1 _c= _v= _t=1. Together, these constraints increase the observed between-case standard deviation to approximately 1.01.0 across the evaluated datasets, preventing the representation from satisfying token-level diversity constraints while remaining effectively constant across geometries. 2.7 Physics Decoder Transfer After Stage 1, the geometry encoder is frozen and a new physics decoder ϕs′ _s is trained for a new physical response space. This follows the parameter-efficient adaptation strategy used elsewhere in representation learning, where a frozen backbone is paired with a small trainable module (Houlsby et al., 2019; Hu et al., 2022). Given g=Eϑ(P,N)g=E_ (P,N), where gradients are not propagated through EϑE_ , the transferred decoder is optimized according to minϕs′[‖Dϕs′(g,q,c)−Fs′(q)‖22]. _ _s E [ \|D_ _s (g,q,c)-F_s (q) \|_2^2 ]. (12) Because the geometry representation is fixed, this experiment tests whether the latent captures geometric information that can be reused by a new physical decoder. Transfer efficiency therefore depends on whether g encodes reusable geometric features rather than task-specific information. 3 Results & Discussion We evaluate the learned geometry representation along three properties: (1) predictive accuracy for full-field and integrated quantities, (2) design information retention and reuse across operating conditions, and (3) transfer to new physical response spaces through swappable decoders. Finally, we evaluate differentiable design optimization with high-fidelity CFD validation. 3.1 Experimental datasets We evaluate D-JEPA on four complementary 3D benchmarks. SHIFT-Wing (Luminary Cloud, 2025b) provides highly parameterized 3D aircraft wing geometries spanning transonic flow regimes, with varied aspect ratios, sweep and spanwise twist distributions. SHIFT-Submarine (Luminary Cloud, 2026) contains parameterized axisymmetric and non-axisymmetric submarine hulls featuring variations in fore-body shape, sail geometry and tail half-angles. SHIFT-SUV (Luminary Cloud, 2025a) covers parameterized automotive geometries that capture complex bluff-body aerodynamic phenomena, including slant angle, backlight geometry and underbody tapering. Finally, AutoHood 3D (Sharma et al., 2025) pairs automotive hood geometries with coupled fluid–structure interaction (FSI) data, in which aerodynamic surface pressure fields act as mechanical loads that generate structural deformation responses. 3.2 Full-field surrogate accuracy We establish that D-JEPA retains strong full-field-prediction accuracy while introducing an explicitly structured geometry representation. Table 1 reports the coefficient of determination (R2R^2) for surface pressure, the three components of wall shear stress (WSSx, WSSy, WSSz) and the integrated force coefficients (CDC_D, CLC_L), for D-JEPA and three baselines: the AeroJEPA predictive-latent model (Giral et al., 2026) and the field-reconstruction surrogates GeoTransolver (Adams et al., 2025) and DoMINO (Ranade et al., 2025). Implementation details for the baselines are given in Appendix F. Across all three aerodynamic benchmarks, D-JEPA maintains or improves field-prediction accuracy relative to AeroJEPA. Surface pressure prediction reaches R2=0.9971R^2=0.9971 on SHIFT-Wing, 0.99670.9967 on SHIFT-Submarine and 0.99370.9937 on SHIFT-SUV. For wall shear stress, D-JEPA improves across every component, with WSSy and WSSz improving on all benchmarks. Relative to the full baseline set, D-JEPA attains the best value in all applicable dataset–channel combinations. D-JEPA also preserves accuracy for integrated aerodynamic observables. On SHIFT-Wing, CDC_D prediction improves from R2=0.973R^2=0.973 with AeroJEPA to 0.9900.990, while CLC_L improves from 0.9520.952 to 0.9700.970. On SHIFT-SUV, D-JEPA matches the best CDC_D prediction at R2=1.000R^2=1.000. These results show that the structured representation preserves the predictive performance required for both spatially resolved fields and integrated engineering quantities. Table 1: Surrogate accuracy across parameterized 3D aerodynamic benchmarks. Full-field R2R^2 is reported for surface pressure and wall-shear-stress components, together with integrated force coefficients CDC_D and CLC_L. Values are mean ± std over three seeds; bold indicates the best value for each dataset and quantity. Model Dataset Pressure WSSx WSSy WSSz CDC_D CLC_L DoMINO SHIFT-Wing 0.9678 ± 0.001 0.8972 ± 0.001 0.9456 ± 0.004 0.9654 ± 0.002 0.940 ± 0.002 0.891 ± 0.001 SHIFT-Submarine 0.9321 ± 0.000 0.9332 ± 0.000 0.9421 ± 0.003 0.9243 ± 0.002 0.973 ± 0.001 – SHIFT-SUV 0.9765 ± 0.000 0.9475 ± 0.000 0.9241 ± 0.002 0.9342 ± 0.002 0.980 ± 0.003 – GeoTransolver SHIFT-Wing 0.9950 ± 0.001 0.9350 ± 0.001 0.9645 ± 0.002 0.9899 ± 0.001 0.960 ± 0.002 0.956 ± 0.001 SHIFT-Submarine 0.9921 ± 0.001 0.9432 ± 0.001 0.9621 ± 0.001 0.9743 ± 0.002 0.990 ± 0.002 – SHIFT-SUV 0.9931 ± 0.001 0.9698 ± 0.001 0.9641 ± 0.001 0.9542 ± 0.001 0.996 ± 0.002 – AeroJEPA SHIFT-Wing 0.9892 ± 0.009 0.9168 ± 0.003 0.9618 ± 0.007 0.9719 ± 0.007 0.973 ± 0.011 0.952 ± 0.046 SHIFT-Submarine 0.9937 ± 0.000 0.9532 ± 0.001 0.9602 ± 0.002 0.9764 ± 0.001 0.987 ± 0.003 – SHIFT-SUV 0.9870 ± 0.001 0.9352 ± 0.001 0.9522 ± 0.001 0.9437 ± 0.001 1.000 ± 0.000 – D-JEPA SHIFT-Wing 0.9971 ± 0.001 0.9376 ± 0.002 0.9849 ± 0.001 0.9900 ± 0.002 0.990 ± 0.002 0.970 ± 0.005 SHIFT-Submarine 0.9967 ± 0.001 0.9691 ± 0.000 0.9766 ± 0.000 0.9809 ± 0.001 0.995 ± 0.002 – SHIFT-SUV 0.9937 ± 0.001 0.9747 ± 0.002 0.9649 ± 0.003 0.9782 ± 0.002 1.000 ± 0.000 – 3.3 Geometry representation and transfer capabilities Having established full-field surrogate accuracy, we evaluate the properties that distinguish D-JEPA from a conventional predictive surrogate: case-level representation diversity, design-variable recoverability, reuse across operating conditions and cross-physics decoder transfer. 3.3.1 Preventing case-level representation collapse Design recoverability requires the latent to distinguish different geometries, but token-level diversity alone does not guarantee variation across design cases. We therefore evaluate the effect of the representation constraints on between-case variation. Table 2 summarizes this ablation. Without case-level constraints, the between-case standard deviation of the target representation remains near zero (0.00020.0002–0.00040.0004) across all three benchmarks. Adding token-level SIGReg alone produces essentially no change, showing that token-level diversity does not prevent case-level collapse. Case-level constraints substantially increase between-case variation. The variance hinge and target-reconstruction terms increase the standard deviation to approximately 1.01.0 across all three geometry families, reaching 1.031.03–1.061.06 on SHIFT-Wing and SHIFT-Submarine and approximately 0.990.99 on SHIFT-SUV. Importantly, these changes occur with only modest changes in forward pressure prediction, indicating that the additional constraints primarily alter the structure of the representation rather than its predictive capacity. The complete diagnostics, including latent and reconstruction losses, are reported in Appendix D. Table 2: Case-level variation in the target representation under progressively stronger representation constraints. Values report stdcase(meanmZt)std_case(mean_mZ_t), measuring variation across geometry cases. Dataset None Token SIGReg Case SIGReg + Variance + Target Recon SHIFT-Wing 0.0003 0.0003 0.6211 1.0346 1.0317 SHIFT-Submarine 0.0004 0.0005 0.7566 1.0633 1.0620 SHIFT-SUV 0.0002 0.0002 0.0003 0.9892 0.9889 3.3.2 Design-variable recoverability A central objective of D-JEPA is to retain information about the underlying geometric design variables in the learned latent. We therefore evaluate whether the design parameters can be recovered using a linear probe from g. Table A.5 in Appendix G reports recovery R2R^2 for every design variable of SHIFT-SUV, SHIFT-Wing and SHIFT-Submarine. D-JEPA achieves near-perfect recoverability across the parameterized shape dimensions, with mean R2R^2 scores of 0.989 on SHIFT-SUV, 0.998 on SHIFT-Wing and 0.998 on SHIFT-Submarine. As a representation-level reference, we apply the identical probe to the AeroJEPA context latent, mean-pooled to the same 128 dimensions as g. The corresponding mean R2R^2 values are 0.683, 0.801 and 0.882, respectively, giving gains of 0.306, 0.197 and 0.116. Thus, D-JEPA retains substantially more linearly recoverable design information across all three geometry families. This structured decodability provides a direct representation-level basis for design navigation and complements the full-field prediction results: the latent is not only useful for predicting physical fields, but also retains explicit information about the geometric variables defining the input shapes. We further perform component ablations to isolate the sources of this recoverability. Here Learned-only denotes the point-transformer branch alone, with the parameter-free global and regional descriptors removed, so that the comparison isolates the contribution of the coordinate-aware descriptors. As shown in Table 3, removing the design loss ℒdisL_dis substantially reduces design recovery across all three datasets, while having a comparatively smaller effect on pressure prediction. Removing the latent prediction loss ℒlatL_lat produces a smaller but consistent reduction in design and pressure R2R^2 across the three datasets. The learned-only encoder retains substantial design information, indicating that recoverability is not solely due to the explicit geometric descriptors. Table 3: Component ablation of D-JEPA across the SHIFT datasets, showing the contributions of latent prediction and design supervision to design recoverability and pressure prediction. Learned-only removes the parameter free global and regional descriptors. Design R2 ↑ Pressure R2 ↑ Dataset Full −ℒlat-L_lat Learned-only −ℒdis-L_dis Full −ℒlat-L_lat Learned-only −ℒdis-L_dis SHIFT-Wing 0.998 0.993 0.976 0.722 0.9971 0.9912 0.9885 0.9823 SHIFT-Submarine 0.998 0.982 0.757 0.564 0.9967 0.9896 0.9852 0.9872 SHIFT-SUV 0.989 0.984 0.976 0.622 0.9937 0.9851 0.9801 0.9821 3.3.3 Operating-condition sweeps (same physics, new conditions) As the geometry encoder EϑE_ receives neither the condition vector c nor the physics label s, the latent representation g can be computed once per geometry and reused across operating conditions. We evaluate this capability on SHIFT-Wing across a transonic Mach sweep (Mach∈[0.50,0.90]Mach∈[0.50,0.90]). Ten Mach values are used during training, while Mach =0.7750=0.7750 and Mach =0.8125=0.8125 are held out entirely. Importantly, both held-out values lie within the trained Mach range and therefore test interpolation across operating conditions rather than extrapolation. Table 4 reports pressure prediction R2R^2 across all trained and held-out conditions. The frozen geometry representation maintains R2≥0.9902R^2≥ 0.9902 across all evaluated conditions, including R2=0.9922R^2=0.9922 and 0.99440.9944 at the two held-out Mach numbers. A single geometry representation therefore remains reusable across the continuous operating sweep without additional geometry-encoder training. Table 4: SHIFT-Wing pressure prediction accuracy (R2R^2) across trained and held-out Mach numbers using a single frozen geometry representation g. The held-out Mach values are interpolation points within the trained Mach range. Bold values denote held-out conditions. Trained operating conditions Held-out Mach 0.5000 0.6000 0.7000 0.7500 0.8000 0.8250 0.8375 0.8500 0.8750 0.9000 0.7750 0.8125 Pressure R2R^2 0.9902 0.9935 0.9948 0.9944 0.9941 0.9956 0.9960 0.9967 0.9952 0.9939 0.9922 0.9944 Std 0.001 0.002 0.001 0.003 0.002 0.001 0.001 0.002 0.001 0.002 0.001 0.002 3.3.4 Swappable physics decoders (cross-physics transfer) We next test whether the same geometry representation can be reused when the target physical response changes. We evaluate this capability on AutoHood 3D by transferring a geometry encoder pretrained on aerodynamic pressure fields to structural displacement prediction. In the D-JEPA transfer setting, the pretrained aerodynamic geometry encoder is completely frozen and only a lightweight structural FiLM decoder is trained, requiring 0.56 M trainable parameters. In contrast, the from-scratch baseline trains the full geometry encoder and structural decoder end-to-end, requiring 2.30 M trainable parameters. Table 5 tracks structural prediction R2R^2 across training budgets. The swappable-decoder formulation reduces trainable parameters by 4.1×4.1× and provides substantially faster adaptation at low training budgets. After 10 epochs, the transferred representation reaches R2=0.3316R^2=0.3316, compared with 0.03960.0396 from scratch. At 100 epochs, it reaches 0.89010.8901 versus 0.87400.8740, and the transferred model remains ahead through 300 epochs, by which point the two approaches are within 0.0070.007 of each other. The benefit demonstrated here is therefore parameter-efficient and rapid adaptation: the aerodynamic geometry representation can be reused for this CFD-to-structural transfer while training only a lightweight physics-specific decoder. Table 5: Structural displacement prediction accuracy (R2R^2) on AutoHood 3D comparing frozen geometry encoder transfer versus training from scratch. Training epoch budget Frozen geometry encoder (0.56 M params) From scratch (2.30 M params) 10 0.3316 ± 0.020 0.0396 ± 0.007 100 0.8901 ± 0.013 0.8740 ± 0.034 200 0.9327 ± 0.002 0.9297 ± 0.012 300 0.9385 ± 0.001 0.9313 ± 0.002 3.4 Design optimization & CFD validation Finally, we evaluate the representation in a downstream engineering workflow using gradient-based design optimization, with selected designs validated against high-fidelity CFD. The differentiable optimization path and design-space constraints are detailed in Appendix H. To assess whether surrogate-guided optimization produces physically consistent design rankings, we perform direct high-fidelity CFD validation for SHIFT-Submarine and SHIFT-Wing. Equivalent high-fidelity CFD validation for SHIFT-SUV was not possible because the corresponding high-fidelity CFD setup and sufficient geometry and operating-condition information were not available. For each benchmark, we evaluate four candidates selected from the optimization trajectories. OPT-τ is the predicted best design within the nominal design radius τ, representing a “safe” optimum. OPT-Wide is the predicted best design with the radius relaxed to 1.8τ1.8τ, explicitly testing extrapolation beyond the nominal region. ANTI is the predicted worst design within τ and serves as a negative control, while MID is a held-out design with a mid-range predicted objective and provides a buildable sanity check. For SHIFT-Submarine, CFD is evaluated at ρ=998.2kg/m3ρ=998.2\,kg/m^3, V=5.0m/sV=5.0\,m/s, q=12,478Paq=12,478\,Pa and Aref=19.27m2A_ref=19.27\,m^2. Predicted drag agrees with CFD within 0.60.6–2.0%2.0\%. For SHIFT-Wing, the relative error in integrated lift ranges from 0.720.72–10.74%10.74\%. Despite these differences in absolute error, the surrogate exactly preserves the CFD ranking of all four candidates in both benchmarks, with rankings 1→11→ 1, 2→22→ 2, 3→33→ 3 and 4→44→ 4, as reported in Table 6. These results provide a direct high-fidelity check that the surrogate preserves the relative ordering of optimized and control designs, including the extrapolative OPT-Wide case. Table 6: High-fidelity CFD validation of surrogate-optimized designs. Integrated drag is reported for SHIFT-Submarine and integrated lift for SHIFT-Wing. Predicted and CFD rankings agree for all evaluated designs. Benchmark Design candidate Predicted [N] CFD [N] Relative error Predicted → CFD rank SHIFT-Submarine (drag) OPT-wide 645.8 637.69 1.3% 1 → 1 OPT-tau 677.0 688.36 1.7% 2 → 2 MID 718.6 723.08 0.6% 3 → 3 ANTI 758.7 773.75 2.0% 4 → 4 SHIFT-Wing (lift) OPT-Wide 3,426,080 3,193,866 7.27% 1 → 1 OPT-Tau 3,108,536 3,086,237 0.72% 2 → 2 MID 2,008,248 2,249,932 10.74% 3 → 3 ANTI 1,434,578 1,386,571 3.46% 4 → 4 The experiments show that D-JEPA provides a reusable geometry representation for accurate simulation, transfer and design optimization, improving full-field prediction over the evaluated field-reconstruction and predictive-latent baselines while maintaining strong integral accuracy and enabling reuse across design spaces, operating conditions and downstream physics without retraining the geometry encoder. Design-variable recovery further provides a differentiable path from the physical objective back to the design parameters, which supports gradient-based design exploration. However, the evaluation remains limited to a small set of aerodynamic and structural benchmarks, with physics transfer demonstrated only in the CFD-to-structural setting and optimization restricted to differentiable surrogates with predefined objectives; broader multi-physics validation, diverse geometries and constrained real-world design problems remain important directions for future work. 4 Conclusions We present D-JEPA, a geometry-centric representation learning framework designed to separate reusable geometric information from physics- and condition-specific prediction. Across three challenging parameterized three-dimensional aerodynamic benchmarks, D-JEPA maintains or improves full-field surrogate accuracy relative to the evaluated baselines while maintaining high accuracy for integral aerodynamic quantities. 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It is intended to provide sufficient implementation and evaluation detail to reproduce the experiments reported in the main paper. Appendix A Dataset Specifications and Preprocessing Pipelines We evaluate D-JEPA across four benchmark datasets spanning parameterized three-dimensional physical systems: SHIFT-Wing (Luminary Cloud, 2025b), SHIFT-Submarine (Luminary Cloud, 2026), SHIFT-SUV (Luminary Cloud, 2025a) and AutoHood 3D (Sharma et al., 2025). Each case is represented by an unstructured spatial point cloud P with associated outward surface normals N, solver-generated physical fields, underlying geometric design parameters and external operating conditions when applicable. Table A.1 summarizes the benchmark characteristics. Table A.1: Benchmark dataset properties, parameter dimensions, operating conditions and surface targets. A dash indicates that the quantity is not applicable. Dataset Cases Design parameters Operating conditions Surface fields SHIFT-Wing 1,966 7 angle of attack, Mach pressure, WSSx, WSSy, WSSz SHIFT-Submarine 4,000 7 fixed pressure, WSSx, WSSy, WSSz SHIFT-SUV 1,996 7 fixed pressure, WSSx, WSSy, WSSz AutoHood 3D 12,000 – load case pressure, displacement SHIFT-Wing is parameterized by seven geometric variables comprising aspect ratio, quarter-chord sweep angle Λc/4 _c/4 and five spanwise twist variables. SHIFT-Submarine contains seven parameters controlling cross-sectional geometry, fore-body shape and length, mid-body length, aft-tail half-angle, sail-length ratio and strake-length ratio. SHIFT-SUV contains seven parameters describing hood height, hood-base height, windshield inclination, total vehicle height, backlight angle, rear tapering ratio and front-rear planform curvature. The explicit design parameters are not provided to the physics decoder. Instead, they are used only by the auxiliary design-recoverability objective during training. At inference, the decoder receives the learned geometry representation g, query location q, local surface normal n(q)n(q) and operating condition vector c. This separation allows the design variables to be evaluated as information retained by the geometry representation rather than as direct decoder inputs. AutoHood 3D provides a cross-physics evaluation in which the same geometric representation can support different physical response spaces. The geometry is associated with an aerodynamic pressure field and a structural displacement response. D-JEPA therefore uses the same geometry representation while allowing physics-specific decoders to predict the corresponding outputs. For SHIFT-Submarine and SHIFT-SUV, the operating condition vector is empty because the evaluated flow conditions are fixed. Operating-condition conditioning is explicitly evaluated on SHIFT-Wing through variation in Mach number and angle of attack. Appendix B Detailed Model Architecture B.1 Geometry encoder The geometry encoder EϑE_ maps an unstructured point cloud and its surface normals to a compact geometry representation, as given in Eq. (A.1): g=Eϑ(P,N)∈ℝ128.g=E_ (P,N) ^128. (A.1) The encoder first processes the input geometry into M=512M=512 context tokens with feature dimension d=128d=128. Local spatial neighborhoods use kaggregate=16k_aggregate=16 nearest neighbors for message passing and kattn=16k_attn=16 neighbors for local self-attention. The backbone contains six transformer blocks with eight attention heads and an MLP expansion factor of four. The resulting token matrix Z∈ℝ512×128Z ^512× 128 is summarized using channel-wise mean and maximum pooling, as given in Eq. (A.2): sloc=[meanmZm;maxmZm]∈ℝ256.s_loc= [ _mZ_m\,;\, _mZ_m ] ^256. (A.2) The local representation is augmented with non-parametric global and regional descriptors to provide explicit information about large-scale geometry. B.2 Global and regional geometric descriptors The global descriptor sglobs_glob contains 15 geometric statistics computed directly from the spatial coordinates. These include per-axis coordinate means and standard deviations, oriented bounding-box dimensions, eigenvalues of the spatial covariance matrix and off-diagonal covariance terms. The regional descriptor sregs_reg discretizes the canonical bounding domain into a 5×5×55× 5× 5 voxel grid. For each voxel, local coordinate statistics, surface-normal statistics and point-density information are computed. The resulting descriptor contains 125×13=1,625125× 13=1,625 features. The three representations are concatenated as in Eq. (A.3): sfused=[sloc;sglob;sreg]∈ℝ1896,s_fused=[s_loc\,;\,s_glob\,;\,s_reg] ^1896, (A.3) and passed through an MLP to produce the final 128-dimensional geometry representation g. Neither the operating condition vector c nor the physics identifier s is provided to the geometry encoder. Thus, the geometry representation is computed solely from geometric information and can subsequently be reused when the operating condition or physical response space changes. B.3 JEPA representation-learning pathway D-JEPA uses latent prediction as an auxiliary representation-learning objective. During training, a context representation is used to predict target latent tokens through the JEPA predictor. The resulting latent prediction objective encourages the geometry representation to retain information useful for predicting the target representation. The JEPA predictor is not used as the physics-field decoder. Instead, the learned geometry representation g is passed directly to the physics-specific decoder for downstream field prediction. This separation allows the JEPA pathway to shape the geometry representation while keeping the physical output mapping independently swappable. B.4 FiLM-modulated physics decoder For a physical response space s, the decoder is written as in Eq. (A.4): F^s(q)=Dϕs(g,q,c), F_s(q)=D_ _s(g,q,c), (A.4) where g is the geometry representation, q is the query location, n(q)n(q) is the local surface normal and c denotes the operating condition. Query coordinates are encoded using Fourier features across 16 frequency octaves (Tancik et al., 2020), producing a 99-dimensional positional representation. The decoder contains four residual FiLM blocks with hidden width 256. The geometry representation modulates the query pathway through learned affine scale and shift parameters rather than being treated only as a concatenated input (Perez et al., 2018). This factorization separates geometry representation learning from the mapping from geometry and operating conditions to a particular physical response space. Consequently, a new decoder can be trained while keeping the geometry encoder fixed. For the aerodynamic benchmarks, the decoder predicts four surface quantities: pressure and the three components of wall shear stress (WSSx, WSSy and WSSz). Appendix C Training Objectives and Optimization C.1 Composite training objective D-JEPA is trained end-to-end using a composite objective that combines latent prediction, physical reconstruction, design-variable recovery, representation regularization and target reconstruction, as given in Eq. (A.5): ℒ=ℒlat+λrℒrec+λdℒdis+λsℒsig+λcℒsig-case+λvℒvar+λtℒrec-t.L=L_lat+ _rL_rec+ _dL_dis+ _sL_sig+ _cL_sig -case+ _vL_var+ _tL_rec -t. (A.5) The latent prediction term ℒlatL_lat aligns predicted latent representations with target-encoder representations. The physical reconstruction loss ℒrecL_rec trains the physics decoder to predict the target physical field. The design-recoverability loss ℒdisL_dis trains a linear probe to recover the known design variables from the geometry representation. The token-level SIGReg term ℒsigL_sig (Balestriero and LeCun, 2025) encourages diversity within the target-token representation. Because token-level diversity does not by itself guarantee variation between different geometry cases, D-JEPA also evaluates case-level regularization. The case-level SIGReg term ℒsig-caseL_sig -case and variance constraint ℒvarL_var encourage variation across geometry cases, while the target-reconstruction term ℒrec-tL_rec -t provides an additional connection between target representations and local physical coordinates. The resulting objective is designed to jointly maintain predictive accuracy, design information and non-collapsed geometry representations. The corresponding weights are listed in Table A.2. Table A.2: Hyperparameter weights used for the composite D-JEPA objective. Loss term Latent Recon. Design Token Case Var. Tgt. rec. (ℒlatL_lat) (λrℒrec _rL_rec) (λdℒdis _dL_dis) (λsℒsig _sL_sig) (λcℒsig-case _cL_sig -case) (λvℒvar _vL_var) (λtℒrec-t _tL_rec -t) Weight 1.0 1.0 1.0 0.01 1.0 1.0 1.0 C.2 Optimization schedule AeroJEPA and D-JEPA use matched optimization settings unless otherwise specified. The primary training configuration is summarized in Table A.3. The batch size is 16 for SHIFT-Wing, SHIFT-SUV and SHIFT-Submarine, selected to accommodate the memory requirements of the underlying solver meshes. Table A.3: Optimization configuration and training hyperparameters. Setting Value Optimizer AdamW (Loshchilov and Hutter, 2019) Base LR 1×10−31× 10^-3 Weight decay 1×10−31× 10^-3 LR schedule Cosine decay Gradient clipping 10.0 Epochs 400 Precision FP32 Seeds 42, 43, 44 Appendix D Representation-Collapse Diagnostics A central requirement of the geometry representation is that different geometry cases should provide sufficiently distinct learning signals to avoid case-level collapse. We therefore measure between-case variation in the target representation using Eq. (A.6): σcase=stdcase(meanmZt), _case= _case ( _mZ_t ), (A.6) where ZtZ_t denotes the target token representation and the mean is taken over tokens before computing the standard deviation across geometry cases. This diagnostic distinguishes case-level variation from token-level diversity: a target representation may contain diverse tokens within an individual geometry while remaining nearly identical across different geometries. Table A.4 extends the summary in Table 2 with the latent and reconstruction losses and the corresponding downstream metrics. Two observations are relevant beyond the between-case standard deviation itself. First, the latent loss LlatL_lat is smallest in exactly the configurations where σcase _case is near zero, so it cannot be used on its own as a training diagnostic. Second, the downstream metrics vary only modestly across configurations, because LrecL_rec and the design probe act on the geometry representation g while the collapse is measured in the target tokens ZtZ_t. Case-level variation therefore has to be monitored directly. Table A.4: Complete representation-collapse diagnostics. The case-level standard deviation measures between-case variation, while LlatL_lat and LrecL_rec report latent prediction and physical reconstruction losses. Pressure R2R^2 and design-recovery R2R^2 quantify downstream performance. Dataset Configuration σcase _case LlatL_lat LrecL_rec Pressure R2R^2 Design R2R^2 SHIFT-Wing None 0.0003 0.0000 0.0354 0.9906 0.992 Token SIGReg 0.0003 0.0000 0.0352 0.9906 0.992 Case SIGReg 0.6211 0.0403 0.0373 0.9893 0.987 + Variance 1.0346 0.0498 0.0381 0.9886 0.986 + Target Recon. 1.0317 0.0447 0.0379 0.9971 0.998 SHIFT-Submarine None 0.0004 0.0000 0.0348 0.9896 0.982 Token SIGReg 0.0005 0.0000 0.0352 0.9897 0.982 Case SIGReg 0.7566 0.0221 0.0361 0.9892 0.980 + Variance 1.0633 0.0221 0.0360 0.9894 0.982 + Target Recon. 1.0620 0.0192 0.0362 0.9967 0.998 SHIFT-SUV None 0.0002 0.0000 0.0537 0.9726 0.975 Token SIGReg 0.0002 0.0000 0.0534 0.9729 0.979 Case SIGReg 0.0003 0.0000 0.0529 0.9734 0.982 + Variance 0.9892 0.1493 0.0547 0.9718 0.967 + Target Recon. 0.9889 0.1526 0.0551 0.9937 0.989 These results motivate the use of case-level constraints in the final D-JEPA training objective. They also illustrate why token-level diversity alone is insufficient as a diagnostic for a geometry representation: variation must be evaluated across geometry cases. Appendix E Evaluation Protocols All benchmark evaluations use fixed 10% validation partitions generated with constant random seeding. Validation metrics are evaluated on held-out geometry cases that are not used for model training. Field prediction accuracy is quantified using the coefficient of determination R2R^2, computed independently for each physical output variable and averaged across held-out cases. Reported uncertainty values are standard deviations across three independent training seeds. Integrated force coefficients, including drag CDC_D and lift CLC_L, are computed by direct integration of predicted pressure and wall-shear-stress fields over the complete solver surface. No secondary regression head is used to predict the integrated quantities. Design-variable recoverability is evaluated using a standardized, five-fold cross-validated ridge-regression probe applied to frozen 128-dimensional geometry representations g (Alain and Bengio, 2017). Out-of-fold R2R^2 values are reported to avoid evaluating the probe on the same samples used for fitting. Qualitative field comparisons are performed on dense, high-resolution solver meshes rather than only on the sparse point samples used during training. Representative median and worst-case held-out examples are shown in Figures A.1, A.2 and A.3. Figure A.1: Qualitative surface pressure prediction for SHIFT-SUV. Predicted pressure and absolute point-wise error are shown for representative median and worst-case held-out geometries. Figure A.2: Qualitative surface pressure prediction for SHIFT-Submarine. Predicted pressure and absolute point-wise error are shown for representative median and worst-case held-out geometries. Figure A.3: Qualitative surface pressure prediction for SHIFT-Wing. Predicted pressure and absolute point-wise error are shown for representative median and worst-case held-out geometries. Appendix F Baseline Models and Comparative Benchmarks F.1 AeroJEPA baseline AeroJEPA (Giral et al., 2026) is used as the primary representation-learning baseline. The comparison retains the same geometry backbone components, point-sampling procedure, optimizer settings and primary reconstruction objective where applicable. This controls for differences in model capacity and focuses the comparison on the geometry representation and additional D-JEPA training objectives. For design recoverability comparison, the pooled 128-dimensional AeroJEPA context representation is evaluated using the same linear probing procedure applied to D-JEPA. F.2 GeoTransolver and DoMINO baselines GeoTransolver (Adams et al., 2025) and DoMINO (Ranade et al., 2025) are evaluated using their official implementations on the same dataset splits. GeoTransolver uses 20 transformer layers, hidden width 256, eight attention heads, 128 slices and a 60,000-point surface representation under the corresponding optimization settings. All baseline metrics are computed on held-out validation cases. The resulting field-prediction and integrated-force comparisons are reported in Table 1 of the main text. Appendix G Supplemental Experimental Results G.1 Design-variable recoverability A central property of D-JEPA is that the geometry representation retains information about the underlying design variables. We evaluate this property independently of the physics decoder by freezing the learned 128-dimensional geometry representation and fitting a linear ridge probe. Table A.5 reports the recovery accuracy for every design variable. Mean R2R^2 values are 0.989 for SHIFT-SUV, 0.998 for SHIFT-Wing and 0.998 for SHIFT-Submarine. Table A.5: Linear-probe recovery accuracy (R2R^2) for individual geometric design parameters from the frozen 128-dimensional geometry representation g. SHIFT-SUV SHIFT-Wing SHIFT-Submarine Parameter R2R^2 Parameter R2R^2 Parameter R2R^2 hood_height 0.995 AR (aspect ratio) 0.999 cross_section_shape 1.000 hood_base_height 0.987 Lambda_c_4 (quarter-chord sweep) 0.998 fore_body_shape 0.996 windshield_angle 0.986 twist_20 0.997 fore_length 0.999 vehicle_height 0.991 twist_k 0.997 mid_length 0.999 backlight_angle 0.990 twist_60 0.999 aft_tail_half_angle 0.997 r_tapering 0.988 twist_80 0.999 sail_length_ratio 0.997 fr_plainview 0.986 twist_t 0.998 strake_length_ratio 0.996 Mean 0.989 Mean 0.998 Mean 0.998 Individual design-parameter R2R^2 values range from 0.986 to 0.995 for SHIFT-SUV, 0.997 to 0.999 for SHIFT-Wing and 0.996 to 1.000 for SHIFT-Submarine. These results indicate that the learned geometry representation retains substantial linearly recoverable information about the underlying parameterization. Figure A.4 shows the corresponding per-parameter probe correlations. Figure A.4: Design-variable probe recoverability across benchmark datasets. True design parameters are compared with values recovered from the frozen geometry representation using linear probes. G.2 Continuous operating-condition generalization Because the geometry representation does not receive the operating condition vector, the same geometry representation can be reused across different flow conditions while the decoder conditions the field prediction on c. On SHIFT-Wing, the evaluated Mach range is 0.50–0.90. Ten Mach states are used during training, while Mach 0.7750 and Mach 0.8125 are held out entirely. Pressure prediction remains above R2=0.990R^2=0.990 across the evaluated conditions, with R2=0.9922R^2=0.9922 and R2=0.9944R^2=0.9944 at the two held-out Mach numbers. Figure A.5 shows the accuracy across the full sweep. Figure A.5: Operating-condition generalization across Mach numbers on SHIFT-Wing. Surface pressure R2R^2 is evaluated across trained and held-out Mach conditions using the same geometry representation. G.3 Cross-physics decoder transfer To evaluate reuse across physical response spaces, a pre-trained aerodynamic geometry encoder is frozen and transferred to AutoHood 3D structural displacement prediction, following the frozen-backbone strategy common in parameter-efficient adaptation (Houlsby et al., 2019; Hu et al., 2022). The geometry encoder and its projection layers remain fixed, while only the new physics decoder is trained. The transfer configuration contains 0.56 M trainable decoder parameters, compared with 2.30 M for a model trained entirely from scratch, a 4.1×4.1× reduction. Per-epoch results are reported in Table 5 of the main text. Appendix H Differentiable Design Optimization D-JEPA supports differentiable design optimization because gradients can be propagated from a design objective through the physics decoder and learned geometry representation back to the underlying design variables. Specifically, for design variables θ, the differentiable path is θ→G(θ)→(P,N)→Eϑ(P,N)=g→Dϕs(g,q,c)→F^→J(F^),θ→ G(θ)→(P,N)→ E_ (P,N)=g→ D_ _s(g,q,c)→ F→ J( F), where G(θ)G(θ) generates the corresponding geometry and J denotes the design objective computed from the predicted physical field. We evaluate this capability as a downstream application of the learned surrogate rather than as a separate representation-learning objective. Gradients are computed using automatic differentiation through the geometry representation and physics decoder. For SHIFT-Wing, SHIFT-SUV and SHIFT-Submarine, optimization trajectories show strong alignment between the optimized objective and latent search directions, with objective correlation values of approximately R2=0.99R^2=0.99, 0.990.99 and 0.980.98, respectively. To keep optimization within the learned design space, design-variable bounds are imposed through linear design probes together with a k-nearest-neighbor trust-region constraint. The candidate designs produced by this procedure are the ones validated against high-fidelity CFD in Table 6. Appendix I Computational Hardware and Runtime Requirements The primary hardware configurations and approximate wall-clock runtimes are summarized in Table A.6. Runtime values are provided to contextualize the computational requirements of the reported experiments. For decoder-transfer experiments, freezing the geometry encoder reduces the number of trainable parameters and eliminates backward-gradient storage through the frozen representation. A forward pass through the frozen encoder is still required. Consequently, transfer efficiency arises from both reduced optimizer memory requirements and the removal of backward computation through the geometry encoder. Table A.6: Primary computational hardware configurations and approximate wall-clock runtimes across pipeline stages. Execution pipeline stage Compute hardware Wall-clock time D-JEPA/AeroJEPA training (400 epochs) 1 × NVIDIA T4 (16 GB) 4–8 hours / dataset Decoder transfer fine-tuning (400 epochs) 1 × NVIDIA T4 (16 GB) ∼ 2 hours GeoTransolver baseline (500 epochs) 8 × NVIDIA H100 (80 GB) ∼ 12 hours / dataset Mach-condition fine-tuning (200 epochs) 1 × NVIDIA T4 (16 GB) ∼ 3 hours The reported runtimes depend on dataset size, solver mesh resolution, batching and hardware utilization and should therefore be interpreted as approximate execution measurements rather than hardware-independent complexity estimates. Appendix J Limitations The current evaluation is limited to a small set of parameterized engineering datasets, primarily covering aerodynamic and structural response fields, with cross-physics transfer demonstrated in a single CFD-to-structural setting. Although the results show strong design-variable recoverability, the degree of recoverability varies across geometry families, suggesting that some geometric degrees of freedom may be less explicitly represented in the learned latent space. The operating-condition transfer experiments are also evaluated over a limited range of conditions and do not establish robustness to substantially different flow regimes or boundary conditions. Similarly, the optimization experiments use differentiable surrogate models and predefined objectives within the evaluated design spaces, whereas practical engineering problems may involve constraints, discrete variables, noisy objectives and stronger out-of-distribution conditions. Broader multi-physics benchmarks, more diverse geometry families and constrained real-world optimization therefore remain important directions for future evaluation.