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Conservation laws determine what physical learning remembers
Bijaya Dangol
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
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Summary
This paper analyzes how conservation laws in physical learning rules (Equilibrium Propagation, Coupled Learning, Adjoint Coupled Learning) determine the inductive bias, initialization memory, and generalization of resistive networks. It proves that EP and CL conserve conductance mass and are trajectory-equivalent for single outputs, while AL dissipates mass. In linear circuits, initialization scale is inert, but in nonlinear circuits (e.g., with diodes), conservative rules permanently retain initialization memory, whereas the dissipative AL rule partially erases it. At matched training loss, dissipative rules often generalize worse than conservative ones.
Entities (6)
Relation Signals (7)
Coupled Learning → conserves → Conductance Mass
confidence 95% · In the small-nudge limit EP and CL exactly conserve the conductance mass K
Equilibrium Propagation → conserves → Conductance Mass
confidence 95% · In the small-nudge limit EP and CL exactly conserve the conductance mass K
Adjoint Coupled Learning → dissipates → Conductance Mass
confidence 95% · We prove that AL does not conserve the mass but dissipates it at exactly twice its own loss.
Conservation → determines → Inductive Bias
confidence 90% · We show that conservation also governs the inductive bias of these rules.
Equilibrium Propagation → trajectoryequivalentto → Coupled Learning
confidence 90% · For a single output we prove that EP and CL are trajectory equivalent
Dissipation → causes → Partial Erasure of Memory
confidence 85% · the conservative rules retain this memory permanently while the dissipative rule partially erases it.
Dissipation → correlateswith → Generalization Penalty
confidence 85% · the penalty correlates with the mass dissipated en route
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Abstract
Abstract:Physical learning rules such as equilibrium propagation (EP), coupled learning (CL), and adjoint coupled learning (AL) train resistive networks through local measurements. In the small-nudge limit EP and CL exactly conserve the conductance mass K = (1/2) sum_e kappa_e^2, a property that stabilizes training. We show that conservation also governs the inductive bias of these rules. For a single output we prove that EP and CL are trajectory equivalent, so single-output experiments cannot distinguish what the two rules learn. We prove that AL does not conserve the mass but dissipates it at exactly twice its own loss. In linear circuits we prove that the conserved mass has no functional consequence: all three vector fields are homogeneous in the conductances, so the selected solution is independent of the initialization scale. Fixed nonlinear elements break this protection. In diode circuits the learned input-output function depends on the initialization scale by up to about forty percent, an effect absent in linear controls, and the conservative rules retain this memory permanently while the dissipative rule partially erases it. At matched training loss the dissipative rule typically generalizes worse than the conservative rules, although it reaches low training loss faster; the penalty correlates with the mass dissipated en route and fades in larger circuits, where little mass is lost. The conservation structure of a local learning rule thus sets its initialization memory, its training speed, and, where dissipation is appreciable, its generalization; it should be treated as a design parameter of physical learning machines.
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- Source: https://arxiv.org/abs/2608.00097v1
- Canonical: https://arxiv.org/abs/2608.00097v1
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Conservation laws determine what physical learning remembers Bijaya Dangol Independent researcher Abstract Physical learning rules such as equilibrium propagation (EP), coupled learning (CL), and adjoint coupled learning (AL) train resistive networks through local measurements. In the small-nudge limit EP and CL exactly conserve the conductance mass K=12∑eκe2K= 12 _e _e^2, a property that stabilizes training. We show that conservation also governs the inductive bias of these rules. For a single output we prove that EP and CL are trajectory equivalent, so single-output experiments cannot distinguish what the two rules learn. We prove that AL does not conserve the mass but dissipates it at exactly twice its own loss. In linear circuits we prove that the conserved mass has no functional consequence: all three vector fields are homogeneous in the conductances, so the selected solution is independent of the initialization scale. Fixed nonlinear elements break this protection. In diode circuits the learned input-output function depends on the initialization scale by up to about forty percent, an effect absent in linear controls, and the conservative rules retain this memory permanently while the dissipative rule partially erases it. At matched training loss the dissipative rule typically generalizes worse than the conservative rules, although it reaches low training loss faster; the penalty correlates with the mass dissipated en route and fades in larger circuits, where little mass is lost. The conservation structure of a local learning rule thus sets its initialization memory, its training speed, and, where dissipation is appreciable, its generalization; it should be treated as a design parameter of physical learning machines. I Introduction A network of adjustable resistors can be trained to compute: contrastive local rules compare two equilibrium states of the circuit, a free state and a nudged state, and adjust each conductance using only quantities measurable at that edge [1, 2]. Equilibrium propagation (EP) nudges by applying a small force proportional to the output error [3]; coupled learning (CL) clamps the outputs a small step toward their targets [1]. Both rules have trained laboratory networks [4, 5], and both are candidates for learning hardware whose energy cost is set by physics rather than by digital arithmetic. A convergence theory exists for these rules: in linear circuits EP and CL reach a solution under a coercivity condition [6], and in the small-nudge limit both exactly conserve the conductance mass K=12∑eκe2K= 12 _e _e^2 [7]. The same analysis introduces a third rule, adjoint coupled learning (AL), which restores the gradient-flow structure that CL lacks [6]. Convergence, however, is only half of learning. Trainable edges outnumber output constraints, so the parameter configurations that satisfy a training objective form a manifold, and a convergence theorem says that training reaches this manifold, not where it lands. The landing point is what determines the learned function on inputs never seen in training. For gradient descent in deep networks that selection question is the subject of the implicit-bias literature; for physical learning it has not been asked. Conservation laws are the natural lens for the question. Gradient flow in deep networks also has conserved quantities tied to architectural symmetries [8, 9], but practical training breaks them through discrete steps, weight decay, and normalization, and the memory of the initialization fades as training proceeds [10]. Physical learning sits at the opposite pole. The conservation of K is exact along the continuous-time EP and CL flows and survives the removal of edges whose conductance reaches zero, so a physically trained network carries a permanent imprint of its initialization. We ask whether that imprint changes the learned input-output map. The answer has a structure that the three rules expose cleanly. For a single output the question cannot even be posed: the EP and CL flows have identical orbits up to a time reparametrization (Proposition 1), which explains why no rule-dependent bias has been observed and forces any comparison to start at two outputs. There the conservation structure separates the rules, because AL does not conserve the mass at all: along its flow K˙=−2Φ∗ K=-2 ^*, twice its own loss (Proposition 2). Conservation alone, however, guarantees nothing about function. In linear circuits the mass is remembered but inert, since the scaling map κ↦cκ cκ sends learning orbits to learning orbits for every rule (Proposition 3), and the same proposition says where scale can act: in circuits with fixed nonlinear elements, where the joint homogeneity breaks. There the memory becomes functional. The learned function shifts by up to about forty percent with initialization scale while linear controls sit at the numerical floor, permanently for the conservative rules and only partially for the dissipative one (Sec. V). The same dissipation carries a cost: at matched training loss AL generalizes worse than EP and CL in a significant majority of paired comparisons, by a penalty that correlates with the mass dissipated and fades in larger circuits where little mass is lost (Sec. VI). I Setup We follow the framework and notation of Refs. [6, 7]. A circuit is a connected graph on N nodes with M trainable edges; edge e carries a conductance κe>0 _e>0 and an incidence vector de∈ℝNd_e ^N. Node voltages x∈ℝNx ^N minimize the co-content, which for a linear circuit is the dissipated power G(x;κ)=12⟨x,L(κ)x⟩,L(κ)=∑eκedede⊤.G(x;κ)= 12 x,L(κ)\,x , L(κ)= _e _e\,d_ed_e . (1) Inputs are imposed as voltage constraints P⊤x=vP x=v on a set of input nodes that includes ground, and outputs are read at disjoint nodes through Q⊤xQ x with Q∈ℝN×OQ ^N× O. The free state is x0=argminG(x;κ):P⊤x=vx_0= \G(x;κ):P x=v\, and for a target w the error is r=Q⊤x0−wr=Q x_0-w with loss Φ=12‖r‖2 = 12\|r\|^2. The three rules perturb the output nodes in different ways. EP adds a force ηQrη\,Qr to the energy. CL imposes the constraint Q⊤xη=(1+η)Q⊤x0−ηwQ x_η=(1+η)\,Q x_0-η w. AL replaces the free state by the doubly constrained reference state x0∗=argminG:P⊤x=v,Q⊤x=wx_0^*= \G:P x=v,\ Q x=w\, whose output constraint force defines the adjoint error μ0 _0, and nudges by ημ0η _0 [6]. In each case the physically measurable update is Δκe∝η−1[∂κeG(xη)−∂κeG(xref)] _e η^-1[ _ _eG(x_η)- _ _eG(x_ref)]. We work throughout in the idealized continuous-time, zero-nudge limit, where the three flows take the common form κ˙e=(de⊤y)(de⊤xref) κ_e=(d_e y)(d_e x_ref) with yEP=RQr,yCL=RQDr,yAL=RALQDALμ0.y_EP=R\,Qr, y_CL=R\,QDr, y_AL=R_AL\,QD_AL\, _0. (2) Here R is the input-constrained resolvent of Ref. [6], D=(Q⊤RQ)−1D=(Q R\,Q)^-1 is a discrete Dirichlet-to-Neumann operator, and the AL quantities are evaluated at the reference state. EP is exact gradient flow of Φ . CL is not the gradient flow of any loss; its error is premultiplied by the state-dependent operator D. AL is exact gradient flow of the adjoint loss Φ∗=12‖μ0‖2 ^*= 12\| _0\|^2 [6]. Following Ref. [7], an edge whose conductance reaches zero is removed and stays removed. Because M>OM>O in all cases of interest, the solution set =κ:Q⊤x0(κ)=wS=\κ:Q x_0(κ)=w\ is generically a manifold of dimension M−OM-O. The convergence results of Ref. [6] guarantee, under a coercivity condition, that training reaches S. Our subject is where on S each flow lands, and what the landing point computes on held-out inputs. For a linear circuit the trained behavior is summarized by the input-output map AC(κ)∈ℝO×IA_C(κ) ^O× I defined by Q⊤x0=AC(κ)vQ x_0=A_C(κ)\,v. The map is 0-homogeneous in κ, and crossbar circuits realize exactly the row-stochastic maps [7]. I Conservation structure of the three rules Our first result explains why selection differences between EP and CL have gone unnoticed. In the single-output setting common to most demonstrations, there are none to see. Proposition 1 (Single-output trajectory equivalence). Let O=1O=1, with inputs and outputs disjoint and G(⋅;κ)G(·;κ) strictly convex on the input-constraint surface. Then κ˙CL=D(κ)κ˙EP κ_CL=D(κ)\, κ_EP with D(κ)=(Q⊤R(κ)Q)−1D(κ)=(Q R(κ)\,Q)^-1 a positive scalar. The two flows have identical orbits up to the time reparametrization dτ/dt=D(κ(t))dτ/dt=D(κ(t)), and in particular identical ω-limit sets. Any selection difference between EP and CL requires O≥2O≥ 2. Proof. For O=1O=1 the operator D is a scalar, and it is positive: R is positive semidefinite with kerR=range(P) R=range(P), and Q∉range(P)Q (P) because inputs and outputs are disjoint, so Q⊤RQ>0Q R\,Q>0. By Eq. (2) the two fields differ pointwise by the positive continuous scalar D(κ)D(κ), and a time change maps one flow onto the other. ∎ Remark 1. The proposition holds for general separable energies, not only linear circuits, since Eq. (2) is derived at that generality in Ref. [6]. In particular, single-output benchmarks, the configuration of most published demonstrations, cannot reveal an inductive-bias difference between EP and CL. The conservation law of Ref. [7] states that EP and CL conserve K=∑ifi(κi)K= _if_i( _i) whenever the energy separates as G=∑igi(κi)G~i(x)G= _ig_i( _i)\, G_i(x), with fi′=gi/gi′f_i =g_i/g_i ; for conductance training, K=12∑eκe2K= 12 _e _e^2. AL was introduced after that analysis and its conservation behavior was not addressed. It is qualitatively different. Proposition 2 (AL dissipates the conserved mass). Under the separability hypothesis above, the zero-nudge AL flow satisfies K˙=−∥μ0∥2=− 2Φ∗. K\;=\;- _0 ^2\;=\;-\,2\, ^*. (3) K decreases strictly whenever the objective is not yet met. Proof. Writing the AL flow as κ˙i=⟨y,∂κi∇xG(x0∗)⟩ κ_i= y, _ _i _xG(x_0^*) , K˙=∑ifi′(κi)κ˙i K= _if_i ( _i)\, κ_i =⟨y,∑ifi′(κi)∂κi∇xG(x0∗)⟩ = y, _if_i ( _i)\, _ _i _xG(x_0^*) =⟨y,∇xG(x0∗)⟩, = y, _xG(x_0^*) , using the separability identity of Ref. [7] at the reference state. The reference-state stationarity conditions give ∇xG(x0∗)=−Pλ0−Qμ0 _xG(x_0^*)=-P _0-Q _0, and the linearized system defining y imposes P⊤y=0P y=0 and Q⊤y=μ0Q y= _0 [6]. Hence K˙=−⟨Q⊤y,μ0⟩=−∥μ0∥2 K=- Q y, _0 =- _0 ^2. ∎ Remark 2. Proposition 2 is the zero-nudge counterpart of the finite-nudge drift K˙=−ηG(y;κ) K=-η\,G(y;κ) that Ref. [7] identifies for one-sided nudging. AL therefore carries, even at zero nudge, the parameter-scale drift that Ref. [7] attributes to non-conserving nudging. In hardware whose conductances have bounded operating ranges this is a design consideration in itself. Below it also turns out to govern initialization memory and generalization. We use the exact rate −2Φ∗-2 ^* as the step-control law for our AL integrator (Appendix A). A conserved quantity restricts each trajectory to a sphere ‖κ‖=const\|κ\|=const, and one might expect the initialization norm to index the learned function. In linear circuits it cannot. Proposition 3 (Initialization scale is inert in linear circuits). In a linear resistive circuit the vector fields of EP, CL, and AL are homogeneous in κ of degrees −1-1, 0, and +1+1: FEP(cκ)=c−1FEP(κ),FCL(cκ)=FCL(κ),FAL(cκ)=cFAL(κ),c>0. aligned F_EP(cκ)&=c^-1F_EP(κ),\\ F_CL(cκ)&=F_CL(κ),\\ F_AL(cκ)&=c\,F_AL(κ), aligned c>0. (4) For each rule the map κ↦cκ cκ sends learning orbits to learning orbits, so the ω-limit from cκ0c _0 is c times the ω-limit from κ0 _0, and since AC(cκ)=AC(κ)A_C(cκ)=A_C(κ), the selected input-output map is the same from both initializations. The conserved mass is functionally inert in linear circuits, and initialization affects selection only through its direction. Proof. The free state x0x_0 is 0-homogeneous in κ: the voltage constraints do not involve κ, and rescaling G by c leaves its constrained minimizer unchanged. The input-constraint multipliers are 11-homogeneous. For EP, y solves Ly+Pμ=QrLy+Pμ=Qr with P⊤y=0P y=0 and r 0-homogeneous, so y→c−1y→ c^-1y under the scaling. For CL, y is pinned by the κ-independent constraints P⊤y=0P y=0 and Q⊤y=rQ y=r, so y is 0-homogeneous. For AL, μ0 _0 is 11-homogeneous and Q⊤y=μ0Q y= _0 forces y→cy→ c\,y. In each case κ˙e=(de⊤y)(de⊤xref) κ_e=(d_e y)(d_e x_ref) inherits the degree of y. A homogeneous vector field of any degree has a scale-invariant direction field, F(cκ)/‖F(cκ)‖=F(κ)/‖F(κ)‖F(cκ)/\|F(cκ)\|=F(κ)/\|F(κ)\|, so scaling maps orbits to time-reparametrized orbits. Edge removal commutes with the scaling, since zero crossings scale with κ. The 0-homogeneity of ACA_C completes the argument. ∎ In linear circuits the conserved norm is therefore remembered but inert. The proposition also identifies where scale can act: any fixed, untrained circuit element breaks the joint homogeneity and, as noted in Ref. [7], the conservation law itself. Section V tests exactly this. IV Solution selection in linear circuits The propositions constrain selection but do not quantify it. We measure it first where they are exact. Circuits are connected random graphs with N=10N=10 to 1212 nodes, M=19M=19 to 2727 trainable edges, three input nodes including ground, and O=2O=2 to 44 outputs. Targets are constructed from a reference conductance vector, so the solution manifold is nonempty and of dimension M−OM-O. Generation details and seeds are in the repository (Appendix A). Each run integrates the three flows from a common initialization to their ω-limits and compares the selected maps AC(κ∞)A_C( _∞) by the relative Frobenius distance δ(A,B)=‖A−B‖/‖A‖δ(A,B)=\|A-B\|/\|A\|. As a numerical floor we re-integrate EP with a different initial step size; the floor is δ≈3×10−9δ≈ 3× 10^-9 in the median. Table 1: Selection in linear circuits at O=2O=2 (36 convergent runs over three topologies, twelve initializations each). Distances are relative Frobenius distances between selected input-output maps from a common initialization. The direction spread is the median distance to the mean map over 40 random initialization directions on a fixed task (one topology), shown for EP; CL and AL give 0.164 and 0.183. median min max numerical floor 3.1×10−93.1× 10^-9 6.8×10−96.8× 10^-9 δ(AEP,ACL)δ(A_EP,A_CL) 1.8×10−51.8× 10^-5 1.4×10−91.4× 10^-9 5.8×10−45.8× 10^-4 δ(AEP,AAL)δ(A_EP,A_AL) 1.2×10−31.2× 10^-3 8.1×10−68.1× 10^-6 1.3×10−21.3× 10^-2 δ(ACL,AAL)δ(A_CL,A_AL) 1.3×10−31.3× 10^-3 8.2×10−68.2× 10^-6 1.3×10−21.3× 10^-2 direction spread (EP) 1.6×10−11.6× 10^-1 The selection differences predicted by Eq. (2) are real (Table 1). With two outputs, EP and CL land on measurably different points of the solution manifold, with a median δ of 1.8×10−51.8× 10^-5, four orders of magnitude above the floor, and AL lands roughly a hundred times farther from either. Consistent with the mechanism, which is D deviating from a scalar, the EP-CL gap correlates with the strength of the effective coupling between output nodes (Spearman rank correlation 0.410.41, n=36n=36 pooled over O=2,3,4O=2,3,4), though it does not grow monotonically with the number of outputs. Real, however, is not the same as large. Over random initialization directions at fixed task, the selected maps spread by δ≈0.16δ≈ 0.16, which exceeds every rule effect by two to four orders of magnitude: in linear circuits the inductive bias of physical learning is set almost entirely by where training starts, and the rule contributes a small systematic correction. Nor does the correction matter for generalization here. Training on a single input-output pair against a ground-truth circuit and evaluating the full map, the three rules generalize identically within statistical resolution, with median relative errors 0.08790.0879, 0.08790.0879, and 0.08810.0881 over 30 runs; a three-pair training set determines the 2×32× 3 map completely and all rules recover it, which confirms identifiability. Linear circuits therefore provide a clean null: rule choice moves the selected solution but not its quality. V Initialization memory in nonlinear circuits Proposition 3 localizes where initialization scale can matter: circuits with fixed, untrained elements. This is also the physically realistic case, since laboratory contrastive networks contain diodes and other non-adjustable components [5]. We therefore study circuits in which a random subset of edges is replaced by fixed rectifiers, ideal diodes with a small reverse leakage ϵ=10−4ε=10^-4 and forward conductance of order one. The co-content remains convex and piecewise quadratic, and the zero-nudge flows of Eq. (2) carry over with the Hessian of G at the reference state in place of L (Appendix A). The trained behavior is no longer a matrix, so we compare functions by their outputs on a fixed 5×55× 5 grid of test inputs spanning both polarities, which demonstrably switches the rectifiers. Table 2: Initialization-norm memory. Relative distance between the function learned from initialization cκ0c\, _0 and from κ0 _0, with the same direction and the same single-pair task, under EP; CL agrees with EP to three digits except where noted in the text. All runs converged to training loss below 10−1110^-11. Controls are the same graphs with all edges linear and trainable. The direction-spread yardstick, the median over four random directions at c=1c=1, is 6.3×10−26.3× 10^-2, 2.6×10−22.6× 10^-2, and 1.2×10−11.2× 10^-1 for the three topologies. c=1/4c=1/4 c=1/2c=1/2 c=2c=2 c=4c=4 diode circuit, topology 1 7.5×10−27.5× 10^-2 4.4×10−24.4× 10^-2 4.5×10−24.5× 10^-2 8.0×10−28.0× 10^-2 diode circuit, topology 2 1.1×10−11.1× 10^-1 5.7×10−25.7× 10^-2 5.2×10−25.2× 10^-2 9.1×10−29.1× 10^-2 diode circuit, topology 3 1.3×10−11.3× 10^-1 8.9×10−28.9× 10^-2 2.1×10−12.1× 10^-1 3.8×10−13.8× 10^-1 linear control (max over topologies) 2.3×10−62.3× 10^-6 4.9×10−74.9× 10^-7 1.5×10−71.5× 10^-7 1.6×10−71.6× 10^-7 The effect is large and clean (Table 2). Rescaling the initialization, with the same direction, the same task, and convergence to the same training loss, changes the learned function by four to thirty-eight percent, growing with distance from the reference scale in every topology and for every rule. The all-linear controls sit five orders of magnitude lower, at the numerical floor, exactly as Proposition 3 requires. The effect is due entirely to the fixed nonlinear elements, whose operating points the conductance scale sets. Measured against the natural yardstick, the spread induced by changing the initialization direction, the scale effect is comparable or larger. Because EP and CL conserve ‖κ‖\|κ\| exactly, the dependence is not a transient. The initialization scale is imprinted on the learned function permanently. In deep networks initialization effects wash out with continued training [10]; a physically trained circuit cannot forget how conductive it was born. The dissipative rule can. In the topology with the largest memory, AL’s learned function moves with the initialization scale markedly less than EP’s or CL’s: at c=2c=2 its distance from the c=1c=1 function is 0.150.15 against 0.210.21 to 0.220.22 for the conservative rules, and at c=4c=4 it is 0.260.26 against 0.380.38 to 0.400.40. This is Proposition 2 at work. AL’s mass decay drives trajectories from different initial spheres toward one another and erases part of the scale memory that the conservative rules are bound to keep. With untrained elements present, EP and CL themselves conserve K only approximately, as anticipated in Ref. [7]. We measure a median relative drift of 1.5×10−31.5× 10^-3 over these runs, occasionally larger in extreme-scale runs, so the conservative rules’ memory is exact in the linear limit and slightly leaky in realistic circuits. Figure 1: Initialization-norm memory. Relative distance between the function learned from initialization cκ0c\, _0 and from κ0 _0 (same direction, same task, EP), for the three N=8N=8 diode topologies of Table 2, their all-linear controls (maximum over topologies), and two N=50N=50 diode circuits. Fixed nonlinear elements separate the learned functions by five to six orders of magnitude relative to the scale-invariant linear case. The memory effect is not an artifact of small circuits (Fig. 1). At N=50N=50 nodes, with 75 trainable edges and 25 rectifiers, the same protocol gives function changes of 11 to 18 percent at c=1/4c=1/4 and c=4c=4. The all-linear controls converge to zero as the integration budget grows, and their finite-time deviations follow the degree ladder of Proposition 3: the rule whose field slows under the scaling is the one that lags, in the direction the degree predicts. VI Generalization at matched training loss Do the rules differ in the quality of the solutions they select? Comparisons of learned functions at the end of training are confounded by convergence depth: the rules reach different training losses in equal time, and test-error differences then reflect optimization speed rather than selection. The confound is large enough to invert conclusions. At a fixed integration budget AL appears to generalize two to four times better than EP on the tasks below; at matched training loss the ordering reverses. We therefore integrate each rule until its training loss first crosses fixed checkpoints, 3×10−63× 10^-6 and 3×10−73× 10^-7, and evaluate the learned functions at the crossings. Tasks are three-pair training sets from ground-truth diode circuits, with six topologies and six initializations each, giving 36 paired runs. Every run reached both checkpoints. Table 3: Generalization at matched training loss in nonlinear circuits (n=36n=36 paired runs over six topologies). Test error is the relative distance to the ground-truth function on the input grid, reported as the pooled median. “AL worse” counts paired runs in which AL’s test error exceeds EP’s, and the penalty is the median of the paired relative difference (AL−EP)/EP(AL-EP)/EP. The residual floor 2L 2L bounds the function resolution at checkpoint L. checkpoint EP CL AL AL worse penalty floor L=3×10−6L=3× 10^-6 0.03430.0343 0.03470.0347 0.03470.0347 29/3629/36 +6.7%+6.7\% 2.4×10−32.4× 10^-3 L=3×10−7L=3× 10^-7 0.02150.0215 0.02170.0217 0.02330.0233 26/3626/36 +3.5%+3.5\% 7.7×10−47.7× 10^-4 At matched loss the dissipative rule loses on average (Table 3). AL’s test error exceeds EP’s in 29 of 36 paired runs at the first checkpoint (two-sided sign test, p=3×10−4p=3× 10^-4) and 26 of 36 at the second (p=0.011p=0.011), with median paired penalties of 6.76.7 and 3.53.5 percent. The effect is topology-dependent: four of the six topologies show AL worse in essentially every run, one shows no consistent ordering, and one shows the reverse. The functions AL selects differ from EP’s by δ≈3δ≈ 3 to 6×10−36× 10^-3, above the residual floors, so the differences are selection rather than incomplete convergence. EP and CL remain statistically indistinguishable: CL’s error exceeds EP’s in 21 and 18 of 36 runs, consistent with a fair coin, and their function difference sits at the floor. AL meanwhile reaches any given training loss fastest of the three; its final residuals at fixed budget are one to two orders of magnitude below EP’s. Faster training, a generalization penalty at matched loss, and weaker initialization memory all follow from the same property of the rule, the dissipation of K. Figure 2: AL’s generalization penalty against the fraction of conductance mass it dissipated before reaching the training-loss checkpoint. Filled circles: N=8N=8, all 36 paired runs at L=3×10−7L=3× 10^-7. Open squares: N=50N=50 runs at L=3×10−6L=3× 10^-6, which reach the checkpoint after an order of magnitude less dissipation and show no significant penalty. Two further measurements support this reading and bound it (Fig. 2). Across the 36 paired runs, the penalty correlates with the fraction of mass AL dissipated before reaching the checkpoint: Spearman rank correlation 0.500.50 at L=3×10−7L=3× 10^-7 (p≈0.002p≈ 0.002), weaker at the shallower checkpoint (0.170.17). Topology-level residuals remain; the reversed topology sits at low dissipation, as the mechanism requires, but one high-penalty topology dissipates little. In N=50N=50 circuits the same checkpoints are reached after an order of magnitude less dissipation (median fraction 0.30.3 percent against 3.33.3 percent at N=8N=8), and there the penalty is consistent with zero: AL is worse in 9 of 16 paired runs with a median penalty below one percent. Both observations are consistent with the dissipation mechanism, though the evidence is correlational: the rule appears penalized in proportion to the mass it burns, and larger circuits reach the training objective before burning much. VII Discussion In Ref. [7] the conserved mass plays a stability role: it explains why training does not drift out of a device’s operating range, and symmetric clamping is prescribed to protect it. That analysis concerns the scale of the parameters during training; ours concerns the function the parameters select. Our results give the same structure a second role: it is a controllable inductive bias. A designer choosing among EP, CL, AL, and clamping schemes is choosing how much the trained network will remember its fabrication state, how fast it will train, and, at least in the circuits studied here, how well its selected solution will generalize. Single-output benchmarks cannot distinguish EP from CL at all. Conservative rules on nonlinear substrates carry a permanent, functionally significant imprint of the initial conductance scale, which is a calibration burden if uncontrolled and possibly a feature if exploited. The dissipative rule trades a modest generalization penalty for faster training and reduced sensitivity to initialization. Three limitations bound our claims. All experiments are on simulated circuits of at most fifty nodes; the propositions hold at any size, and the memory effect persists at the largest size we test, but hardware-scale magnitudes are open. The matched-loss penalty is modest, several percent in the median, and not universal: one of six topologies shows the reverse ordering. The mechanism is supported correlationally, through the dissipation-penalty relation and its predicted weakening with system size, but topology-level residuals remain unexplained; a derivation is the natural next step, together with a selection theory for the nonlinear circuits where the interesting behavior lives. In machine learning, conservation laws of training dynamics have been catalogued but are broken by practical optimizers, and initialization memory is transient [8, 9, 10]. Physical learning realizes the opposite design point exactly. Conservation is native to the rule, the memory it protects is permanent, and both are consequences of the same local physics that does the learning. Data and code availability All results are reproducible from the repository at https://github.com/dangoldbj/physical-learning-memory, which contains the simulation package, the validation suite, and one script per experiment with pinned seeds. References Stern et al. [2021] M. Stern, D. Hexner, J. W. Rocks, and A. J. Liu, Supervised learning in physical networks: From machine learning to learning machines, Physical Review X 11, 021045 (2021). Stern and Murugan [2023] M. Stern and A. Murugan, Learning without neurons in physical systems, Annual Review of Condensed Matter Physics 14, 417 (2023). Scellier and Bengio [2017] B. Scellier and Y. Bengio, Equilibrium propagation: Bridging the gap between energy-based models and backpropagation, Frontiers in Computational Neuroscience 11, 24 (2017). Dillavou et al. [2022] S. Dillavou, M. Stern, A. J. Liu, and D. J. Durian, Demonstration of decentralized physics-driven learning, Physical Review Applied 18, 014040 (2022). Dillavou et al. [2024] S. Dillavou, B. D. Beyer, M. Stern, A. J. Liu, M. Z. Miskin, and D. J. Durian, Machine learning without a processor: Emergent learning in a nonlinear analog network, Proceedings of the National Academy of Sciences 121, e2319718121 (2024). McGinnis et al. [2026a] J. A. McGinnis, X. Li, and Y. Mori, Coercivity and local convergence of physical learning in linear circuits (2026a), arXiv:2606.15443 [math.OC] . McGinnis et al. [2026b] J. A. McGinnis, A. G. Kline, and Y. Mori, A conservation law for equilibrium propagation and coupled learning (2026b), arXiv:2606.15444 [math.OC] . Kunin et al. [2021] D. Kunin, J. Sagastuy-Brena, S. Ganguli, D. L. K. Yamins, and H. Tanaka, Neural mechanics: Symmetry and broken conservation laws in deep learning dynamics, in International Conference on Learning Representations (ICLR) (2021). Marcotte et al. [2023] S. Marcotte, R. Gribonval, and G. Peyré, Abide by the law and follow the flow: Conservation laws for gradient flows, in Advances in Neural Information Processing Systems (NeurIPS) (2023). Teney et al. [2024] D. Teney, A. M. Nicolicioiu, V. Hartmann, and E. Abbasnejad, Neural redshift: Random networks are not random functions, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2024). Appendix A Numerical methods States. Free, nudged, and reference states are computed from the KKT systems of the constrained energy minimizations. In linear circuits these are single bordered linear solves. In diode circuits the co-content is convex and piecewise quadratic, and states are computed by semismooth Newton iteration on the equality-constrained KKT system, with tolerance 10−1210^-12 and warm starts along trajectories. Rectifiers carry a reverse leakage ϵ=10−4ε=10^-4, which keeps the Hessian nondegenerate on connected circuits. Flows. The zero-nudge flows are integrated with fourth-order Runge-Kutta. Step control exploits the theory. For EP and CL the drift of K, exactly conserved in the linear case, serves as the local error metric. For AL the step is controlled against the exact dissipation rate K˙=−2Φ∗ K=-2 ^* of Proposition 2. In nonlinear circuits, step-doubling error control is used instead. An integration step that would drive a conductance negative is shortened to the zero crossing and the edge is removed, following Ref. [7]. Matched-loss protocol. Each rule is integrated once per task, with snapshots taken the first time the training loss crosses each prescribed level. All comparisons in Sec. VI are between snapshots at identical levels. The residual floor quoted there is 2L 2L, the function-space resolution implied by training loss L. Validation. The implementation reproduces the closed-form voltage-divider dynamics of Ref. [7] to 10−1010^-10, conserves K along EP and CL to 10−1210^-12 over full trajectories, and confirms Proposition 1 numerically: single-output EP and CL ω-limits agree to 2×10−92× 10^-9 in relative distance. The nonlinear module reduces exactly to the linear one when no diodes are present and reproduces the scale invariance of Proposition 3 to 10−810^-8 in its all-linear control configuration.