Minimum-Action Learning: Energy-Constrained Symbolic Model Selection for Physical Law Identification from Noisy Data

Martin G. Frasch

cs.LGMar 16, 2026

Minimum-Action Learning: Energy-Constrained Symbolic Model Selection for Physical Law Identification from Noisy Data

Authors: Martin G. Frasch

Organizations: Institute on Human Development and Disability, University of Washington, Seattle, WA 98195, USA · Health Stream Analytics, LLC, Seattle, WA, USA

Abstract

Identifying physical laws from noisy observational data is a central challenge in scientific machine learning. We present Minimum-Action Learning (MAL), a framework that selects symbolic force laws from a pre-specified basis library by minimizing a Triple-Action functional combining trajectory reconstruction, architectural sparsity, and energy-conservation enforcement. A wide-stencil acceleration-matching technique reduces noise variance by 10,000x, transforming an intractable problem (SNR ~0.02) into a learnable one (SNR ~1.6); this preprocessing is the critical enabler shared by all methods tested, including SINDy variants. On two benchmarks -- Kepler gravity and Hooke's law -- MAL recovers the correct force law with Kepler exponent p = 3.01 +/- 0.01 at ~0.07 kWh (40% reduction vs. prediction-error-only baselines). The raw correct-basis rate is 40% for Kepler and 90% for Hooke; an energy-conservation-based criterion discriminates the true force law in all cases, yielding 100% pipeline-level identification. Basis library sensitivity experiments show that near-confounders degrade selection (20% with added r^{-2.5} and r^{-1.5}), while distant additions are harmless, and the conservation diagnostic remains informative even when the correct basis is absent. Direct comparison with noise-robust SINDy variants, Hamiltonian Neural Networks, and Lagrangian Neural Networks confirms MAL's distinct niche: interpretable, energy-constrained model selection that combines symbolic basis identification with dynamical rollout validation.

Figures & tables

Figure 1: Trajectory reconstruction and Hamiltonian conservation. (A) Comparison of ground-truth Keplerian orbits (blue) vs. MAL reconstruction (red) for a test orbit rolled out over 5 orbital periods from initial conditions using the identified $r^{-2}$ force law. Slight enlargement is attributable to the 6% deficit in recovered $GM$ . (B) Energy conservation error $\Delta H$ remains bounded, enforced by the Noether-symmetry term $\mathcal{L}_{\mathrm{Symmetry}}$ in the triple-action functional, which constrains the parameter trajectory $\theta(t)$ to satisfy $dH/dt\approx 0$ . Implementation: the force is computed via NoetherForceBasis.forward() , which evaluates $f(r)=\sum_{i}A_{i}\theta_{i}\phi_{i}(r)$ with gates $A_{i}=\mathrm{softmax}(\texttt{A\_logits}/\tau)$ .

Figure 2: Soft-to-discrete architectural crystallization. (A) Evolution of gate activation probabilities $A_{i}$ from uniform (epoch 1, soft state) to one-hot selection of the $r^{-2}$ basis (epoch 200, discrete state). This transition occurs on a soft-to-discrete architecture manifold where A_logits are sharpened via softmax temperature decay ( $\tau:1\to 0.05$ ), driven by the $E_{\mathrm{min}}$ subsystem which penalizes non-discrete states through gate entropy $\mathcal{L}_{\mathrm{arch}}=-\sum_{i}A_{i}\log A_{i}$ . (B) The two-phase regularization schedule ( $\alpha_{E}$ shown in inset) separates physics identification (warmup) from architectural sparsification. Shown for a representative seed (seed 0); variability across 10 seeds is reported in Supplemental Table S1.

Figure 3: Energy efficiency and training dynamics. (A) Training loss components vs. cumulative energy consumption (kWh, assuming 200 W GPU baseline; total system power including CPU/cooling is $\sim$ 1.5 $\times$ this value). Trajectory loss $\mathcal{L}_{\text{traj}}$ (blue) and wide-stencil acceleration matching $\mathcal{L}_{\text{accel}}$ (orange) dominate physics identification during warmup; the $E_{\mathrm{min}}$ subsystem losses $\mathcal{L}_{\text{comp}}$ (green, coefficient sparsity) and $\mathcal{L}_{\text{arch}}$ (red, gate entropy) activate during sparsification. (B) Temperature schedule $\tau$ (purple) and regularization weight $\alpha_{E}$ (brown) implement the two-phase protocol. The $E_{\mathrm{min}}$ subsystem penalizes high-entropy architectural configurations, driving the soft-to-discrete transition shown in Fig. 2 . Implementation: loss components are computed in minaction_loss() , with $\mathcal{L}_{\text{comp}}=\langle|A_{i}\theta_{i}|\rangle$ and $\mathcal{L}_{\text{arch}}=-\sum_{i}A_{i}\log A_{i}$ .

Stride $s$	Noise $\sigma_{a}$	Signal $\|a\|$	SNR	Training
1 (naive)	15.7	0.25	0.016	Failed
5	0.63	0.25	0.40	Partial
10	0.16	0.25	1.6	Success
20	0.039	0.25	6.4	Success

Table 1: Noise reduction via wide-stencil differentiation

	Kepler ( $r^{-2}$ )	Hooke ( $r$ )
Basis library	$\{r^{-2},r^{-1},r,1,r^{-3}\}$
Direct selection rate	4/10 (40%)	9/10 (90%)
Pipeline rate ∗	10/10 (100%)	10/10 (100%)
Biased init rate	10/10 (100%)	—
Recovered coeff.	$\hat{\theta}_{0}=0.94$	$\hat{k}=0.98$
Kepler exponent $\hat{p}$	$3.01\pm 0.01$	—

Table 2: Summary of MAL pipeline performance across benchmarks

Figure 4: Energy-conservation-based model selection and schedule geometry. (Left) Phase-space trajectory of $(\alpha_{E},\tau)$ during training, color-coded by epoch. Red diamonds mark epochs where the ratio $\alpha_{E}/\tau$ passes through integer values (3:1, 2:1, 1:1), coinciding with major transitions in gate selectivity (onset, sparsification, crystallization). These coincidences arise from the designed schedule geometry; whether they reflect deeper dynamical principles remains an open question (see Discussion). (Right) Energy conservation $\sigma_{H}$ by selected basis: correct $r^{-2}$ models conserve $H$ over long-horizon rollouts, while incorrect bases violate Hamiltonian dynamics despite matching short-term trajectories. This provides an energy-conservation-based diagnostic for model selection.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Seed	Basis	$\hat{p}$	Onset	Sparse	Frozen	Span	$\gamma$
0	$r^{-2}$	2.995	116	138	156	40	1.125
1	$r^{-2}$	3.001	117	136	152	35	1.141
2	$r^{-2}$	3.003	133	153	169	36	1.139
3	$r^{-3}$	3.004	136	155	171	35	1.140
4	$r^{-3}$	3.004	127	143	159	32	1.153
5	$r^{-3}$	3.002	116	143	161	45	1.110

Appendix

Table 1: Ten-seed robustness sweep (full data from tsparse_sweep_results.json )

Selected basis	$n$ (seeds)	$\bar{\sigma}_{H}$	Traj. MSE
$r^{-2}$ (correct)	4	$0.017\pm 0.016$	5.8
$r^{-1}$	3	$0.056\pm 0.023$	6.1
$r^{-3}$	3	$0.11\pm 0.017$	$1.2\times 10^{3}$

Appendix

Table 2: Energy conservation as energy-conservation diagnostic

Seed	Basis	$\hat{k}$	Onset	Frozen	Span	Reject gravity?
0	$r^{-3}$	0.040	125	164	39	Yes
1	$r$	0.980	113	157	44	Yes
2	$r$	0.980	106	148	42	Yes
3	$r$	0.980	104	147	43	Yes
4	$r$	0.980	100	144	44	Yes
5	$r$	0.980	123	165	42	Yes

Appendix

Table 3: Hooke’s law 10-seed sweep results

Intervention	$r^{-2}$ rate	Description	Key insight
Standard (baseline)	4/10	Default training protocol	—
Extended warmup (100 ep.)	4/10	Double warmup duration	More exploration doesn’t help
Lower noise ( $\sigma=0.005$ )	4/10	Half observation noise	Cleaner data doesn’t help
Biased $\alpha_{\text{logits}}$	10/10	$[1.5,0,0,0,0]$ init	Gate competition is bottleneck

Appendix

Table 4: Basis selection intervention experiments (Kepler, 10 seeds each)

∗ Biased init. or energy-conservation post-selection.
Method	$s$	$r^{-2}$ rate	$\overline{GM}$	Time
SINDy (naive)	1	0/10	—	${<}0.01$ s
SINDy (wide)	10	10/10	1.05–2.74	${<}0.01$ s
GP-SINDy (wide)	10	8/10	0.97–1.88	${\sim}6$ s
Ens.-SINDy (wide)	10	10/10	1.07–2.68	${\sim}0.4$ s
MAL	10	4/10 (10/10 ∗ )	${\sim}0.94$	${\sim}835$ s

Appendix

Table 5: SINDy variant comparison on Kepler benchmark (10 seeds)

Method	Params	Time	Energy	$\sigma_{H}$	Symbolic?	Status
HNN	17,281	113 s	0.006 kWh	$4.1\times 10^{-4}$	No	Converged
LNN	17,281	242 s	0.013 kWh	$1.9\times 10^{-2}$	No	Hessian singular
MAL	$\sim$ 20	835 s	0.046 kWh	$0.017$	Yes	Converged

Appendix

Table 6: Structure-preserving neural network comparison

Experiment	$K$	$r^{-2}$	$\bar{C}_{\mathrm{gate}}$	$\bar{\sigma}_{H}$ (corr.)	$\bar{\sigma}_{H}$ (incorr.)
Standard	5	5/10	0.97	0.132	0.183
Confounders	7	2/10	1.00	0.091	0.158
Expanded	8	5/10	0.96	0.134	0.138
Missing	4	—	0.96	—	0.156

Appendix

Table 7: Basis library sensitivity (Kepler, 10 seeds each)

Figure 1: Robustness: Seed 137 orbit reconstruction. Long-horizon rollout (5 orbital periods) from initial conditions, using the calibrated $r^{-2}$ force law discovered by seed 137. Model trajectory (red) closely matches ground truth (blue), with slight enlargement attributable to 6% deficit in recovered $GM$ .

Figure 2: Robustness: Seed 137 training curves. Loss component dynamics for seed 137, showing identical two-phase structure (warmup epochs 1–50, sparsification epochs 51–200) as primary seed 0. Onset occurs at epoch 121, within one SD of the mean (117 $\pm$ 9).

Figure 3: Robustness: Seed 137 gate evolution. Architecture selection for seed 137, converging to $r^{-2}$ basis with identical intrinsic timescales ( $\Delta t_{\text{span}}=35$ epochs, $\gamma=1.141$ ).

Figure 4: Robustness: Seed 137 energy conservation. Hamiltonian variance $\sigma_{H}=0.019$ for seed 137, within the $r^{-2}$ group mean $0.017\pm 0.016$ , confirming energy-conservation selection criterion.

Figure 5: Multi-panel discovery summary. (A) Orbit comparison: model rollout vs. ground truth for test orbit. (B) Architecture gate evolution over 200 epochs. (C) Learned force coefficients $\theta_{i}$ before and after calibration. (D) Kepler exponent fit: $T^{2}\propto a^{3.01}$ .

Figure 6: Architectural modularity analysis. (A) Modularity index $Q$ (Herfindahl–Hirschman concentration on effective gate contributions $A_{i}|\theta_{i}|$ ) over training epochs for MAL (blue, mean $\pm$ SD, $N=10$ seeds) vs. teacher-forcing-only baselines (red dashed, $N=10$ ). Vertical dashed lines mark the warmup-to-sparsification transition. MAL modularity rises sharply during sparsification, reaching $Q=0.99\pm 0.02$ by epoch 200, while teacher-forcing remains diffuse ( $Q=0.14\pm 0.04$ ). (B) Final modularity by group: models selecting the correct $r^{-2}$ basis, models selecting other bases, and teacher-forcing baselines (permutation test $p<0.001$ , $n=10{,}000$ ). Both MAL groups achieve near-maximal $Q$ regardless of which basis is selected, confirming that energy-constrained training drives architectural sparsification independent of outcome. (C) Final gate probability distributions $A_{i}$ for all 10 MAL seeds (S0–S9) and 10 teacher-forcing baselines (B0–B9). MAL seeds crystallize to one-hot gate vectors (single dominant basis per seed), while baselines maintain diffuse distributions across all five bases.

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Machine Learning Hamiltonian Dynamical Systems with Sparse and Noisy Data

Vedanta Thapar, Abhinav Gupta

The Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom · Center for Theoretical Physics, St. Stephen’s College, University of Delhi, Delhi 110007, India

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Machine learning has become a powerful tool for discovering governing laws of dynamical systems from data. However, most existing approaches degrade severely when observations are sparse, noisy, or irregularly sampled. In this work, we address the problem of learning symbolic representations of nonlinear Hamiltonian dynamical systems under extreme data scarcity by explicitly incorporating physical structure into the learning architecture. We introduce Adaptable Symplectic Recurrent Neural Networks (ASRNNs), a parameter-cognizant, structure-preserving model that combines Hamiltonian learning with symplectic recurrent integration, avoiding time derivative estimation, and enabling stable learning under noise. We demonstrate that ASRNNs can accurately predict long-term dynamics even when each training trajectory consists of only two irregularly spaced time points, possibly corrupted by correlated noise. Leveraging ASRNNs as structure-preserving data generators, we further enable symbolic discovery using independent regression methods (SINDy and PySR), recovering exact symbolic equations for polynomial systems and consistent polynomial approximations for non-polynomial Hamiltonians. Our results show that such architectures can provide a robust pathway to interpretable discovery of Hamiltonian dynamics from sparse and noisy data.

Abstract

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