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
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 ΔH remains bounded, enforced by the Noether-symmetry term LSymmetry in the triple-action functional, which constrains the parameter trajectory θ(t) to satisfy dH/dt≈0 . Implementation: the force is computed via NoetherForceBasis.forward() , which evaluates f(r)=∑iAiθiϕi(r) with gates Ai=softmax(A_logits/τ) .
Figure 2: Soft-to-discrete architectural crystallization. (A) Evolution of gate activation probabilities Ai 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 ( τ:1→0.05 ), driven by the Emin subsystem which penalizes non-discrete states through gate entropy Larch=−∑iAilogAi . (B) The two-phase regularization schedule ( αE 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 ∼ 1.5 × this value). Trajectory loss Ltraj (blue) and wide-stencil acceleration matching Laccel (orange) dominate physics identification during warmup; the Emin subsystem losses Lcomp (green, coefficient sparsity) and Larch (red, gate entropy) activate during sparsification. (B) Temperature schedule τ (purple) and regularization weight αE (brown) implement the two-phase protocol. The Emin 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 Lcomp=⟨∣Aiθi∣⟩ and Larch=−∑iAilogAi .
Stride s
Noise σa
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.
θ^0=0.94
k^=0.98
Kepler exponent p^
3.01±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 (αE,τ) during training, color-coded by epoch. Red diamonds mark epochs where the ratio αE/τ 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 σH 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
p^
Onset
Sparse
Frozen
Span
γ
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)
σˉH
Traj. MSE
r−2 (correct)
4
0.017±0.016
5.8
r−1
3
0.056±0.023
6.1
r−3
3
0.11±0.017
1.2×103
Appendix
Table 2: Energy conservation as energy-conservation diagnostic
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 ± 9).
Figure 3: Robustness: Seed 137 gate evolution. Architecture selection for seed 137, converging to r−2 basis with identical intrinsic timescales ( Δtspan=35 epochs, γ=1.141 ).
Figure 4: Robustness: Seed 137 energy conservation. Hamiltonian variance σH=0.019 for seed 137, within the r−2 group mean 0.017±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 θi before and after calibration. (D) Kepler exponent fit: T2∝a3.01 .
Figure 6: Architectural modularity analysis. (A) Modularity index Q (Herfindahl–Hirschman concentration on effective gate contributions Ai∣θi∣ ) over training epochs for MAL (blue, mean ± 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±0.02 by epoch 200, while teacher-forcing remains diffuse ( Q=0.14±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 Ai 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.
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.
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
Conservation laws are fundamental to understanding dynamical systems, but discovering them from data remains challenging due to parameter variation, non-polynomial invariants, local minima, and false positives on chaotic systems. We introduce NGCG, a neural-symbolic pipeline that decouples dynamics learning from invariant discovery and systematically addresses these challenges. A multi-restart variance minimiser learns a near-constant latent representation; system-specific symbolic extraction (polynomial Lasso, log-basis Lasso, explicit PDE candidates, and PySR) yields closed-form expressions; a strict constancy gate and diversity filter eliminate spurious laws. On a benchmark of nine diverse systems including Hamiltonian and dissipative ODEs, chaos, and PDEs, NGCG achieves consistent discovery (DR=1.0, FDR=0.0, F1=1.0) on all four systems with true conservation laws, with constancy two to three orders of magnitude lower than the best baseline. It is the only method that succeeds on the Lotka--Volterra system, and it correctly outputs no law on all five systems without invariants. Extensive experiments demonstrate robustness to noise (σ=0.1), sample efficiency (50--100 trajectories), insensitivity to hyperparameters, and runtime under one minute per system. A Pareto analysis shows that the method provides a range of candidate expressions, allowing users to trade complexity for constancy. NGCG achieves strong performance relative to prior methods for data-driven conservation-law discovery, combining high accuracy with interpretability.
Rahul D Ray
Department of Electronics and Electrical Engineering, BITS Pilani, Hyderabad Campus
Before a machine can discover a physical law, it must discover what its measurements are: which observations live on cells, which are intensive or extensive, which sectors are dual, and which distinctions are merely gauge. We introduce physical representation-language discovery, the problem of recovering this hidden ontology directly from anonymous controlled experiments. We give an identifiability theory and constructive polynomial-time procedure that recovers a carrier and differential sequence, measurement types and orientation twist, noninvertible refinement semantics, primal-dual Maxwell diagrams, and the residual equivalences that no permitted experiment can break. The theory turns material nuisance into a commutant, uses refinement to separate quantities from coordinates, and selects physics only after its representation has been recovered. For a certified finite experiment family, we prove an end-to-end two-stage measurement bound and a matching minimax rate in dimension, accuracy, and confidence. Blind Maxwell experiments recover complete primal/relative-dual ontologies on regular and unstructured carriers under jointly corrupted observations; an independent unstructured RLC system demonstrates that the result is not specific to Maxwell. The framework scales to tens of thousands of cells per carrier, while stress audits demonstrate robustness across severe physical regimes - including non-Markovian memory, nonlinearities, nonlocality, and complex constitutive hysteresis. A public FDTD audit demonstrates the emergence of anonymous curl structure from incomplete field data, while characterizing the informational prerequisites for complete recovery. The goal is to move scientific ML from learning laws in a human-supplied language to discovering the language in which laws become expressible, establishing exact theoretical limits on observational identifiability.