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

Technical Deep Dive: Minimum-Action Learning for Physical Law Identification

Overview

The paper presents Minimum-Action Learning (MAL), a framework that embeds biological energy-optimization principles in neural architecture search to enable energy-constrained identification of physical laws from noisy data. Key contributions include:

• Incorporating a Triple-Action functional combining information maximization, energy minimization, and symmetry enforcement to drive model selection over a pre-specified basis library.
• A wide-stencil acceleration matching technique that reduces noise variance by 10,000x, transforming an intractable problem (SNR ≈ 0.02) into a learnable one (SNR ≈ 1.6).
• A two-phase "bimodal glial-neural optimization" (BGNO) training schedule that encourages the emergence of sparse, modular architectures through energy-driven gate sharpening.
• Demonstration of 100% pipeline identification accuracy on Kepler's inverse-square law and Hooke's linear restoring force, with the energy-conservation criterion providing a robust model selection diagnostic.

Problem & Context

• Identifying physical laws from noisy observational data is a central challenge in scientific machine learning. Existing approaches like symbolic regression and Hamiltonian/Lagrangian neural networks have limitations in robustness to noise and interpretability.
• The authors propose that embedding principles of action minimization (inspired by the physics of least action and biological metabolic constraints) can improve the efficiency and interpretability of physical law identification.

Methodology

Key components of the MAL framework:

Noether Force Basis: Forces are parameterized as radial functions $F(r) = f(r)r̂$, where $f(r) = \sum{i=1}^5 Ai \thetai \phii(r)$ and $\phii \in \{r^{-2}, r^{-1}, r, 1, r^{-3}\}$ are a pre-specified basis library. Learnable gates $Ai$ select among the basis functions.

Triple-Action Objective: The loss function combines 3 terms:

• $L{I{max}}$: Trajectory reconstruction and wide-stencil acceleration matching to maximize information extraction from noisy data.
• $L{E{min}}$: Architectural complexity penalties (gate entropy, coefficient sparsity) to minimize energy.
• $L_{Symmetry}$: Enforcement of energy conservation, inspired by Noether's theorem.

Bimodal Glial-Neural Optimization (BGNO): A two-phase training schedule with soft gate exploration followed by energy-driven architectural sharpening, drawing inspiration from the brain's 1:1 glia-to-neuron ratio and evidence that glial cells mediate metabolic constraints during development.

Data & Experimental Setup

• Synthetic Keplerian orbits in 2D with Gaussian noise ($\sigma = 0.01 \times$ median semi-major axis) added to position measurements.
• 16 orbits for training, 15 for validation, 15 for testing.
• Wide-stencil differentiation ($s=10$) used to reduce acceleration noise variance by 10,000x.

Results

Kepler Benchmark:

• 4 out of 10 seeds directly selected the correct $r^{-2}$ basis; the remaining 6 selected $r^{-3}$ or $r^{-1}$.
• All 10 seeds recovered Kepler exponent $p \approx 3.0$, but the energy-conservation criterion discriminated the correct $r^{-2}$ models, achieving 100% pipeline-level identification.
• Biased initialization (favoring $r^{-2}$) achieved 10/10 correct selection.
• Training time: 835 seconds, energy consumption: ~0.07 kWh (40% reduction vs. baselines).

Hooke's Law Benchmark:

• 9 out of 10 seeds directly selected the correct $r$ basis, with the energy-conservation criterion rejecting all gravity-family potentials.
• Recovered spring constant $\hat{k} = 0.980 \pm 0.001$ (true: $k = 1.0$).
• Crystallization timescales comparable to Kepler.

Interpretation

• Embedding metabolic constraints ($L{E{min}}$) alongside information maximization improves the efficiency and interpretability of physical law identification, paralleling biological principles of energy-constrained optimization.
• The wide-stencil preprocessing is a critical enabler shared across methods, but MAL's Noetherian validation provides an additional layer of dynamical consistency checking.
• The soft-to-discrete gate sharpening and intrinsic crystallization timescales exhibit structural parallels to patterns observed in biological systems under metabolic constraints, though the connections remain analogical rather than formally established.

Limitations & Uncertainties

• MAL requires a pre-specified basis library, limiting it to model selection rather than open-ended discovery. Expanding to construct novel functional forms requires integrating with symbolic regression methods.
• Only two benchmarks (Kepler, Hooke) with synthetic data have been tested so far. Generalization to more complex physical systems, dissipative dynamics, and real observational data remains to be demonstrated.
• The coincidence between gate transitions and the training schedule's integer-ratio passages is currently correlational; whether the dynamics are genuinely sensitive to these passages requires further investigation.

What Comes Next

• Extending MAL to more complex physical systems, dissipative dynamics, and real-world datasets.
• Investigating whether the training schedule's integer-ratio coincidences reflect deeper dynamical principles akin to frequency-locking phenomena in biological coordination.
• Exploring connections between MAL's architectural sparsification, modularity in biological networks, and unified frameworks for energy-constrained optimization across scales.
• Integrating MAL's energy-conservation-based model selection with open-ended symbolic regression methods to enable truly autonomous physical discovery.