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

arXiv: https://arxiv.org/abs/2603.16951 · HTML: https://mlsanity.com/paper/2603.16951/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 46 references that are in the map.
- First posted on arXiv 16 Mar 2026.

## Builds on (most cited first, 11 of 11)

- [Energy and Policy Considerations for Deep Learning in NLP](https://mlsanity.com/paper/1906.02243/) (2019)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Carbon Emissions and Large Neural Network Training](https://mlsanity.com/paper/2104.10350/) (2021)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [SNAS: Stochastic Neural Architecture Search](https://mlsanity.com/paper/1812.09926/) (2018)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [Rediscovering orbital mechanics with machine learning](https://mlsanity.com/paper/2202.02306/) (2022)
- [Noether's Learning Dynamics: Role of Symmetry Breaking in Neural Networks](https://mlsanity.com/paper/2105.02716/) (2021)
- [Noether's razor: Learning Conserved Quantities](https://mlsanity.com/paper/2410.08087/) (2024)

## Built on by (most cited first, 1 of 1)

- [minAction.net: Energy-First Neural Architecture Design -- From Biological Principles to Systematic Validation](https://mlsanity.com/paper/2604.24805/) (2026)

## Abstract (arXiv)

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.