从噪声数据中自动找出符合物理定律的力公式，还能保证能量守恒。
Minimum-Action Learning: Energy-Constrained Symbolic Model Selection for Physical Law Identification from Noisy Data
- 通过最小化三重作用泛函，结合轨迹重建、稀疏性与能量守恒筛选符号力公式。
- 在开普勒和胡克定律测试中准确恢复真实力律，能量守恒判据实现100%识别率。
- 适合需要可解释性与物理一致性验证的科学机器学习研究者。
从噪声观测数据中识别物理定律是科学机器学习的核心挑战。本文提出最小作用量学习（MAL）框架，通过最小化融合轨迹重建、结构稀疏性与能量守恒的三重作用泛函，从预设基函数库中选择符号力公式。采用宽窗加速匹配技术，将噪声方差降低10,000倍，使信噪比从~0.02提升至~1.6，显著改善可学习性；该预处理为所有方法（包括SINDy变体）的关键前提。在开普勒引力与胡克定律两个基准上，MAL以~0.07 kWh能耗（较仅优化预测误差的基线降低40%）准确恢复力律，开普勒指数为3.01±0.01。原始基函数正确率分别为40%（开普勒）与90%（胡克），而基于能量守恒的判别准则在所有情况下均能精准识别真解，实现100%流水线级识别。基函数库敏感性实验表明，近似混淆项（如r^{-2.5}、r^{-1.5}）会降低选择性能（降至20%），远距离添加则无影响，且即使正确基函数缺失，守恒诊断仍具判别力。与鲁棒噪声处理的SINDy变体、哈密顿神经网络及拉格朗日神经网络直接对比，证实MAL的独特定位：兼具符号基识别与动力学回放验证的可解释、能量约束模型选择。
原文摘要 · Abstract (English)
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.
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