用诺特定理让模型自动学习物理守恒量，提升预测精度。
Noether's razor: Learning Conserved Quantities
- 将对称性建模为可学习的守恒量，通过贝叶斯选择联合训练。
- 在多振子和多体系统中准确识别守恒量与对称群，提升测试性能。
- 无需手动调参，自动避免守恒律退化为常数，适合物理启发建模。
对称性在机器学习中已被证明有助于提升泛化能力和整体性能。近期学习动力系统的方法依赖于建模底层哈密顿量以保证能量守恒。这些方法可通过数学物理中的经典定理——诺特定理联系起来：系统的对称性对应守恒量。本文利用诺特定理，将对称性参数化为可学习的守恒量，并通过近似贝叶斯模型选择，直接从训练数据中联合学习守恒量及其对应对称性。以变分下界作为训练目标，该目标天然具备奥卡姆剃刀效应，能自动防止守恒律退化为平凡常数，无需额外人工正则化。我们在n-谐振子和n-体系统上进行了验证，结果表明方法能正确识别出守恒量及对应的U(n)和SE(n)对称群，显著提升了测试数据上的整体性能与预测准确性。
原文摘要 · Abstract (English)
Symmetries have proven useful in machine learning models, improving generalisation and overall performance. At the same time, recent advancements in learning dynamical systems rely on modelling the underlying Hamiltonian to guarantee the conservation of energy. These approaches can be connected via a seminal result in mathematical physics: Noether's theorem, which states that symmetries in a dynamical system correspond to conserved quantities. This work uses Noether's theorem to parameterise symmetries as learnable conserved quantities. We then allow conserved quantities and associated symmetries to be learned directly from train data through approximate Bayesian model selection, jointly with the regular training procedure. As training objective, we derive a variational lower bound to the marginal likelihood. The objective automatically embodies an Occam's Razor effect that avoids collapse of conservation laws to the trivial constant, without the need to manually add and tune additional regularisers. We demonstrate a proof-of-principle on $n$-harmonic oscillators and $n$-body systems. We find that our method correctly identifies the correct conserved quantities and U($n$) and SE($n$) symmetry groups, improving overall performance and predictive accuracy on test data.
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