Noether剃刀：学习守恒量
Noether's razor: Learning Conserved Quantities
AI总结：
该研究利用Noether定理将对称性参数化为可学习的守恒量，通过近似贝叶斯模型选择推导变分下界进行联合训练，自动避免守恒定律坍缩，在谐振子和多体系统中准确识别守恒量与对称群并提升预测精度。
摘要：
对称性已被证明在机器学习模型中非常有用，能够提升泛化能力和整体性能。与此同时，学习动力系统的最新进展依赖于对底层Hamiltonian进行建模以保证能量守恒。这些方法可以通过数学物理中的一项开创性成果联系起来：Noether定理，该定理指出动力系统中的对称性对应于守恒量。这项工作利用Noether定理将对称性参数化为可学习的守恒量。随后，我们通过近似贝叶斯模型选择，允许守恒量及其相关的对称性直接从训练数据中学习，并与常规训练过程联合进行。作为训练目标，我们推导出了边缘似然的变分下界。该目标自动体现了Occam剃刀效应，避免了守恒定律坍缩为平凡的常数，而无需手动添加和调整额外的正则化项。我们在$n$谐振子和$n$体系统上展示了原理验证。我们发现，我们的方法正确地识别出了正确的守恒量以及U($n$)和SE($n$)对称群，提升了测试数据上的整体性能和预测精度。
英文摘要：
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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