Reproduce “Machine-learning hidden symmetries” on OpenResearch

About this paper

Ziming Liu, Max Tegmark

A machine learning algorithm can automatically discover hidden symmetries in physical systems by minimizing a loss function based on the violation of generalized symmetry-encoding partial differential equations, subsequently translating the learned coordinate transformations into interpretable symbolic expressions. The method successfully rediscovered complex transformations for systems ranging from classical mechanics, such as the Kepler problem and double pendulum, to general relativity, including the FRW metric (k=-1) and the Schwarzschild black hole's Gullstrand-Painlevé coordinates.

Key insights

• Hidden symmetries can be systematically discovered by reframing the problem as minimizing the violation of specific PDEs that define the desired symmetric properties.
• Invertible neural networks can effectively parameterize the vast space of possible non-linear coordinate transformations required to reveal these hidden symmetries.
• The concept of symmetry can be broadly generalized beyond traditional invariances to include properties like Hamiltonicity and Modularity, significantly expanding the scope of automated discovery.

Summary by alphaXiv.