Visualizing a neural network that develops quantum perturbation theory

Abstract

Motivated by the question whether the empirical fitting of data by neural networks can yield the same structure of physical laws, we apply neural networks to a quantum-mechanical two-body scattering problem with short-range potentials—a problem that by itself plays an important role in many branches of physics. After training, the neural network can accurately predict s-wave scattering length, which governs the low-energy scattering physics. By visualizing the neural network, we show that it develops perturbation theory order by order when the potential depth increases, without solving the Schrödinger equation or obtaining the wave function explicitly. The result provides an important benchmark to the machine-assisted physics research or even automated machine learning physics laws.