Visualizing Neural Network Developing Perturbation Theory

Abstract: In this letter, motivated by the question that whether the empirical fitting of data by neural network can yield the same structure of physical laws, we apply the neural network to a simple quantum mechanical two-body scattering problem with short-range potentials, which by itself also plays an important role in many branches of physics. We train a neural network to accurately predict $ s $-wave scattering length, which governs the low-energy scattering physics, directly from the scattering potential without solving Schr\"odinger equation or obtaining the wavefunction. After analyzing the neural network, it is shown that the neural network develops perturbation theory order by order when the potential increases. This provides an important benchmark to the machine-assisted physics research or even automated machine learning physics laws.