# Learning and Interpreting Potentials for Classical Hamiltonian Systems

## TL;DR

This paper presents a method to learn and interpret potential energy functions from trajectories of classical Hamiltonian systems, combining neural networks with equation discovery to produce interpretable models.

## Contribution

The paper introduces a novel approach that learns neural potentials and extracts closed-form algebraic expressions, enabling interpretable modeling of classical Hamiltonian systems.

## Key findings

- Accurately learns neural potentials matching ground truth.
- Successfully extracts interpretable algebraic expressions.
- Demonstrates effectiveness on multiple classical systems.

## Abstract

We consider the problem of learning an interpretable potential energy function from a Hamiltonian system's trajectories. We address this problem for classical, separable Hamiltonian systems. Our approach first constructs a neural network model of the potential and then applies an equation discovery technique to extract from the neural potential a closed-form algebraic expression. We demonstrate this approach for several systems, including oscillators, a central force problem, and a problem of two charged particles in a classical Coulomb potential. Through these test problems, we show close agreement between learned neural potentials, the interpreted potentials we obtain after training, and the ground truth. In particular, for the central force problem, we show that our approach learns the correct effective potential, a reduced-order model of the system.