Exact conservation laws for neural network integrators of dynamical systems

TL;DR

This paper introduces a neural network architecture that inherently incorporates conservation laws using Noether's Theorem, improving the prediction accuracy for dynamical systems with known conserved quantities.

Contribution

It presents a novel neural network design that enforces conservation laws exactly, leveraging Noether's Theorem, which enhances modeling of physical systems with known invariants.

Findings

Improved prediction accuracy for dynamical systems with conservation laws.

Exact enforcement of conservation laws in neural network models.

Demonstrated effectiveness on three physical system models.

Abstract

The solution of time dependent differential equations with neural networks has attracted a lot of attention recently. The central idea is to learn the laws that govern the evolution of the solution from data, which might be polluted with random noise. However, in contrast to other machine learning applications, usually a lot is known about the system at hand. For example, for many dynamical systems physical quantities such as energy or (angular) momentum are exactly conserved. Hence, the neural network has to learn these conservation laws from data and they will only be satisfied approximately due to finite training time and random noise. In this paper we present an alternative approach which uses Noether's Theorem to inherently incorporate conservation laws into the architecture of the neural network. We demonstrate that this leads to better predictions for three model systems: the motion of a non-relativistic particle in a three-dimensional Newtonian gravitational potential, the motion of a massive relativistic particle in the Schwarzschild metric and a system of two interacting particles in four dimensions.