Solving physics-based initial value problems with unsupervised machine learning

TL;DR

This paper introduces an unsupervised deep learning framework to solve physics-based initial value problems, effectively modeling complex mechanical systems and conserving physical properties without supervised data.

Contribution

The authors develop a novel neural network approach capable of solving nonlinear, coupled, and chaotic dynamical systems in an unsupervised manner, advancing physics-informed machine learning.

Findings

Successfully modeled classical mechanics systems including pendulums and celestial systems.

Neural networks conserved physical properties like energy and stationary action.

Demonstrated effectiveness on nonlinear and coupled dynamical systems.

Abstract

Initial value problems -- a system of ordinary differential equations and corresponding initial conditions -- can be used to describe many physical phenomena including those arise in classical mechanics. We have developed a novel approach to solve physics-based initial value problems using unsupervised machine learning. We propose a deep learning framework that models the dynamics of a variety of mechanical systems through neural networks. Our framework is flexible, allowing us to solve non-linear, coupled, and chaotic dynamical systems. We demonstrate the effectiveness of our approach on systems including a free particle, a particle in a gravitational field, a classical pendulum, and the H\'enon--Heiles system (a pair of coupled harmonic oscillators with a non-linear perturbation, used in celestial mechanics). Our results show that deep neural networks can successfully approximate solutions to these problems, producing trajectories which conserve physical properties such as energy and those with stationary action. We note that probabilistic activation functions, as defined in this paper, are required to learn any solutions of initial value problems in their strictest sense, and we introduce coupled neural networks to learn solutions of coupled systems.