Neural DAEs: Constrained neural networks

TL;DR

This paper explores how adding algebraic trajectory constraints to neural networks improves their performance in modeling dynamical systems, with practical methods tested on physical simulations.

Contribution

It introduces methods for incorporating algebraic constraints into neural networks, inspired by differential-algebraic equations, and evaluates their effectiveness in physical system simulations.

Findings

Constraint methods improve inference accuracy

Projection methods are effective and easy to implement

Limited impact on training performance

Abstract

This article investigates the effect of explicitly adding auxiliary algebraic trajectory information to neural networks for dynamical systems. We draw inspiration from the field of differential-algebraic equations and differential equations on manifolds and implement related methods in residual neural networks, despite some fundamental scenario differences. Constraint or auxiliary information effects are incorporated through stabilization as well as projection methods, and we show when to use which method based on experiments involving simulations of multi-body pendulums and molecular dynamics scenarios. Several of our methods are easy to implement in existing code and have limited impact on training performance while giving significant boosts in terms of inference.