Taylor-Model Physics-Informed Neural Networks (PINNs) for Ordinary Differential Equations

TL;DR

This paper introduces higher-order physics-informed neural networks (PINNs) that incorporate symbolic differentiation and Taylor series to improve the accuracy of neural models for solving parametric ODEs, especially under uncertainty.

Contribution

The paper proposes a novel class of higher-order PINNs combining neural networks with symbolic Taylor series and Lie derivatives to better model solutions of parametric ODEs.

Findings

Higher-order PINNs improve accuracy on challenging ODE benchmarks.

The models effectively handle parametric uncertainties in physical systems.

The approach is useful for control applications involving uncertain ODEs.

Abstract

We study the problem of learning neural network models for Ordinary Differential Equations (ODEs) with parametric uncertainties. Such neural network models capture the solution to the ODE over a given set of parameters, initial conditions, and range of times. Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for learning such models that combine data-driven deep learning with symbolic physics models in a principled manner. However, the accuracy of PINNs degrade when they are used to solve an entire family of initial value problems characterized by varying parameters and initial conditions. In this paper, we combine symbolic differentiation and Taylor series methods to propose a class of higher-order models for capturing the solutions to ODEs. These models combine neural networks and symbolic terms: they use higher order Lie derivatives and a Taylor series expansion obtained symbolically, with the remainder term modeled as a neural network. The key insight is that the remainder term can itself be modeled as a solution to a first-order ODE. We show how the use of these higher order PINNs can improve accuracy using interesting, but challenging ODE benchmarks. We also show that the resulting model can be quite useful for situations such as controlling uncertain physical systems modeled as ODEs.