Physics-Informed Learning Machines for Partial Differential Equations: Gaussian Processes Versus Neural Networks

Abstract

We review and compare physics-informed learning models built upon Gaussian processes and deep neural networks for solving forward and inverse problems governed by linear and nonlinear partial differential equations. We define a unified data model on which Gaussian processes, physics-informed Gaussian processes, neural networks, and physics-informed neural networks are based. We develop continuous-time and discrete-time models to facilitate different application scenarios. We present a connection between a Gaussian process and an infinitely wide neural network, which enables us to obtain a ``best'' kernel, which is determined directly by the data. We demonstrate the implementation of physics-informed Gaussian processes and physics-informed neural networks using a pedagogical example. Additionally, we compare physics-informed Gaussian processes and physics-informed neural networks for two nonlinear partial differential equations, i.e. the 1D Burgers' equation and the 2D Navier--Stokes, and provide guidance in choosing the proper machine learning model according to the problem type, i.e. forward or inverse problem, and the availability of data. These new methods for solving partial differential equations governing multi-physics problems do not require any grid, and they are simple to implement, and agnostic to specific application. Hence, we expect that variants and proper extensions of these methods will find broad applicability in the near future across different scientific disciplines but also in industrial applications.