# Difference between revisions of "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations"

Cameron Meaney

## Introduction

In recent years, there has been an enormous growth in the amount of data and computing power available to researchers. Unfortunately, for many real-world scenarios, the cost of data acquisition is simply too high to collect an amount of data sufficient to guarantee robustness or convergence of training algorithms. In such situations, researchers are faced with the challenge of trying to generate results based on partial or incomplete datasets. Regularization techniques or methods which can artificially inflate the dataset become particularly useful in these situations; however, such techniques are often highly dependent of the specifics of the problem.

Luckily, in important real-world scenarios that we endeavor to analyze, there is often a wealth of existing information from which we can draw. This existing information commonly manifests in the form of a mathematical model, particularly a set of partial differential equations (PDEs). In this paper, the authors provide a technique for incorporating the information of a physical system contained in a PDE into the optimization of a deep neural network. This technique is most useful in situations where established PDE models exist, but where our amount of available data is too small for neural network training. In essence, the accompanying PDE model can be used as a regularization agent, constraining the space of acceptable solutions to help the optimization converge more quickly and more accurately.

## Problem Setup

Consider the following general PDE

\begin{align*} u_t + N[u;\lambda] = 0 \end{align*}

where the $u$ is the function we wish to find, subscripts denote partial derivatives, and $\vec{\lambda}$ is the set of parameters on which the PDE depends. This general form encompasses a wide array of PDEs used across the physical sciences including conservation laws, diffusion processes, advection-diffusion-reaction systems, and kinetic equations. Suppose that we have noisy measurements of the PDE solution, $u$ scattered across the spatio-temporal input domain. Then, we are interested in answering two questions about the physical system:

(1) Given fixed model parameters $\vec{\lambda}$, what can be said about the unknown hidden state $u(t,x)$?

and

(2) What parameters $\vec{\lambda}$ best describe the observed data?