The Neural Network Approach to Inverse Problems in Differential Equations

TL;DR

This paper introduces a neural network framework for solving inverse differential equations, providing error analysis, sensitivity analysis, and demonstrating the advantages of neural networks in this context.

Contribution

It presents a novel neural network-based approach for inverse problems in differential equations, including error estimation and automatic differentiation for sensitivity analysis.

Findings

Numerical examples confirm the convergence and error predictions.

The framework effectively utilizes neural networks' approximation and regularization capabilities.

Automatic differentiation simplifies gradient computations in inverse problems.

Abstract

We proposed a framework for solving inverse problems in differential equations based on neural networks and automatic differentiation. Neural networks are used to approximate hidden fields. We analyze the source of errors in the framework and derive an error estimate for a model diffusion equation problem. Besides, we propose a way for sensitivity analysis, utilizing the automatic differentiation mechanism embedded in the framework. It frees people from the tedious and error-prone process of deriving the gradients. Numerical examples exhibit consistency with the convergence analysis and error saturation is noteworthily predicted. We also demonstrate the unique benefits neural networks offer at the same time: universal approximation ability, regularizing the solution, bypassing the curse of dimensionality and leveraging efficient computing frameworks.