Numerical Approaches for Identifying the Time-Dependent Potential Coefficient in the Diffusion Equation

TL;DR

This paper investigates the inverse problem of determining a time-dependent potential in a diffusion equation, establishing theoretical results and comparing three numerical methods including a neural network approach.

Contribution

It introduces and compares three numerical methods for solving the inverse problem, including a novel physics-informed neural network technique.

Findings

All methods accurately recover the potential in noiseless data.

The methods demonstrate robustness against noisy measurements.

The neural network approach offers competitive efficiency and flexibility.

Abstract

We address the inverse problem of identifying a time-dependent potential coefficient in a one-dimensional diffusion equation subject to Dirichlet boundary conditions and a nonlocal integral overdetermination constraint reflecting spatially averaged measurements. After establishing well-posedness for the forward problem and deriving an a priori estimate that ensures uniqueness and continuous dependence on the data, we prove existence and uniqueness for the inverse problem. To compute numerically the unknown coefficient, we propose and compare three numerical methods: an integration-based scheme, a Newton-Raphson iterative solver, and a physics-informed neural network (PINN). Numerical experiments on both exact and noisy data demonstrate the accuracy, robustness, and efficiency of each approach.