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

TL;DR

This work addresses the inverse problem of identifying a time-dependent potential $p(t)$ and the corresponding state $u(x,t)$ for a 1D diffusion equation with Dirichlet boundaries and a nonlocal integral overdetermination. It establishes well-posedness for the direct problem, derives a priori estimates, and proves existence and uniqueness for the inverse problem via a Schauder fixed-point framework. Three numerical strategies are developed and compared: an integration-based method, a Newton–Raphson solver, and a physics-informed neural network (PINN), with convergence and stability analyses for the forward problem. Numerical experiments on manufactured data, including noise, show that Newton–Raphson achieves the highest accuracy in noiseless settings, the integration-based approach is robust and simple but less accurate, and PINNs offer a flexible, mesh-free alternative with strong noise tolerance. The results highlight different trade-offs between accuracy, stability, and computational cost, suggesting potential for hybrid schemes that combine fast convergence with noise-robust mesh-free learning in more complex or higher-dimensional regimes.

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.

Figures (9)

• Figure 1: Architecture of the PINN for simultaneous approximation of $u(x,t)$ and $p(t)$.
• Figure 2: Analytical vs numerical solutions of $p(t)$ and $u(x, t)$ at $T=1$ using the Integration-based approach.
• Figure 3: Analytical vs numerical solutions of $p(t)$ and $u(x, t)$ at $T=1$ using the Newton--Raphson-based approach.
• Figure 4: Analytical vs numerical solutions of $p(t)$ and $u(x, t)$ at $T=1$ using PINN approach.
• Figure 5: Comparison of reconstructed $p(t)$ from all three approaches against the exact solution.

Theorems & Definitions (22)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Remark 2.4
• Theorem 2.5: Arzelà–Ascoli
• Definition 2.6: Newton–Raphson Method
• Definition 2.7: Physics-Informed Neural Network Raissi2019
• Theorem 3.1: Existence and Uniqueness
• proof
• Theorem 3.2: A priori estimate