Solving the inverse potential problem in the parabolic equation by the deep neural networks method

TL;DR

This paper introduces a deep neural network approach to solve an inverse potential problem in parabolic equations, proving uniqueness, designing a new loss function with regularization, and providing theoretical error estimates and numerical results.

Contribution

The work presents a novel DNN-based reconstruction scheme for inverse potential problems, including proof of uniqueness, a new loss function with regularization, and rigorous generalization error analysis.

Findings

The proposed method effectively reconstructs the unknown potential.

The new loss function improves regularity and stability in the inverse problem.

Numerical results demonstrate the method's accuracy and robustness.

Abstract

In this work, we consider an inverse potential problem in the parabolic equation, where the unknown potential is a space-dependent function and the used measurement is the final time data. The unknown potential in this inverse problem is parameterized by deep neural networks (DNNs) for the reconstruction scheme. First, the uniqueness of the inverse problem is proved under some regularities assumption on the input sources. Then we propose a new loss function with regularization terms depending on the derivatives of the residuals for partial differential equations (PDEs) and the measurements. These extra terms effectively induce higher regularity in solutions so that the ill-posedness of the inverse problem can be handled. Moreover, we establish the corresponding generalization error estimates rigorously. Our proofs exploit the conditional stability of the classical linear inverse source problems, and the mollification on the noisy measurement data which is set to reduce the perturbation errors. Finally, the numerical algorithm and some numerical results are provided.