The Hadamard-PINN for PDE inverse problems: Convergence with distant initial guesses

Abstract

This paper presents the Hadamard-Physics-Informed Neural Network (H-PINN) for solving inverse problems in partial differential equations (PDEs), specifically the heat equation and the Korteweg–de Vries (KdV) equation. H-PINN addresses challenges in convergence and accuracy when initial parameter guesses are far from their actual values. The training process is divided into two phases: data fitting and parameter optimization. This phased approach is based on Hadamard’s conditions for well-posed problems, which emphasize that the uniqueness of a solution relies on the specified initial and boundary conditions. The model is trained using the Adam optimizer, along with a combined learning rate scheduler. To ensure reliability and consistency, we repeated each numerical experiment five times across three different initial guesses. Results showed significant improvements in parameter accuracy compared to the standard PINN, highlighting H-PINN’s effectiveness in scenarios with substantial deviations in initial guesses.