Accelerating Physics-Informed Neural Network Training with Prior Dictionaries

TL;DR

This paper introduces PD-PINNs, a variant of Physics-Informed Neural Networks that incorporates prior dictionaries to improve convergence speed and provides theoretical error bounds for solving elliptic PDEs.

Contribution

The paper proposes PD-PINNs, integrating task-specific dictionaries to enhance representation and training efficiency, along with theoretical error bounds for elliptic PDEs.

Findings

PD-PINNs achieve faster convergence than traditional PINNs.

In numerical simulations, prior dictionaries significantly improve training speed.

Theoretical bounds relate neural network errors to PDE and boundary condition losses.

Abstract

Physics-Informed Neural Networks (PINNs) can be regarded as general-purpose PDE solvers, but it might be slow to train PINNs on particular problems, and there is no theoretical guarantee of corresponding error bounds. In this manuscript, we propose a variant called Prior Dictionary based Physics-Informed Neural Networks (PD-PINNs). Equipped with task-dependent dictionaries, PD-PINNs enjoy enhanced representation power on the tasks, which helps to capture features provided by dictionaries so that the proposed neural networks can achieve faster convergence in the process of training. In various numerical simulations, compared with existing PINN methods, combining prior dictionaries can significantly enhance convergence speed. In terms of theory, we obtain the error bounds applicable to PINNs and PD-PINNs for solving elliptic partial differential equations of second order. It is proved that under certain mild conditions, the prediction error made by neural networks can be bounded by expected loss of PDEs and boundary conditions.