State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning

TL;DR

This review emphasizes the importance of experiment design in Physics-Informed Neural Networks (PINNs) for solving complex PDEs, demonstrating that sampling strategies significantly impact accuracy, with Hammersley sampling outperforming others.

Contribution

The paper provides a comprehensive comparison of different experiment design schemes for PINNs across five PDEs, highlighting the superiority of Hammersley sampling.

Findings

Hammersley sampling-based PINN outperforms other sampling strategies.

Experiment design significantly affects PINN accuracy.

PINNs can effectively replace traditional numerical methods for PDEs.

Abstract

This paper presents a comprehensive review of the design of experiments used in the surrogate models. In particular, this study demonstrates the necessity of the design of experiment schemes for the Physics-Informed Neural Network (PINN), which belongs to the supervised learning class. Many complex partial differential equations (PDEs) do not have any analytical solution; only numerical methods are used to solve the equations, which is computationally expensive. In recent decades, PINN has gained popularity as a replacement for numerical methods to reduce the computational budget. PINN uses physical information in the form of differential equations to enhance the performance of the neural networks. Though it works efficiently, the choice of the design of experiment scheme is important as the accuracy of the predicted responses using PINN depends on the training data. In this study, five different PDEs are used for numerical purposes, i.e., viscous Burger's equation, Shr\"{o}dinger equation, heat equation, Allen-Cahn equation, and Korteweg-de Vries equation. A comparative study is performed to establish the necessity of the selection of a DoE scheme. It is seen that the Hammersley sampling-based PINN performs better than other DoE sample strategies.