A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity

TL;DR

This paper demonstrates a novel approach combining Physics Informed Neural Networks with classical analytical methods like Airy stress functions to efficiently solve biharmonic equations in elasticity, achieving high accuracy with minimal parameters.

Contribution

It introduces a new method integrating Airy stress functions with PINNs to solve complex biharmonic PDEs more accurately and efficiently.

Findings

Enriching feature space with Airy stress functions improves PINN accuracy.

PINNs can effectively solve biharmonic PDEs with fewer parameters.

The approach outperforms traditional methods in speed and precision.

Abstract

We explore an application of the Physics Informed Neural Networks (PINNs) in conjunction with Airy stress functions and Fourier series to find optimal solutions to a few reference biharmonic problems of elasticity and elastic plate theory. Biharmonic relations are fourth-order partial differential equations (PDEs) that are challenging to solve using classical numerical methods, and have not been addressed using PINNs. Our work highlights a novel application of classical analytical methods to guide the construction of efficient neural networks with the minimal number of parameters that are very accurate and fast to evaluate. In particular, we find that enriching feature space using Airy stress functions can significantly improve the accuracy of PINN solutions for biharmonic PDEs.