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

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 a minimal number of parameters that are very accurate and fast to evaluate. In particular, we find that enriching the feature space using Airy stress functions can significantly improve the accuracy of PINN solutions for biharmonic PDEs.

Automated summary

We explore the application of Physics-Informed Neural Networks (PINNs) with Airy stress functions and Fourier series in finding optimal solutions to reference biharmonic problems in elasticity and elastic plate theory. Our work demonstrates a novel application of classical analytical methods in constructing efficient neural networks with minimal parameters, which are accurate and fast in evaluation. We find that enriching the feature space with Airy stress functions can significantly improve the accuracy of PINN solutions for biharmonic PDEs.