Physics-Informed Holomorphic Neural Networks (PIHNNs): Solving Linear Elasticity Problems

TL;DR

This paper introduces physics-informed holomorphic neural networks (PIHNNs) for solving linear elasticity boundary value problems, leveraging complex analysis to improve efficiency, accuracy, and applicability to complex geometries.

Contribution

The paper develops a novel PIHNN framework that inherently satisfies elasticity equations using holomorphic functions, with a universal approximation theorem and tailored initialization for enhanced training.

Findings

PIHNNs outperform standard PINNs in training efficiency.

PIHNNs require fewer boundary evaluations and less memory.

The method effectively handles complex, multiply-connected geometries.

Abstract

We propose physics-informed holomorphic neural networks (PIHNNs) as a method to solve boundary value problems where the solution can be represented via holomorphic functions. Specifically, we consider the case of plane linear elasticity and, by leveraging the Kolosov-Muskhelishvili representation of the solution in terms of holomorphic potentials, we train a complex-valued neural network to fulfill stress and displacement boundary conditions while automatically satisfying the governing equations. This is achieved by designing the network to return only approximations that inherently satisfy the Cauchy-Riemann conditions through specific choices of layers and activation functions. To ensure generality, we provide a universal approximation theorem guaranteeing that, under basic assumptions, the proposed holomorphic neural networks can approximate any holomorphic function. Furthermore, we suggest a new tailored weight initialization technique to mitigate the issue of vanishing/exploding gradients. Compared to the standard PINN approach, noteworthy benefits of the proposed method for the linear elasticity problem include a more efficient training, as evaluations are needed solely on the boundary of the domain, lower memory requirements, due to the reduced number of training points, and $C^\infty$ regularity of the learned solution. Several benchmark examples are used to verify the correctness of the obtained PIHNN approximations, the substantial benefits over traditional PINNs, and the possibility to deal with non-trivial, multiply-connected geometries via a domain-decomposition strategy.