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

Matteo Calafà, Emil Hovad, Allan P. Engsig-Karup, Tito Andriollo

TL;DR

The paper introduces physics-informed holomorphic neural networks (PIHNNs) to solve 2D linear elasticity problems by leveraging the Kolosov-Muskhelishvili holomorphic representation, reducing the solution to learning two holomorphic functions. By enforcing holomorphic outputs and using boundary-only loss terms, PIHNNs achieve fast training with low memory, while providing smooth, $C^{\infty}$-regular solutions and facilitating domain decomposition for multiply-connected domains. A universal approximation theorem for holomorphic networks justifies the representational capacity, and a tailored complex-valued weight initialization stabilizes training. Benchmark experiments on rings, plates with holes, and irregular BCs demonstrate higher accuracy and efficiency than standard real-valued PINNs, and domain decomposition effectively handles complex geometries with modest overhead. The approach offers a compact, interpretable framework for elasticity problems that can be extended to other holomorphic-representable PDEs and to higher dimensions with future work.

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.

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

TL;DR

The paper introduces physics-informed holomorphic neural networks (PIHNNs) to solve 2D linear elasticity problems by leveraging the Kolosov-Muskhelishvili holomorphic representation, reducing the solution to learning two holomorphic functions. By enforcing holomorphic outputs and using boundary-only loss terms, PIHNNs achieve fast training with low memory, while providing smooth, -regular solutions and facilitating domain decomposition for multiply-connected domains. A universal approximation theorem for holomorphic networks justifies the representational capacity, and a tailored complex-valued weight initialization stabilizes training. Benchmark experiments on rings, plates with holes, and irregular BCs demonstrate higher accuracy and efficiency than standard real-valued PINNs, and domain decomposition effectively handles complex geometries with modest overhead. The approach offers a compact, interpretable framework for elasticity problems that can be extended to other holomorphic-representable PDEs and to higher dimensions with future work.

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 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.
Paper Structure (20 sections, 1 theorem, 46 equations, 10 figures, 3 tables, 1 algorithm)

This paper contains 20 sections, 1 theorem, 46 equations, 10 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Governing equations : complex representation of linear elasticity equations
  3. Holomorphic Neural Networks
  4. Universal approximation theorem
  5. Neural Network architecture
  6. Activation functions
  7. Overall structure
  8. Definition of loss function
  9. Weight initialization
  10. Benchmark examples
  11. Simply-connected domain with regular BCs
  12. Circular ring under uniform pressure
  13. Plate with circular hole under uniaxial tension
  14. Simply-connected domain with irregular BCs
  15. Clamped plate under shear
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\phi \in H(\mathbb{C})$ be an entire non-polynomial activation function. Furthermore, let $D\subset \mathbb{C}$ be a complex simply-connected domain. Then, a 1-layer neural network with activation function $\phi$ satisfies the universal approximation property compactly on the space of $H(D)$ fu where

Figures (10)

  • Figure 1: Surface plots of two possible activation functions, exponential (top) and 1/2-order $-\cos(\sqrt{z})$ (bottom). Both functions are entire and monotonic in the neighborhood of the origin. It can be noted the growth of $-\cos(\sqrt{z})$ is slower compared to that of the exponential.
  • Figure 2: Simplified diagram of the proposed network (vanilla version). The input contains a batch of complex coordinates from points on the boundary of the domain and feeds two independent fully-connected networks. Once the holomorphic functions $\varphi,\psi$ are computed, automatic differentiation and Kolosov-Muskhelishvili formulae allow to obtain stresses and displacements.
  • Figure 3: Ring subjected to uniform negative pressure on the outer boundary. \ref{['fig:geometry_test1']}: Geometry and boundary conditions. \ref{['fig:test1_loss']}: training loss (red) and test loss (blue) during the training of the HNN. \ref{['fig:test1_phiR_exact']},\ref{['fig:test1_phiI_exact']},\ref{['fig:test1_psiR_exact']},\ref{['fig:test1_phiI_exact']},\ref{['fig:test1_psiI_exact']}: real and imaginary parts of the analytical functions from \ref{['eq:ring_exact_sol']}. \ref{['fig:test1_phiR']},\ref{['fig:test1_phiI']},\ref{['fig:test1_psiR']},\ref{['fig:test1_psiI']}: corresponding solutions obtained from the training of the network.\ref{['fig:test1_phi_error']},\ref{['fig:test1_psi_error']}: errors as difference of learned and exact solutions. To highlight the comparison, learned and exact solutions have the same color ranges.
  • Figure 4: Sampled training points for the test in \ref{['fig:results_test2']}. As explained in \ref{['sec:definitionofcostfunction']}, the loss function is computed from a finite set of uniformly sampled coordinates. Specifically, the number of sampled points on each edge is proportional to its length.
  • Figure 5: Plate with circular hole under uniaxial tension. \ref{['fig:geometry_test2']}: schematic of the geometry and BCs. \ref{['fig:test2_loss']}: training loss (red) and test loss (blue) during the training of the HNN. The same test is run with $\beta=\beta_1$ (black) and $\beta=\beta_3$ (gray). \ref{['fig:test2_xx_exact']},\ref{['fig:test2_yy_exact']},\ref{['fig:test2_xy_exact']}: numerical stresses obtained from the finite element method. \ref{['fig:test2_xx']},\ref{['fig:test2_xy']},\ref{['fig:test2_xy']}: stresses obtained from the training of the network. \ref{['fig:test2_xx_error']},\ref{['fig:test2_yy_error']},\ref{['fig:test2_xy_error']}: errors as difference of learned and numerical solutions. To highlight the comparison, learned and numerical stresses have the same color ranges. Stress units are MPa.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Remark 1