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Discontinuous hybrid neural networks for the one-dimensional partial differential equations

Xiaoyu Wang, Long Yuan, Yao Yu

TL;DR

This work introduces discontinuous hybrid neural networks (DHNN) for time-harmonic PDEs in one dimension, integrating a variational-type loss with interface jump and boundary constraints to improve accuracy and efficiency. Nonlinear parameters are updated with RMSprop while linear parameters are solved via discontinuous Galerkin in a staged algorithm, with an adaptive loss-balancing factor to maintain stability. The authors prove convergence in the loss functional and provide an a posteriori error estimate, then demonstrate DHNN on Poisson and Helmholtz problems, showing faster convergence and higher stability than VPINN, especially at large wavenumbers. The approach offers a scalable, accurate alternative for solving PDEs with discontinuous architectures and variational formulations, reducing computational cost while maintaining precision.

Abstract

A feedforward neural network, including hidden layers, motivated by nonlinear functions (such as Tanh, ReLU, and Sigmoid functions), exhibits uniform approximation properties in Sobolev space, and discontinuous neural networks can reduce computational complexity. In this work, we present a discontinuous hybrid neural network method for solving the partial differential equations, construct a new hybrid loss functional that incorporates the variational of the approximation equation, interface jump stencil and boundary constraints. The RMSprop algorithm and discontinuous Galerkin method are employed to update the nonlinear parameters and linear parameters in neural networks, respectively. This approach guarantees the convergence of the loss functional and provides an approximate solution with high accuracy.

Discontinuous hybrid neural networks for the one-dimensional partial differential equations

TL;DR

This work introduces discontinuous hybrid neural networks (DHNN) for time-harmonic PDEs in one dimension, integrating a variational-type loss with interface jump and boundary constraints to improve accuracy and efficiency. Nonlinear parameters are updated with RMSprop while linear parameters are solved via discontinuous Galerkin in a staged algorithm, with an adaptive loss-balancing factor to maintain stability. The authors prove convergence in the loss functional and provide an a posteriori error estimate, then demonstrate DHNN on Poisson and Helmholtz problems, showing faster convergence and higher stability than VPINN, especially at large wavenumbers. The approach offers a scalable, accurate alternative for solving PDEs with discontinuous architectures and variational formulations, reducing computational cost while maintaining precision.

Abstract

A feedforward neural network, including hidden layers, motivated by nonlinear functions (such as Tanh, ReLU, and Sigmoid functions), exhibits uniform approximation properties in Sobolev space, and discontinuous neural networks can reduce computational complexity. In this work, we present a discontinuous hybrid neural network method for solving the partial differential equations, construct a new hybrid loss functional that incorporates the variational of the approximation equation, interface jump stencil and boundary constraints. The RMSprop algorithm and discontinuous Galerkin method are employed to update the nonlinear parameters and linear parameters in neural networks, respectively. This approach guarantees the convergence of the loss functional and provides an approximate solution with high accuracy.
Paper Structure (18 sections, 3 theorems, 31 equations, 13 figures)

This paper contains 18 sections, 3 theorems, 31 equations, 13 figures.

Table of Contents

  1. Introduction
  2. The model and discontinuous neural networks
  3. The model
  4. Discontinuous neural networks
  5. A discontinuous hybrid neural network algorithm
  6. Updating linear parameters
  7. Updating nonlinear parameters
  8. Updating the loss balancing factor $\tau$
  9. Algorithm overview
  10. Convergence estimates
  11. Numerical examples
  12. Possion equations
  13. A homogeneous case
  14. An inhomogeneous case
  15. Helmholtz equations
  16. ...and 3 more sections

Key Result

Lemma 4.1

Suppose that $1\leq p <\infty$, $0\leq s <\infty$, and that $\Omega\subset R^d$ is compact, then $V_{n}^{\sigma}({\cal T}_h)$ is dense in $W^{s,p}({\cal T}_h):=\{v\in L^p(\Omega), D^\alpha v\in L^p(\Omega_k) \quad \forall |\alpha|\leq s,k=1,\cdots N\}$. In particular, for any given function $f\in W^

Figures (13)

  • Figure 1: The values of the loss functional for the DHNN. Left: inner iterations. Right: outer iterations.
  • Figure 2: The values of the loss functional for the DHNN. Left: inner iterations. Right: outer iterations.
  • Figure 3: Comparison of the loss functional values between the VPINN and the DHNN methods. Left: sigmoid activation function. Right: tanh activation function.
  • Figure 4: The point-wise error distributions for both methods. Left: sigmoid activation function. Right: tanh activation function.
  • Figure 5: The values of the loss functional.
  • ...and 8 more figures

Theorems & Definitions (3)

  • Lemma 4.1
  • Theorem 4.1
  • Theorem 4.2