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

2025/05/15 by Xiaoyu Wang, Long Yuan, Wang, Xiaoyu +3
Computer Science · Physics and Astronomy · #65N30 #65N55 #68T07 #FOS: Mathematics #Model Reduction and Neural Networks #Neural Networks and Applications #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2505.09911

openalex publication_date 2025/05/15 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

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.

Citations

• A discontinuous plane wave neural network method for Helmholtz equation and time-harmonic Maxwell's equations
• Physics-Informed Neural Networks for Heat Transfer Problems
• Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control
• Variational Physics-Informed Neural Networks For Solving Partial Differential Equations
• Plane Wave Discontinuous Galerkin methods for the Helmholtz equation and Maxwell equations in Anisotropic Media
• Topological properties of the set of functions generated by neural networks of fixed size

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