A Variational Physics-Informed Neural Network Framework Using Petrov-Galerkin Method for Solving Singularly Perturbed Boundary Value Problems
This addresses numerical stability and accuracy issues in computational physics for researchers dealing with challenging PDEs, though it is incremental as it builds on existing VPINN methods.
The paper tackled solving singularly perturbed boundary value problems and parabolic PDEs by proposing a Variational Physics-Informed Neural Network framework that integrates the Petrov-Galerkin method with neural networks, resulting in significantly improved accuracy in L2 and maximum norms compared to standard VPINN approaches.
This work proposes a Variational Physics-Informed Neural Network (VPINN) framework that integrates the Petrov-Galerkin formulation with deep neural networks (DNNs) for solving one-dimensional singularly perturbed boundary value problems (BVPs) and parabolic partial differential equations (PDEs) involving one or two small parameters. The method adopts a nonlinear approximation in which the trial space is defined by neural network functions, while the test space is constructed from hat functions. The weak formulation is constructed using localized test functions, with interface penalty terms introduced to enhance numerical stability and accurately capture boundary layers. Dirichlet boundary conditions are imposed via hard constraints, and source terms are computed using automatic differentiation. Numerical experiments on benchmark problems demonstrate the effectiveness of the proposed method, showing significantly improved accuracy in both the $L_2$ and maximum norms compared to the standard VPINN approach for one-dimensional singularly perturbed differential equations (SPDEs).