The paper introduces MNO, an FMM-inspired neural operator that jointly maps PDE coefficients, source terms, and boundary conditions to the solution, and shows it works on 1D Poisson, Darcy flow, and a nonlinear BVP.
A Review of Neural Network Solvers for Second-order Boundary Value Problems
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abstract
Deep learning-based partial differential equation(PDE) solvers have received much attention in the past few years. Methods of this category can solve a wide range of PDEs with high accuracy, typically by transforming the problems into highly nonlinear optimization problems of neural network parameters. This work reviews several deep learning solvers proposed a few years ago, including PINN, WAN, DRM, and VPINN. Numerical results are provided to make comparisons amongst them and address the importance of loss formulation and the optimization method. A rigorous error analysis for PINN is also presented. Finally, we discuss the current limitations and bottlenecks of these methods.
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MNO : A Multi-modal Neural Operator for Parametric Nonlinear BVPs
The paper introduces MNO, an FMM-inspired neural operator that jointly maps PDE coefficients, source terms, and boundary conditions to the solution, and shows it works on 1D Poisson, Darcy flow, and a nonlinear BVP.