RBF-RLS outperforms PINNs on PDEs with Dirac deltas via weak-form integration, delivering consistent forward and inverse solutions for linear transport problems in porous media and rivers.
The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems
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abstract
We propose a deep learning based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz method is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions. The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning. We illustrate the method on several problems including some eigenvalue problems.
years
2026 2verdicts
UNVERDICTED 2representative citing papers
Random neural networks achieve a dimension-free approximation rate of 1/2 for sufficiently regular time-dependent Sobolev functions and can efficiently approximate solutions to Porous Medium Equations and Compressible Navier-Stokes Equations.
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Physics-Informed Neural Networks and Radial Basis Functions for PDEs with Dirac Delta Sources
RBF-RLS outperforms PINNs on PDEs with Dirac deltas via weak-form integration, delivering consistent forward and inverse solutions for linear transport problems in porous media and rivers.
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Random Neural Network Expressivity for Non-Linear Partial Differential Equations
Random neural networks achieve a dimension-free approximation rate of 1/2 for sufficiently regular time-dependent Sobolev functions and can efficiently approximate solutions to Porous Medium Equations and Compressible Navier-Stokes Equations.