Multi-scale Deep Neural Networks for Solving High Dimensional PDEs
arXiv:1910.11710
Abstract
In this paper, we propose the idea of radial scaling in frequency domain and activation functions with compact support to produce a multi-scale DNN (MscaleDNN), which will have the multi-scale capability in approximating high frequency and high dimensional functions and speeding up the solution of high dimensional PDEs. Numerical results on high dimensional function fitting and solutions of high dimensional PDEs, using loss functions with either Ritz energy or least squared PDE residuals, have validated the increased power of multi-scale resolution and high frequency capturing of the proposed MscaleDNN.
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- Implicit bias with Ritz-Galerkin method in understanding deep learning for solving PDEs
- BINet: Learning to Solve Partial Differential Equations with Boundary Integral Networks
- An Upper Limit of Decaying Rate with Respect to Frequency in Deep Neural Network
- A priori generalization error for two-layer ReLU neural network through minimum norm solution