Multi-scale Deep Neural Networks for Solving High Dimensional PDEs
This work addresses the computational bottleneck of solving high-dimensional PDEs, which is crucial for fields like physics and finance, though it appears incremental as it builds on existing DNN methods with specific modifications.
The paper tackled the challenge of solving high-dimensional PDEs by proposing a multi-scale deep neural network (MscaleDNN) that uses radial scaling and compact support activation functions to improve multi-scale resolution and high-frequency capture, with numerical results validating its effectiveness in high-dimensional function fitting and PDE solutions.
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.