DAS: A deep adaptive sampling method for solving partial differential equations

12/28/2021
by   Kejun Tang, et al.
17

In this work we propose a deep adaptive sampling (DAS) method for solving partial differential equations (PDEs), where deep neural networks are utilized to approximate the solutions of PDEs and deep generative models are employed to generate new collocation points that refine the training set. The overall procedure of DAS consists of two components: solving the PDEs by minimizing the residual loss on the collocation points in the training set and generating a new training set to further improve the accuracy of current approximate solution. In particular, we treat the residual as a probability density function and approximate it with a deep generative model, called KRnet. The new samples from KRnet are consistent with the distribution induced by the residual, i.e., more samples are located in the region of large residual and less samples are located in the region of small residual. Analogous to classical adaptive methods such as the adaptive finite element, KRnet acts as an error indicator that guides the refinement of the training set. Compared to the neural network approximation obtained with uniformly distributed collocation points, the developed algorithms can significantly improve the accuracy, especially for low regularity and high-dimensional problems. We present a theoretical analysis to show that the proposed DAS method can reduce the error bound and demonstrate its effectiveness with numerical experiments.

READ FULL TEXT

page 16

page 18

page 19

page 21

page 22

07/21/2022

A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

Physics-informed neural networks (PINNs) have shown to be an effective t...
01/14/2020

SelectNet: Self-paced Learning for High-dimensional Partial Differential Equations

The residual method with deep neural networks as function parametrizatio...
12/28/2021

Active Learning Based Sampling for High-Dimensional Nonlinear Partial Differential Equations

The deep-learning-based least squares method has shown successful result...
04/03/2022

Implicit-Explicit Error Indicator based on Approximation Order

With the immense computing power at our disposal, the numerical solution...
01/05/2021

A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations

This paper concerns the a priori generalization analysis of the Deep Rit...
06/15/2022

Priori Error Estimate of Deep Mixed Residual Method for Elliptic PDEs

In this work, we derive a priori error estimate of the mixed residual me...
06/04/2019

Encoding Invariances in Deep Generative Models

Reliable training of generative adversarial networks (GANs) typically re...