DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations
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 demonstrate the effectiveness of the proposed DAS method with numerical experiments.
Code (1)
Similar Papers 제목 키워드 기반
Quasi-Random Physics-informed Neural Networks
Physics-informed neural networks have shown promise in solving partial differential equations (PDEs) by integrating physical constraints into neural network training, but their performance is sensitive to the sampling of…
RL-PINNs: Reinforcement Learning-Driven Adaptive Sampling for Efficient Training of PINNs
Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs). However, their performance heavily relies on the strategy used to select training points. C…
Reinforcement Learning (RL)GAS: A Gaussian Mixture Distribution-Based Adaptive Sampling Method for PINNs
With the recent study of deep learning in scientific computation, the Physics-Informed Neural Networks (PINNs) method has drawn widespread attention for solving Partial Differential Equations (PDEs). Compared to traditio…
Incremental LearningMonte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations
We introduce a sampling based machine learning approach, Monte Carlo physics informed neural networks (MC-PINNs), for solving forward and inverse fractional partial differential equations (FPDEs). As a generalization of …
MC-Nonlocal-PINNs: handling nonlocal operators in PINNs via Monte Carlo sampling
We propose, Monte Carlo Nonlocal physics-informed neural networks (MC-Nonlocal-PINNs), which is a generalization of MC-fPINNs in \cite{guo2022monte}, for solving general nonlocal models such as integral equations and non…