7 papers · 1 filter
Deep Neural networks for solving high-dimensional parabolic partial differential equations
Wenzhong Zhang, Zheyuan Hu, Wei Cai +1
The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based metho…
Two-level overlapping additive Schwarz preconditioner for training scientific machine learning applications
Youngkyu Lee, Alena KopaniÄáková, George Em Karniadakis
We introduce a novel two-level overlapping additive Schwarz preconditioner for accelerating the training of scientific machine learning applications. The design of the proposed pre…
Scalable Bayesian Physics-Informed Kolmogorov-Arnold Networks
Zhiwei Gao, George Em Karniadakis
Uncertainty quantification (UQ) plays a pivotal role in scientific machine learning, especially when surrogate models are used to approximate complex systems. Although multilayer p…
Tensor neural networks for high-dimensional Fokker-Planck equations
Taorui Wang, Zheyuan Hu, Kenji Kawaguchi +2
We solve high-dimensional steady-state Fokker-Planck equations on the whole space by applying tensor neural networks. The tensor networks are a linear combination of tensor product…
Two-scale Neural Networks for Partial Differential Equations with Small Parameters
Qiao Zhuang, Chris Ziyi Yao, Zhongqiang Zhang +1
We propose a two-scale neural network method for solving partial differential equations (PDEs) with small parameters using physics-informed neural networks (PINNs). We directly inc…
Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers
Enrui Zhang, Adar Kahana, Alena KopaniÄáková +4
Neural networks suffer from spectral bias having difficulty in representing the high frequency components of a function while relaxation methods can resolve high frequencies effici…