mlsanityData to 7 Oct 2026

Two-scale Neural Networks for Partial Differential Equations with Small Parameters

arXiv 2402.17232 ↗First posted 27 Feb 2024math.NA cs.LG cs.NAcited by 0

Not on the map. Why?

Abstract (arXiv)

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 incorporate the small parameters into the architecture of neural networks. The proposed method enables solving PDEs with small parameters in a simple fashion, without adding Fourier features or other computationally taxing searches of truncation parameters. Various numerical examples demonstrate reasonable accuracy in capturing features of large derivatives in the solutions caused by small parameters.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 26 of its 46 references that are in the map. First posted on arXiv 27 Feb 2024.

Builds on

26 of 46 references.

References · 26

  1. 2018On the Spectral Bias of Neural Networkscited by 1,321
  2. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  3. 2019Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networkscited by 360
  4. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
  5. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 233
  6. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
  7. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 220
  8. 2022Respecting causality is all you need for training physics-informed neural networkscited by 154
  9. 2020Frequency Bias in Neural Networks for Input of Non-Uniform Densitycited by 140
  10. 2017Sobolev Training for Neural Networkscited by 136
  11. 2019Robust Learning with Jacobian Regularizationcited by 127
  12. 2023An Expert's Guide to Training Physics-informed Neural Networkscited by 119
14 more
  1. 2021Efficient training of physics-informed neural networks via importance samplingcited by 109
  2. 2020Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domainscited by 78
  3. 2021DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equationscited by 66
  4. 2022Failure-informed adaptive sampling for PINNscited by 64
  5. 2021Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networkscited by 43
  6. 2019A Phase Shift Deep Neural Network for High Frequency Approximation and Wave Problemscited by 34
  7. 2023Stacked networks improve physics-informed training: applications to neural networks and deep operator networkscited by 21
  8. 2023Adversarial Adaptive Sampling: Unify PINN and Optimal Transport for the Approximation of PDEscited by 18
  9. 2023Failure-informed adaptive sampling for PINNs, Part II: combining with re-sampling and subset simulationcited by 13
  10. 2021Subspace Decomposition based DNN algorithm for elliptic type multi-scale PDEscited by 10
  11. 2022Asymptotic-Preserving Neural Networks for multiscale hyperbolic models of epidemic spreadcited by 10
  12. 2023Ensemble learning for Physics Informed Neural Networks: a Gradient Boosting approachcited by 10
  13. 2022A Dimension-Augmented Physics-Informed Neural Network (DaPINN) with High Level Accuracy and Efficiencycited by 7
  14. 2023Gradient and Uncertainty Enhanced Sequential Sampling for Global Fitcited by 2

Built on by

Not cited yet.