mlsanityData to 14 Oct 2026

Multi-scale Deep Neural Networks for Solving High Dimensional PDEs

arXiv 1910.11710 ↗First posted 25 Oct 2019cs.LG cs.NA math.NAcited by 25

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Abstract (arXiv)

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.

Key facts. Cited by 25 papers in the map (data through 14 Oct 2026). First posted on arXiv 25 Oct 2019.

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Citing papers per year: 2021: 2; 2022: 5; 2023: 1; 2024: 7; 2025: 8; 2026: 2.

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  1. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  2. 2022Overview frequency principle/spectral bias in deep learningcited by 69
  3. 2024On the expressiveness and spectral bias of KANscited by 37
  4. 2025Mitigating Spectral Bias in Neural Operators via High-Frequency Scaling for Physical Systemscited by 31
  5. 2022On the Activation Function Dependence of the Spectral Bias of Neural Networkscited by 28
  6. 2024Addressing Spectral Bias of Deep Neural Networks by Multi-Grade Deep Learningcited by 14
  7. 2022BI-GreenNet: Learning Green's functions by boundary integral networkcited by 13
  8. 2024Learnable Activation Functions in Physics-Informed Neural Networks for Solving Partial Differential Equationscited by 7
  9. 2025ATHENA: Agentic Team for Hierarchical Evolutionary Numerical Algorithmscited by 7
  10. 2021Hierarchical Learning to Solve Partial Differential Equations Using Physics-Informed Neural Networkscited by 5
  11. 2022A Neural Network Approach for Homogenization of Multiscale Problemscited by 5
  12. 2023Neural Control of Parametric Solutions for High-dimensional Evolution PDEscited by 5
13 more
  1. 2021An Upper Limit of Decaying Rate with Respect to Frequency in Deep Neural Networkcited by 4
  2. 2025Fourier Multi-Component and Multi-Layer Neural Networks: Unlocking High-Frequency Potentialcited by 4
  3. 2025Physics-Informed Machine Learning in Biomedical Science and Engineeringcited by 4
  4. 2022Computing non-equilibrium trajectories by a deep learning approachcited by 3
  5. 2024Dilated convolution neural operator for multiscale partial differential equationscited by 3
  6. 2024Transfer Learning on Multi-Dimensional Data: A Novel Approach to Neural Network-Based Surrogate Modelingcited by 3
  7. 2025Sobolev acceleration for neural networkscited by 3
  8. 2025Towards Robust Surrogate Models: Benchmarking Machine Learning Approaches to Expediting Phase Field Simulations of Brittle Fracturecited by 1
  9. 2026Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equationscited by 1
  10. 2024Approximation of Solution Operators for High-dimensional PDEscited by 0
  11. 2025Frequency-adaptive tensor neural networks for high-dimensional multi-scale problemscited by 0
  12. 2025Is the Frequency Principle always valid?cited by 0
  13. 2026Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approachescited by 0