Pith. sign in

REVIEW 6 cited by

Multi-scale Deep Neural Networks for Solving High Dimensional PDEs

Not yet reviewed by Pith; the record is open.

Re-run · record.json Download PDF Read on arXiv ↗

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

arxiv 1910.11710 v1 pith:5I65HYWT submitted 2019-10-25 cs.LG cs.NAmath.NAstat.ML

Wei Cai , Zhi-Qin John Xu This is my paper · ORCID
classification cs.LGcs.NAmath.NAstat.ML
keywords highdimensionalmulti-scalefrequencyfunctionspdesmscalednnactivation
verification ladder T0 review T1 audit T2 compute T3 formal

Signed reviews

No signed human review yet.

0 comments
read the original abstract

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.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Deep learning for the semi-classical limit of the Schr\"odinger equation

    physics.comp-ph 2025-08 conditional novelty 6.0 of 10

    Gaussian wave packet reduction plus MscaleDNNs and physics-informed DeepONets solves the semi-classical Schrödinger equation and learns the initial-to-solution map, with MscaleDNNs one to two orders more accurate than PINNs.

  2. KKANs: Kurkova-Kolmogorov-Arnold Networks and Their Learning Dynamics

    cs.LG 2024-12 conditional novelty 6.0 of 10

    KKANs, a two-block KART-based architecture with MLP inner functions and basis-function outer functions, universally approximate continuous functions and empirically outperform MLP and cKAN baselines in regression, PIN...

  3. ATHENA: Agentic Team for Hierarchical Evolutionary Numerical Algorithms

    cs.LG 2025-12 unverdicted novelty 5.0 of 10

    ATHENA introduces an agentic team framework that autonomously manages the end-to-end computational research lifecycle via a knowledge-driven HENA loop to achieve validation errors of 10^{-14} in scientific computing a...

  4. Neural Multiscale Decomposition for Solving The Nonlinear Klein-Gordon Equation with Time Oscillation

    math.NA 2025-11 reject novelty 5.0 of 10

    NeuralMD solves the oscillatory NKGE by training one network on the slow NLSW envelope and another on the remainder, but its model-selection step requires the exact solution as ground truth.

  5. A Kolmogorov High Order Deep Neural Network for High Frequency Partial Differential Equations in High Dimensions

    math.NA 2025-02 conditional novelty 5.0 of 10

    K-HOrderDNN approximates univariate KST components with high-order networks, reducing basis count from (p+1)^d to about d(p+1) and showing strong accuracy on high-frequency, high-dimensional PDE tests.

  6. On understanding and overcoming spectral biases of deep neural network learning methods for solving PDEs

    math.NA 2025-01 conditional

    A survey of methods to overcome spectral bias in deep neural network solvers for PDEs, with an emphasis on the authors' own MscaleDNN and PhaseDNN approaches.

Pith tools