# Multi-scale Deep Neural Networks for Solving High Dimensional PDEs

arXiv: https://arxiv.org/abs/1910.11710 · HTML: https://mlsanity.com/paper/1910.11710/

## Key facts

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

## Built on by (most cited first, 25 of 25)

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- [Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2602.18515/) (2026)
- [Approximation of Solution Operators for High-dimensional PDEs](https://mlsanity.com/paper/2401.10385/) (2024)
- [Frequency-adaptive tensor neural networks for high-dimensional multi-scale problems](https://mlsanity.com/paper/2508.15198/) (2025)
- [Is the Frequency Principle always valid?](https://mlsanity.com/paper/2508.17323/) (2025)
- [Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches](https://mlsanity.com/paper/2604.16842/) (2026)

## 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.