# Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domains

arXiv: https://arxiv.org/abs/2510.22497 · HTML: https://mlsanity.com/paper/2510.22497/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 43 references that are in the map.
- First posted on arXiv 26 Oct 2025.

## Builds on (most cited first, 27 of 27)

- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains](https://mlsanity.com/paper/2007.11207/) (2020)
- [Neural Network Approximation: Three Hidden Layers Are Enough](https://mlsanity.com/paper/2010.14075/) (2020)
- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory](https://mlsanity.com/paper/2006.15733/) (2020)
- [Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons](https://mlsanity.com/paper/2107.02397/) (2021)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [Can Shallow Neural Networks Beat the Curse of Dimensionality? A mean field training perspective](https://mlsanity.com/paper/2005.10815/) (2020)
- [Finite Expression Method for Solving High-Dimensional Partial Differential Equations](https://mlsanity.com/paper/2206.10121/) (2022)
- [Reproducing Activation Function for Deep Learning](https://mlsanity.com/paper/2101.04844/) (2021)
- [Deep Operator Learning Lessens the Curse of Dimensionality for PDEs](https://mlsanity.com/paper/2301.12227/) (2023)
- [Solving PDEs on Unknown Manifolds with Machine Learning](https://mlsanity.com/paper/2106.06682/) (2021)
- [Residual-based error bound for physics-informed neural networks](https://mlsanity.com/paper/2306.03786/) (2023)
- [A Finite Expression Method for Solving High-Dimensional Committor Problems](https://mlsanity.com/paper/2306.12268/) (2023)
- [Temporal Difference Learning for High-Dimensional PIDEs with Jumps](https://mlsanity.com/paper/2307.02766/) (2023)
- [A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks](https://mlsanity.com/paper/2105.03480/) (2021)
- [Convergence Analysis of the Deep Galerkin Method for Weak Solutions](https://mlsanity.com/paper/2302.02405/) (2023)
- [Solving High-Dimensional Partial Integral Differential Equations: The Finite Expression Method](https://mlsanity.com/paper/2410.00835/) (2024)
- [Optimal Neural Network Approximation for High-Dimensional Continuous Functions](https://mlsanity.com/paper/2409.02363/) (2024)
- [Curse of Dimensionality in Neural Network Optimization](https://mlsanity.com/paper/2502.05360/) (2025)

## Abstract (arXiv)

Solving partial differential equations (PDEs) with highly oscillatory solutions on complex domains remains a challenging and important problem. High-frequency oscillations and intricate geometries often result in prohibitively expensive representations for traditional numerical methods and lead to difficult optimization landscapes for machine learning-based approaches. In this work, we introduce an enhanced Finite Expression Method (FEX) designed to address these challenges with improved accuracy, interpretability, and computational efficiency. The proposed framework incorporates three key innovations: a symbolic spectral composition module that enables FEX to learn and represent multiscale oscillatory behavior; a redesigned linear input layer that significantly expands the expressivity of the model; and an eigenvalue formulation that extends FEX to a new class of problems involving eigenvalue PDEs. Through extensive numerical experiments, we demonstrate that FEX accurately resolves oscillatory PDEs on domains containing multiple holes of varying shapes and sizes. Compared with existing neural network-based solvers, FEX achieves substantially higher accuracy while yielding interpretable, closed-form solutions that expose the underlying structure of the problem. These advantages, often absent in conventional finite element, finite difference, and black-box neural approaches, highlight FEX as a powerful and transparent framework for solving complex PDEs.