Finite Expression Method With TranNet‐Based Function Learning for High‐Dimensional Partial Differential Equations
- 21 August 2026
- journal article
- research article
- Published by Wiley in Numerical Methods for Partial Differential Equations
- Vol. 42 (5) , e70139
Abstract
In this article, we study a machine‐learning‐based solver for high‐dimensional partial differential equations (PDEs). Computing accurate solutions efficiently for such problems remains challenging because of the curse of dimensionality, which severely limits the scalability of classical numerical methods. Our approach builds on the recently developed finite expression method (FEX), which approximates PDE solutions in a function space generated by finitely many analytic expressions. This framework has been shown to achieve high, and in some cases machine‐level, accuracy with polynomial memory complexity and favorable computational cost. We propose an extension of FEX in which the functional pool is generated by shallow neural network operators whose parameters are initialized using the transferable neural network method TransNet. Numerical experiments suggest that the proposed extension is an effective alternative for solving several high‐dimensional PDEs.Keywords
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Funding Information
- U.S. Department of Energy (DE‐SC0025412)
- U.S. Department of Energy (DE‐SC0024703)
- National Science Foundation (DMS‐2142672)
This publication has 38 references indexed in Scilit:
- A multiple transferable neural network method with domain decomposition for elliptic interface problemsJournal of Computational Physics, 2025
- Transferable Neural Networks for Partial Differential EquationsJournal of Scientific Computing, 2024
- Reinforcement learning for combinatorial optimization: A surveyComputers & Operations Research, 2021
- Nonlinear Approximation and (Deep) $\mathrm {ReLU}$ NetworksConstructive Approximation, 2021
- Transfer learning based multi-fidelity physics informed deep neural networkJournal of Computational Physics, 2021
- Weak adversarial networks for high-dimensional partial differential equationsJournal of Computational Physics, 2020
- Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equationsJournal of Computational Physics, 2019
- DGM: A deep learning algorithm for solving partial differential equationsJournal of Computational Physics, 2018
- Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential EquationsCommunications in Mathematics and Statistics, 2017
- Practical Methods of Optimization2000