aiqbench analysis
This research advances scientific machine learning by training multilayer perceptrons to approximate nonlinear finite element solution operators for partial differential equations. By leveraging the PDE energy functional as a loss function, the method enables efficient, parallelizable training that can accelerate or enhance standard computational workflows. The study demonstrates practical benefits in solving complex parametrized problems where traditional methods are computationally expensive.
Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimization
arXiv:2412.04596v3 Announce Type: replace Abstract: We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution operator by a multilayer perceptron (MLP) that takes problem data variables as input and gives a prediction of the finite element solution as output. The variables will typically correspond to parameters in a parametrization of input data such as boundary conditions, coefficients, and right-hand sides. The output will be an approximation of the corresponding finite element solution, thus enabling support and enhancement by the standard finite element method (FEM) both theoretically and practically. We focus on using the corresponding energy functional to a PDE as the loss function since this allows us formulate efficient parallelizable training algorithms based on assembling the energy locally on each element. For large problems, the learning process can thus be made more efficient by using only a small fraction of randomly chosen elements in the mesh in each iteration. The method is evaluated on several relevant test cases, where learning the finite element solution operator turns out to be beneficial, both in its own right but also by combination with standard FEM theory and software.
Abstract:We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution operator by a multilayer perceptron (MLP) that takes problem data variables as input and gives a prediction of the finite element solution as output. The variables will typically correspond to parameters in a parametrization of input data such as boundary conditions, coefficients, and right-hand sides. The output will be an approximation of the corresponding finite element solution, thus enabling support and enhancement by the standard finite element method (FEM) both theoretically and practically. We focus on using the corresponding energy functional to a PDE as the loss function since this allows us formulate efficient parallelizable training algorithms based on assembling the energy locally on each element. For large problems, the learning process can thus be made more efficient by using only a small fraction of randomly chosen elements in the mesh in each iteration. The method is evaluated on several relevant test cases, where learning the finite element solution operator turns out to be beneficial, both in its own right but also by combination with standard FEM theory and software.
Author's Accepted Manuscript (AAM) of article published in SIAM Journal on Scientific Computing. 30 pages, 11 figures (26 subfigures in total)
Machine Learning (cs.LG); Numerical Analysis (math.NA)
G.1.8; I.2.6; J.2
arXiv:2412.04596 [cs.LG]
(or arXiv:2412.04596v3 [cs.LG] for this version)
M. G. Larson, C. Lundholm, and A. Persson, Learning Nonlinear Finite Element Solution Operators Using Multilayer Perceptrons and Energy Minimization, SIAM Journal on Scientific Computing, 48 (2026), C1192-C1219
https://doi.org/10.1137/25M1776007 DOI(s) linking to related resources
Full text under the publisher's open licence · 276 words
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