# Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimization

arXiv: https://arxiv.org/abs/2412.04596 · HTML: https://mlsanity.com/paper/2412.04596/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 62 references that are in the map.
- First posted on arXiv 5 Dec 2024.

## Builds on (most cited first, 24 of 24)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [JAX-FEM: A differentiable GPU-accelerated 3D finite element solver for automatic inverse design and mechanistic data science](https://mlsanity.com/paper/2212.00964/) (2022)
- [Parameterized Physics-informed Neural Networks for Parameterized PDEs](https://mlsanity.com/paper/2408.09446/) (2024)
- [VarNet: Variational Neural Networks for the Solution of Partial Differential Equations](https://mlsanity.com/paper/1912.07443/) (2019)
- [A Physics-Guided Neural Network Framework for Elastic Plates: Comparison of Governing Equations-Based and Energy-Based Approaches](https://mlsanity.com/paper/2010.06050/) (2020)
- [MAgNET: A Graph U-Net Architecture for Mesh-Based Simulations](https://mlsanity.com/paper/2211.00713/) (2022)
- [Hybrid FEM-NN models: Combining artificial neural networks with the finite element method](https://mlsanity.com/paper/2101.00962/) (2021)
- [MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs](https://mlsanity.com/paper/2107.03673/) (2021)
- [Variationally Mimetic Operator Networks](https://mlsanity.com/paper/2209.12871/) (2022)
- [Data-Free Learning of Reduced-Order Kinematics](https://mlsanity.com/paper/2305.03846/) (2023)
- [An extended physics informed neural network for preliminary analysis of parametric optimal control problems](https://mlsanity.com/paper/2110.13530/) (2021)
- [Finite Element Method-enhanced Neural Network for Forward and Inverse Problems](https://mlsanity.com/paper/2205.08321/) (2022)
- [Physics-informed MeshGraphNets (PI-MGNs): Neural finite element solvers for non-stationary and nonlinear simulations on arbitrary meshes](https://mlsanity.com/paper/2402.10681/) (2024)
- [NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEs](https://mlsanity.com/paper/2110.01601/) (2021)
- [On the use of graph neural networks and shape-function-based gradient computation in the deep energy method](https://mlsanity.com/paper/2207.07216/) (2022)
- [Multilevel CNNs for Parametric PDEs](https://mlsanity.com/paper/2304.00388/) (2023)
- [Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations](https://mlsanity.com/paper/2304.04234/) (2023)
- [DeepPhysics: a physics aware deep learning framework for real-time simulation](https://mlsanity.com/paper/2109.09491/) (2021)

## Built on by (most cited first, 1 of 1)

- [Adaptive Fusion Graph Network for 3D Strain Field Prediction in Solid Rocket Motor Grains](https://mlsanity.com/paper/2512.23443/) (2025)

## Abstract (arXiv)

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.