mlsanityData to 13 Oct 2026

NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEs

arXiv 2110.01601 ↗First posted 4 Oct 2021cs.LG cs.NA math.NAcited by 10

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Abstract (arXiv)

We consider a mesh-based approach for training a neural network to produce field predictions of solutions to parametric partial differential equations (PDEs). This approach contrasts current approaches for "neural PDE solvers" that employ collocation-based methods to make point-wise predictions of solutions to PDEs. This approach has the advantage of naturally enforcing different boundary conditions as well as ease of invoking well-developed PDE theory -- including analysis of numerical stability and convergence -- to obtain capacity bounds for our proposed neural networks in discretized domains. We explore our mesh-based strategy, called NeuFENet, using a weighted Galerkin loss function based on the Finite Element Method (FEM) on a parametric elliptic PDE. The weighted Galerkin loss (FEM loss) is similar to an energy functional that produces improved solutions, satisfies a priori mesh convergence, and can model Dirichlet and Neumann boundary conditions. We prove theoretically, and illustrate with experiments, convergence results analogous to mesh convergence analysis deployed in finite element solutions to PDEs. These results suggest that a mesh-based neural network approach serves as a promising approach for solving parametric PDEs with theoretical bounds.

Key facts. Cited by 10 papers in the map (data through 13 Oct 2026). Builds on 35 of its 71 references that are in the map. First posted on arXiv 4 Oct 2021.

Builds on

35 of 71 references.

References · 35

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23 more
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Cited by year

Citing papers per year: 2022: 1; 2023: 1; 2024: 6; 2025: 2.

10 papers.

  1. 2024A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domainscited by 15
  2. 2024Learning smooth functions in high dimensions: from sparse polynomials to deep neural networkscited by 7
  3. 2022CAS4DL: Christoffel Adaptive Sampling for function approximation via Deep Learningcited by 6
  4. 2024Finite Element Neural Network Interpolation. Part I: Interpretable and Adaptive Discretization for Solving PDEscited by 3
  5. 2024Geometry Matters: Benchmarking Scientific ML Approaches for Flow Prediction around Complex Geometriescited by 3
  6. 2024Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticitycited by 2
  7. 20253D Neural Operator-Based Flow Surrogates around 3D geometries: Signed Distance Functions and Derivative Constraintscited by 2
  8. 2025DPG loss functions for learning parameter-to-solution maps by neural networkscited by 2
  9. 2024Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimizationcited by 1
  10. 2023A Deep Learning Framework for Solving Hyperbolic Partial Differential Equations: Part Icited by 0