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Engineering with Computers
Published in:

10-04-2024 | Original Article

NeuFENet: neural finite element solutions with theoretical bounds for parametric PDEs

Authors: Biswajit Khara, Aditya Balu, Ameya Joshi, Soumik Sarkar, Chinmay Hegde, Adarsh Krishnamurthy, Baskar Ganapathysubramanian

Published in: Engineering with Computers | Issue 5/2024

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Abstract

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 pointwise 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.
previous article Cut-PFEM: a Particle Finite Element Method using unfitted boundary meshes
next article Coupling of bond-based peridynamics and continuous density-based topology optimization methods for effective design of three-dimensional structures with discontinuities

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Appendix
Available only for authorised users
Footnotes
1
In contrast, state-of-art neural methods allow us to use basis functions beyond polynomials or Fourier bases and approximate much more complicated mappings. Although such methods can be analyzed theoretically, the estimates are often impractical [25‐27] This is a very active area of research, and we expect tighter estimates in the future.
 
2
While a probability-based definition of \(\omega\) is not needed for defining a parameteric PDE, we choose this definition for two reasons. First, such a formulation allows easy extension to the stochastic PDE case. Second, such a formulation will allow using expectation-based arguments in the analysis of convergence.
 
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Metadata
Title
NeuFENet: neural finite element solutions with theoretical bounds for parametric PDEs
Authors
Biswajit Khara
Aditya Balu
Ameya Joshi
Soumik Sarkar
Chinmay Hegde
Adarsh Krishnamurthy
Baskar Ganapathysubramanian
Publication date
10-04-2024
Publisher
Springer London
Published in
Engineering with Computers / Issue 5/2024
Print ISSN: 0177-0667
Electronic ISSN: 1435-5663
DOI
https://doi.org/10.1007/s00366-024-01955-7

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