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

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

    Research output: Contribution to journalArticlepeer-review

    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.

    Original languageEnglish (US)
    JournalEngineering with Computers
    DOIs
    StateAccepted/In press - 2024

    Keywords

    • Data-free modeling
    • Deep learning
    • Neural solvers
    • Parametric PDE
    • Physics informed learning

    ASJC Scopus subject areas

    • Software
    • Modeling and Simulation
    • General Engineering
    • Computer Science Applications

    Fingerprint

    Dive into the research topics of 'NeuFENet: neural finite element solutions with theoretical bounds for parametric PDEs'. Together they form a unique fingerprint.

    Cite this