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LO1: Learning Operators and surrogate models using scientific machine learning Location: D406 Session Chair: Stefan Frei | |
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Application of learning operators in neural network preconditioning and generative AI models 1: Université de Strasbourg, France; 2: Inria, Bordeaux; 3: Cerfacs, Toulouse
This talk mainly focuses on CNN-FGMRES, a hybrid algorithm that integrates classical numerical linear algebra methods with nonlinear neural operator preconditioning to accelerate the solution of some parametric Partial Differential Equations (PDEs). Specifically, we consider Krylov subspace methods, such as Flexible GMRES (FGMRES), combined with nonlinear preconditioners derived from the trained neural operator models with convolutional neural networks (CNN). The parametric PDEs addressed include the Helmholtz equations, Poisson equations, Darcy flow, and Diffusion-Advection equations, spanning both academic benchmarks and practical datasets. Compared to the classical numerical preconditioners, the trained neural operator preconditioning exhibits significant generalization capabilities in addressing a wide range of numerical and parametric variations. Besides, it owns a matrix-free implementation, indicating it can address some large-scale problems that memory constraints restrict classical numerical preconditioners. In addition, it supports both CPU and GPU implementations. On the other hand, compared to the pure neural network solver, CNN-FGMRES can reach machine precision with much less computational cost than the classical subspace methods. Overall, this work demonstrates the efficiency and flexibility of combining modern neural networks with classical Krylov methods, leveraging the strengths of both to achieve higher attainable accuracy and broader applicability across diverse problem settings. This talk also covers some related follow-ups on generative models for applications to other PDE-based problems.
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