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Learning Neural PDE Solvers with Convergence Guarantees

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arxiv 1906.01200 v1 pith:2B4OZILB submitted 2019-06-04 math.NA cs.NAstat.COstat.ML

Jun-Ting Hsieh , Shengjia Zhao , Stephan Eismann , Lucia Mirabella , Stefano Ermon This is my paper · ORCID
classification math.NAcs.NAstat.COstat.ML
keywords existingsolversapproachconvergencefastguaranteesiterativelearning
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Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be sub-optimal for specific classes of problems. In contrast to existing hand-crafted solutions, we propose an approach to learn a fast iterative solver tailored to a specific domain. We achieve this goal by learning to modify the updates of an existing solver using a deep neural network. Crucially, our approach is proven to preserve strong correctness and convergence guarantees. After training on a single geometry, our model generalizes to a wide variety of geometries and boundary conditions, and achieves 2-3 times speedup compared to state-of-the-art solvers.

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Cited by 3 Pith papers

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  1. Accurate and scalable deep Maxwell solvers using multilevel iterative methods

    physics.comp-ph 2025-09 conditional novelty 7.0 of 10

    A neural subdomain preconditioner plus multilevel domain decomposition solves 2D Maxwell problems up to 200 wavelengths and drives inverse design of large nanophotonic devices.

  2. NeuroForge: A Self-Correcting, Geometry-Native Neural CFD Engine with Calibrated Physics-Residual Trust

    physics.flu-dyn 2026-07 conditional novelty 6.0 of 10

    The steady-RANS residual of a neural CFD prediction is a backbone-robust case-level trust signal but a poor correction objective; a supervised DEQ corrector cuts field MSE on a SOTA backbone without needing residual c...

  3. Accelerating PDE-Constrained Optimization by the Derivative of Neural Operators

    cs.LG 2025-06 conditional novelty 5.0 of 10

    Reference neural operators with a Virtual-Fourier layer learn solution derivatives and a hybrid solver feedback loop accelerates PDE-constrained optimization.

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