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This research introduces PCGBandit, a method that leverages online learning to adaptively optimize solver configurations for transient PDEs without requiring extensive pre-training data. By utilizing feedback from the linear solver itself via bandit algorithms, it achieves immediate speedups in fluid and magnetohydrodynamics simulations within standard open-source frameworks like OpenFOAM. This approach addresses key bottlenecks in ML-for-science by demonstrating practical efficiency gains over classical baselines while minimizing computational overhead during the learning phase.
One-shot acceleration of transient PDE solvers via online-learned preconditioners
arXiv:2509.08765v4 Announce Type: replace-cross Abstract: Data-driven acceleration of scientific computing workflows has been a high-profile aim of machine learning (ML) for science, with numerical simulation of transient partial differential equations (PDEs) being one of the main applications. The focus thus far has been on methods that require classical simulations to train, which when combined with the data-hungriness and optimization challenges of neural networks has caused difficulties in demonstrating a convincing advantage against strong classical baselines. We consider an alternative paradigm in which the learner uses a classical solver's own data to accelerate it, enabling a one-shot speedup of the simulation. Concretely, since transient PDEs often require solving a sequence of related linear systems, the feedback from repeated calls to a linear solver such as preconditioned conjugate gradient (PCG) can be used by a bandit algorithm to online-learn an adaptive sequence of solver configurations (e.g. preconditioners). The method we develop, PCGBandit, is implemented directly on top of the popular open-source software OpenFOAM, which we use to show its effectiveness on a set of fluid and magnetohydrodynamics (MHD) problems.
Abstract:Data-driven acceleration of scientific computing workflows has been a high-profile aim of machine learning (ML) for science, with numerical simulation of transient partial differential equations (PDEs) being one of the main applications. The focus thus far has been on methods that require classical simulations to train, which when combined with the data-hungriness and optimization challenges of neural networks has caused difficulties in demonstrating a convincing advantage against strong classical baselines. We consider an alternative paradigm in which the learner uses a classical solver's own data to accelerate it, enabling a one-shot speedup of the simulation. Concretely, since transient PDEs often require solving a sequence of related linear systems, the feedback from repeated calls to a linear solver such as preconditioned conjugate gradient (PCG) can be used by a bandit algorithm to online-learn an adaptive sequence of solver configurations (e.g. preconditioners). The method we develop, PCGBandit, is implemented directly on top of the popular open-source software OpenFOAM, which we use to show its effectiveness on a set of fluid and magnetohydrodynamics (MHD) problems.
Computational Physics (physics.comp-ph); Machine Learning (cs.LG); Numerical Analysis (math.NA); Machine Learning (stat.ML)
arXiv:2509.08765 [physics.comp-ph]
(or arXiv:2509.08765v4 [physics.comp-ph] for this version)
https://doi.org/10.1016/j.cpc.2026.110304 DOI(s) linking to related resources
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