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arXiv · 2409.20206

SetPINNs: Set-based Physics-informed Neural Networks

Mayank Nagda·Phil Ostheimer·Thomas Specht·Frank Rhein·Fabian Jirasek·Stephan Mandt·Marius Kloft·Sophie Fellenz
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Abstract

Physics-Informed Neural Networks (PINNs) solve partial differential equations using deep learning. However, conventional PINNs perform pointwise predictions that neglect dependencies within a domain, which may result in suboptimal solutions. We introduce SetPINNs, a framework that effectively captures local dependencies. With a finite element-inspired sampling scheme, we partition the domain into sets to model local dependencies while simultaneously enforcing physical laws. We provide a rigorous theoretical analysis showing that SetPINNs yield unbiased, lower-variance estimates of residual energy and its gradients, ensuring improved domain coverage and reduced residual error. Extensive experiments on synthetic and real-world tasks show improved accuracy, efficiency, and robustness.

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Mayank Nagda, Phil Ostheimer, Thomas Specht, Frank Rhein, Fabian Jirasek, Stephan Mandt, Marius Kloft, Sophie Fellenz. 2024-09-30. SetPINNs: Set-based Physics-informed Neural Networks. https://arxiv.org/abs/2409.20206

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