SetPINNs: Set-based Physics-informed Neural Networks
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.
Cite
@article{arxiv.2409.20206,
title = {SetPINNs: Set-based Physics-informed Neural Networks},
author = {Mayank Nagda and Phil Ostheimer and Thomas Specht and Frank Rhein and Fabian Jirasek and Stephan Mandt and Marius Kloft and Sophie Fellenz},
journal= {arXiv preprint arXiv:2409.20206},
year = {2025}
}