SetPINNs: Set-based Physics-informed Neural Networks
This work provides an incremental improvement for researchers and practitioners using PINNs to solve partial differential equations, by addressing the issue of neglecting domain dependencies.
This paper introduces SetPINNs, a framework designed to address the limitations of conventional Physics-Informed Neural Networks (PINNs) by capturing local dependencies within a domain. SetPINNs achieve this by partitioning the domain into sets and modeling local dependencies, leading to improved accuracy, efficiency, and robustness in solving partial differential equations.
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.