SetPINNs: Set-based Physics-informed Neural Networks

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 a domain into sets to model local dependencies while simultaneously enforcing physical laws. We provide rigorous theoretical analysis and bounds to show that SetPINNs provide improved domain coverage over pointwise prediction methods. Extensive experiments across a range of synthetic and real-world tasks show improved accuracy, efficiency, and robustness.