Physics-Informed Neural Networks (PINNs) have emerged as a promising method for approximating solutions to partial differential equations (PDEs) using deep learning. However, PINNs, based on multilayer perceptrons (MLP), often employ point-wise predictions, overlooking the implicit dependencies within the physical system such as temporal or spatial dependencies. These dependencies can be captured using more complex network architectures, for example CNNs or Transformers. However, these architectures conventionally do not allow for incorporating physical constraints, as advancements in integrating such constraints within these frameworks are still lacking. Relying on point-wise predictions often results in trivial solutions. To address this limitation, we propose SetPINNs, a novel approach inspired by Finite Elements Methods from the field of Numerical Analysis. SetPINNs allow for incorporating the dependencies inherent in the physical system while at the same time allowing for incorporating the physical constraints. They accurately approximate PDE solutions of a region, thereby modeling the inherent dependencies between multiple neighboring points in that region. Our experiments show that SetPINNs demonstrate superior generalization performance and accuracy across diverse physical systems, showing that they mitigate failure modes and converge faster in comparison to existing approaches. Furthermore, we demonstrate the utility of SetPINNs on two real-world physical systems.

本研究解决了传统物理信息神经网络（PINNs）在处理偏微分方程（PDEs）时忽略物理系统内隐含依赖的问题。我们提出的SetPINNs方法灵感来源于有限元方法，能够有效捕捉空间和时间依赖，并结合物理约束。实验结果表明，SetPINNs在多个物理系统中表现出更佳的泛化性能和准确性，显著优于现有方法。

基于集合的物理信息神经网络