SetPINNs通过集合采样提升物理信息神经网络的局部依赖建模能力。
SetPINNs: Set-based Physics-informed Neural Networks
- 用有限元思想将域划分为集合，捕捉局部依赖关系。
- 理论证明其残差能量估计更准、方差更低，误差更小。
- 在合成与真实任务中表现更优，适合需高精度解的科学计算场景。
物理信息神经网络（PINNs）利用深度学习求解偏微分方程。然而，传统PINNs仅进行点对点预测，忽略了域内的局部依赖关系，可能导致次优解。我们提出SetPINNs框架，有效建模局部依赖。通过类有限元采样策略，将域划分为多个集合，在建模局部依赖的同时强制满足物理定律。我们提供了严格的理论分析，表明SetPINNs能获得无偏且方差更低的残差能量及其梯度估计，从而实现更好的域覆盖和更小的残差误差。在合成数据与真实世界任务上的大量实验表明，该方法在准确性、效率和鲁棒性上均有显著提升。
原文摘要 · Abstract (English)
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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