基于哈密顿矩阵的自适应采样，提升物理神经网络求解精度。
Provably Accurate Adaptive Sampling for Collocation Points in Physics-informed Neural Networks
- 利用微分方程残差的海森矩阵指导采样点分布
- 在1D与2D方程上验证，求解误差显著降低
- 方法可证明准确，适合高精度科学计算场景
尽管数值模拟已取得显著进展，高效求解偏微分方程（PDE）仍是复杂且昂贵的问题。物理信息神经网络（PINN）通过将PDE嵌入损失函数，并利用自动微分在所谓配点处最小化残差，成为学习代理求解器的有效方式。初始采用均匀采样，近年来配点选择已发展出多种自适应优化策略。本文基于一种新的定积分近似求积方法，提出一种基于PDE残差海森矩阵的可证明准确的配点自适应采样方法。在一组1D与2D PDE上的对比实验表明，该方法能有效提升求解精度。
原文摘要 · Abstract (English)
Despite considerable scientific advances in numerical simulation, efficiently solving PDEs remains a complex and often expensive problem. Physics-informed Neural Networks (PINN) have emerged as an efficient way to learn surrogate solvers by embedding the PDE in the loss function and minimizing its residuals using automatic differentiation at so-called collocation points. Originally uniformly sampled, the choice of the latter has been the subject of recent advances leading to adaptive sampling refinements for PINNs. In this paper, leveraging a new quadrature method for approximating definite integrals, we introduce a provably accurate sampling method for collocation points based on the Hessian of the PDE residuals. Comparative experiments conducted on a set of 1D and 2D PDEs demonstrate the benefits of our method.
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