用准随机采样提升物理信息神经网络的求解精度与稳定性。
Quasi-Random Physics-informed Neural Networks
- 用低差异序列替代随机采样，增强训练点分布均匀性。
- 在高维偏微分方程求解中，收敛速度和精度显著优于传统PINNs。
- 适合高维物理建模、需稳定训练的科学计算场景。
物理信息神经网络（PINNs）通过将物理约束融入神经网络训练，在求解偏微分方程（PDEs）方面展现出潜力，但其性能对采样点分布敏感。基于准蒙特卡洛方法在高维问题中的优异表现，本文提出准随机物理信息神经网络（QRPINNs），采用低差异序列进行采样，而非直接从域中随机采样。理论上，QRPINNs具有比PINNs更优的收敛速率。实验表明，QRPINNs在高维PDE求解中显著优于传统PINNs及部分代表性自适应采样方法。此外，将QRPINNs与自适应采样结合可进一步提升性能。
原文摘要 · Abstract (English)
Physics-informed neural networks have shown promise in solving partial differential equations (PDEs) by integrating physical constraints into neural network training, but their performance is sensitive to the sampling of points. Based on the impressive performance of quasi Monte-Carlo methods in high dimensional problems, this paper proposes Quasi-Random Physics-Informed Neural Networks (QRPINNs), which use low-discrepancy sequences for sampling instead of random points directly from the domain. Theoretically, QRPINNs have been proven to have a better convergence rate than PINNs. Empirically, experiments demonstrate that QRPINNs significantly outperform PINNs and some representative adaptive sampling methods, especially in high-dimensional PDEs. Furthermore, combining QRPINNs with adaptive sampling can further improve the performance.
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