带硬约束的有限基物理信息神经网络用于高穿孔域中的粘性流体流动

中文摘要 AI 辅助

本研究采用物理信息神经网络（PINNs）研究高穿孔域中由斯托克斯方程控制的粘性流体流动。穿孔微结构会产生复杂的边界条件和细尺度流动特征，标准神经网络难以解析。传统PINNs即便结合先进训练技术，随着穿孔数量增加也会出现精度和效率下降的问题，其中一个重要原因是通过惩罚项对边界条件的软约束，这可能导致刚度、梯度冲突以及近边界流动结构解析不良。硬约束通过将边界条件精确编码到网络假设中提供了另一种方案，但可能因近似的全局性质引入不期望的非局部效应。为应对这些挑战，基于域分解和局域化原理的有限基PINNs（FBPINNs）与硬边界约束结合使用，该约束可高效编码与穿孔相关的边界条件。此方法有助于缓解频谱偏差，提高整体精度，且其收敛性仅受穿孔数量的微弱影响，从而提供了一种高效且高度可并行化的神经网络框架。所提方法还得到理论论证支持，具体聚焦于FBPINNs的局域化和近似特性。

英文摘要

In this work, viscous fluid flow governed by the Stokes equations in highly perforated domains is studied using physics-informed neural networks (PINNs). Perforated microstructures induce complex boundary conditions and fine-scale flow features that are difficult for standard neural networks to resolve. Conventional PINNs, even when combined with advanced training techniques, can suffer from a loss of accuracy and efficiency as the number of perforations increases. One important source of this difficulty is the soft enforcement of boundary conditions through penalty terms, which can lead to stiffness, gradient conflicts, and poor resolution of near-boundary flow structures. Hard constraints provide an alternative by encoding boundary conditions exactly into the network ansatz, but may introduce undesirable non-local effects due to the global nature of the approximation. To address these challenges, finite basis PINNs (FBPINNs), which are based on domain decomposition and localisation principles, are used together with hard boundary constraints that efficiently encode perforation-related boundary conditions. This approach helps mitigate spectral bias, improves overall accuracy, and exhibits convergence that is only weakly affected by the number of perforations, thereby providing an efficient and highly parallelisable neural network framework. The proposed approach is further supported with theoretical arguments, specifically focusing on the localisation and approximation properties of FBPINNs.

发表机构

• Korea Advanced Institute of Science and Technology（韩国科学技术院）
• Weierstrass-Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机学研究所）
• Leibniz-Institut für Verbundwerkstoffe GmbH（莱布尼茨复合材料有限公司研究所）

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