用课程学习提升极限学习机，高效求解非线性流体方程。
Curriculum Learning-Driven PIELMs for Fluid Flow Simulations
- 将非线性方程分步转化为复杂度递增的拟线性方程求解
- 在雷诺数100下成功模拟激波与腔流，精度优于传统方法
- 支持物理可解释初始化，适合流体仿真与医学血流预测
本文提出两种基于物理信息极端学习机（PIELM）的新算法，用于求解稳态与非稳态非线性偏微分方程（PDEs）的流体问题。尽管单隐层PIELM在求解线性与拟线性PDE时速度和精度优于深度物理信息神经网络（PINNs），但其扩展至非线性问题仍具挑战。为此，我们引入课程学习策略，将非线性PDE重构成一系列复杂度递增的拟线性PDE。同时，通过径向基函数（RBFs）实现网络参数的物理可解释初始化。所提算法在两个基准不可压流问题上验证：黏性Burgers方程与顶盖驱动腔流。据我们所知，这是首个将PIELM应用于求解Burgers激波解及雷诺数达100的顶盖腔流的工作。作为实际应用，我们将PIELM用于狭窄血管中的血流预测。结果表明，PIELM能高效处理非线性PDE，为线性与非线性PDE提供了一种有前景的替代方案。
原文摘要 · Abstract (English)
This paper presents two novel, physics-informed extreme learning machine (PIELM)-based algorithms for solving steady and unsteady nonlinear partial differential equations (PDEs) related to fluid flow. Although single-hidden-layer PIELMs outperform deep physics-informed neural networks (PINNs) in speed and accuracy for linear and quasilinear PDEs, their extension to nonlinear problems remains challenging. To address this, we introduce a curriculum learning strategy that reformulates nonlinear PDEs as a sequence of increasingly complex quasilinear PDEs. Additionally, our approach enables a physically interpretable initialization of network parameters by leveraging Radial Basis Functions (RBFs). The performance of the proposed algorithms is validated on two benchmark incompressible flow problems: the viscous Burgers equation and lid-driven cavity flow. To the best of our knowledge, this is the first work to extend PIELM to solving Burgers' shock solution as well as lid-driven cavity flow up to a Reynolds number of 100. As a practical application, we employ PIELM to predict blood flow in a stenotic vessel. The results confirm that PIELM efficiently handles nonlinear PDEs, positioning it as a promising alternative to PINNs for both linear and nonlinear PDEs.
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