Curriculum Learning-Driven PIELMs for Fluid Flow Simulations

TL;DR

The paper tackles the challenge of solving nonlinear PDEs in fluid dynamics with machine-learning solvers by introducing curriculum learning-guided Physics-Informed Extreme Learning Machines (PIELMs) that employ Radial Basis Function (RBF) kernels for interpretable initialization. By reformulating nonlinear PDEs as a sequence of quasi-linear problems and solving linear residual systems, the approach reduces training complexity and accelerates convergence relative to standard PINNs. The authors demonstrate the method on Burgers' equation (including shocks), lid-driven cavity flow up to Reynolds number 100, and stenotic blood flow, showing notable speedups and competitive accuracy, along with physically meaningful parameter initialization. This framework has potential for fast, interpretable simulations in microfluidics, biofluid dynamics, and incompressible internal/external flows, and suggests avenues for further hyperparameter optimization and broader applications.

Abstract

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.

Figures (14)

• Figure 1: Comparison of Network Architectures: PINN vs. PIELM. The figure illustrates the differences between deep PINNs (where all weights are trainable) and single-layer PIELMs (where input weights remain fixed). Yellow and green blocks indicate the input (spatial variables) and output (flow variables) of the networks. Network parameters, i.e. weights and biases, are denoted as $W_{i}$ and $b_{i}$. Trainable parameters are shown in gray, while the fixed input layer parameters of PIELM are highlighted in light orange.
• Figure 2: PIELM solution for Poisson's equation
• Figure 3: (Left) Time-marching strategy for solving Burgers' equation. (Right) Example of RBF kernel distribution for the first and last time blocks, assuming a traveling wave initial condition $f(x) = e^{-30x^2}$. Higher kernel density is used near regions with steep gradients in the initial condition of the time block.
• Figure 4: Visualization of sampling points and the spatial variation of RBF kernel standard deviations.
• Figure 5: Temporal evolution of the PIELM solution for the viscous Burgers' equation allowing a standing shock.