Curriculum Learning-Driven PIELMs for Fluid Flow Simulations

2025-03-08

AI Summary

Addressing the challenges of poor convergence and weak physical interpretability in solving steady/unsteady partial differential equations (PDEs) governing nonlinear fluid dynamics, this work proposes a curriculum learning-enhanced physics-informed extreme learning machine (CL-PIELM). The method decomposes strongly nonlinear PDEs into progressively quasi-linear subproblems and employs radial basis functions (RBFs) for physics-driven, interpretable network initialization. This is the first application of PIELM to Burgers’ equation shock solutions and high-Reynolds-number (Re = 100) lid-driven cavity flow simulation. Curriculum learning substantially improves training stability and convergence speed. Benchmark evaluations demonstrate that CL-PIELM outperforms physics-informed neural networks (PINNs) in both accuracy and computational efficiency. Furthermore, CL-PIELM successfully predicts highly nonlinear blood flow in narrow vessels, validating its capability to model real-world complex fluid phenomena and its practical effectiveness.

Technology Category

Machine Learning: Bio-inspired LearningComputer Vision: Low Level & Physics-based VisionSearch and Optimization: Learning to Search

Application Category

Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingGraph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsUser Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• Extend PIELM to solve nonlinear PDEs for fluid flow.
• Introduce curriculum learning for complex quasilinear PDEs.
• Validate PIELM on Burgers' equation and cavity flow.

Innovation

• Curriculum learning for nonlinear PDEs
• RBF-based network initialization
• PIELM for high Reynolds flows