# Curriculum Learning-Driven PIELMs for Fluid Flow Simulations

arXiv: https://arxiv.org/abs/2503.06347 · HTML: https://mlsanity.com/paper/2503.06347/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 31 references that are in the map.
- First posted on arXiv 8 Mar 2025.

## Builds on (most cited first, 9 of 9)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Neural Map: Structured Memory for Deep Reinforcement Learning](https://mlsanity.com/paper/1702.08360/) (2017)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs](https://mlsanity.com/paper/2102.13037/) (2021)
- [Curriculum learning for language modeling](https://mlsanity.com/paper/2108.02170/) (2021)
- [Distributed physics informed neural network for data-efficient solution to partial differential equations](https://mlsanity.com/paper/1907.08967/) (2019)
- [On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elements](https://mlsanity.com/paper/2110.14121/) (2021)
- [EfficientTrain++: Generalized Curriculum Learning for Efficient Visual Backbone Training](https://mlsanity.com/paper/2405.08768/) (2024)

## Built on by (most cited first, 8 of 8)

- [Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients](https://mlsanity.com/paper/2507.10241/) (2025)
- [General Fourier Feature Physics-Informed Extreme Learning Machine (GFF-PIELM) for High-Frequency PDEs](https://mlsanity.com/paper/2510.12293/) (2025)
- [Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEs](https://mlsanity.com/paper/2510.01039/) (2025)
- [Towards Fast Option Pricing PDE Solvers Powered by PIELM](https://mlsanity.com/paper/2510.04322/) (2025)
- [Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation](https://mlsanity.com/paper/2510.04490/) (2025)
- [Soft Partition-based KAPI-ELM for Multi-Scale PDEs](https://mlsanity.com/paper/2601.08719/) (2026)
- [Learning Where the Physics Is: Probabilistic Adaptive Sampling for Stiff PDEs](https://mlsanity.com/paper/2603.06287/) (2026)
- [Meta-Learned Basis Adaptation for Parametric Linear PDEs](https://mlsanity.com/paper/2604.09289/) (2026)

## Abstract (arXiv)

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.