# RandNet-Parareal: a time-parallel PDE solver using Random Neural Networks

arXiv: https://arxiv.org/abs/2411.06225 · HTML: https://mlsanity.com/paper/2411.06225/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 82 references that are in the map.
- First posted on arXiv 9 Nov 2024.

## Builds on (most cited first, 21 of 21)

- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [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)
- [Approximation Bounds for Random Neural Networks and Reservoir Systems](https://mlsanity.com/paper/2002.05933/) (2020)
- [Reservoir Computing Universality With Stochastic Inputs](https://mlsanity.com/paper/1807.02621/) (2018)
- [Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems](https://mlsanity.com/paper/1712.00754/) (2017)
- [Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality](https://mlsanity.com/paper/2106.08900/) (2021)
- [Learning with SGD and Random Features](https://mlsanity.com/paper/1807.06343/) (2018)
- [But How Does It Work in Theory? Linear SVM with Random Features](https://mlsanity.com/paper/1809.04481/) (2018)
- [Optimal Stopping via Randomized Neural Networks](https://mlsanity.com/paper/2104.13669/) (2021)
- [An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions](https://mlsanity.com/paper/2309.07049/) (2023)
- [Differentiable reservoir computing](https://mlsanity.com/paper/1902.06094/) (2019)
- [Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs](https://mlsanity.com/paper/2307.12904/) (2023)
- [Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEs](https://mlsanity.com/paper/2405.05192/) (2024)
- [Infinite-dimensional reservoir computing](https://mlsanity.com/paper/2304.00490/) (2023)
- [The Computational Capacity of LRC, Memristive and Hybrid Reservoirs](https://mlsanity.com/paper/2009.00112/) (2020)
- [A unified analysis framework for iterative parallel-in-time algorithms](https://mlsanity.com/paper/2203.16069/) (2022)
- [Nearest Neighbors GParareal: Improving Scalability of Gaussian Processes for Parallel-in-Time Solvers](https://mlsanity.com/paper/2405.12182/) (2024)

## Built on by (most cited first, 3 of 3)

- [Random Feature Representation Boosting](https://mlsanity.com/paper/2501.18283/) (2025)
- [MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functions](https://mlsanity.com/paper/2605.12368/) (2026)
- [Random Neural Network Expressivity for Non-Linear Partial Differential Equations](https://mlsanity.com/paper/2605.25057/) (2026)

## Abstract (arXiv)

Parallel-in-time (PinT) techniques have been proposed to solve systems of time-dependent differential equations by parallelizing the temporal domain. Among them, Parareal computes the solution sequentially using an inaccurate (fast) solver, and then "corrects" it using an accurate (slow) integrator that runs in parallel across temporal subintervals. This work introduces RandNet-Parareal, a novel method to learn the discrepancy between the coarse and fine solutions using random neural networks (RandNets). RandNet-Parareal achieves speed gains up to x125 and x22 compared to the fine solver run serially and Parareal, respectively. Beyond theoretical guarantees of RandNets as universal approximators, these models are quick to train, allowing the PinT solution of partial differential equations on a spatial mesh of up to 105 points with minimal overhead, dramatically increasing the scalability of existing PinT approaches. RandNet-Parareal's numerical performance is illustrated on systems of real-world significance, such as the viscous Burgers' equation, the Diffusion-Reaction equation, the two- and three-dimensional Brusselator, and the shallow water equation.