Parallel-in-Time Solutions with Random Projection Neural Networks

TL;DR

This paper introduces a parallel-in-time method using Random Projection Neural Networks as coarse propagators, providing theoretical convergence analysis and demonstrating efficiency and accuracy in solving differential equations.

Contribution

It extends the Parareal method by integrating RPNNs, a novel neural architecture with random weights, enhancing computational efficiency without sacrificing accuracy.

Findings

The proposed method converges effectively for Lorenz and Burgers' equations.

RPNNs significantly reduce training time compared to standard neural networks.

The approach maintains high accuracy in solving the SIR system.

Abstract

This paper considers one of the fundamental parallel-in-time methods for the solution of ordinary differential equations, Parareal, and extends it by adopting a neural network as a coarse propagator. We provide a theoretical analysis of the convergence properties of the proposed algorithm and show its effectiveness for several examples, including Lorenz and Burgers' equations. In our numerical simulations, we further specialize the underpinning neural architecture to Random Projection Neural Networks (RPNNs), a 2-layer neural network where the first layer weights are drawn at random rather than optimized. This restriction substantially increases the efficiency of fitting RPNN's weights in comparison to a standard feedforward network without negatively impacting the accuracy, as demonstrated in the SIR system example.