Parallel-in-Time Solutions with Random Projection Neural Networks
This work addresses computational efficiency for solving ODEs in scientific computing, but it is incremental as it builds on existing Parareal methods with a specialized neural network adaptation.
The paper tackles the problem of solving ordinary differential equations more efficiently by extending the Parareal parallel-in-time method with a neural network as a coarse propagator, showing effectiveness in examples like Lorenz and Burgers' equations and achieving increased efficiency without accuracy loss in the SIR system.
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.