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Parallel-in-Time Solutions with Random Projection Neural Networks

arXiv 2408.09756 ↗First posted 19 Aug 2024math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

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.

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References · 12

  1. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  2. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 87
  3. 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74
  4. 2019Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equationscited by 66
  5. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
  6. 2020Approximation Bounds for Random Neural Networks and Reservoir Systemscited by 52
  7. 2020From Fourier to Koopman: Spectral Methods for Long-term Time Series Predictioncited by 50
  8. 2020Hierarchical Deep Learning of Multiscale Differential Equation Time-Stepperscited by 37
  9. 2020Solving Differential Equations Using Neural Network Solution Bundlescited by 20
  10. 2022Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEscited by 19
  11. 2021Parareal Neural Networks Emulating a Parallel-in-time Algorithmcited by 4
  12. 2023Parareal with a physics-informed neural network as coarse propagatorcited by 4

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