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Linear Stability Analysis of Physics-Informed Random Projection Neural Networks for ODEs

arXiv 2408.15393 ↗First posted 27 Aug 2024math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

We present a linear stability analysis of physics-informed random projection neural networks (PI-RPNNs), for the numerical solution of {the initial value problem (IVP)} of (stiff) ODEs. We begin by proving that PI-RPNNs are uniform approximators of the solution to ODEs. We then provide a constructive proof demonstrating that PI-RPNNs offer consistent and asymptotically stable numerical schemes, thus convergent schemes. In particular, we prove that multi-collocation PI-RPNNs guarantee asymptotic stability. Our theoretical results are illustrated via numerical solutions of benchmark examples including indicative comparisons with the backward Euler method, the midpoint method, the trapezoidal rule, the 2-stage Gauss scheme, and the 2- and 3-stage Radau schemes.

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References · 16

  1. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  2. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 135
  3. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 129
  4. 2022Physics-Informed Deep Neural Operator Networkscited by 113
  5. 2020The Random Feature Model for Input-Output Maps between Banach Spacescited by 90
  6. 2020Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equationscited by 81
  7. 2017Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equationscited by 72
  8. 2021Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionalitycited by 34
  9. 2023Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processescited by 31
  10. 2022Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approachcited by 25
  11. 2022Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEscited by 19
  12. 2024RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operatorscited by 16
4 more
  1. 2022A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networkscited by 14
  2. 2024Random Projection Neural Networks of Best Approximation: Convergence theory and practical applicationscited by 11
  3. 2023Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learningcited by 10
  4. 2024Fast training of accurate physics-informed neural networks without gradient descentcited by 8

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