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Machine learning for a class of partial differential equations with multi-delays based on numerical Gaussian processes

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  • Zhang, Wenbo
  • Gu, Wei

Abstract

Delay partial differential equations (PDEs) are widely utilized in many fields, such as climate prediction and epidemiology. But observation data in real world is often noisy and discrete. And in order to expand the applications of delay PDEs, we consider numerical Gaussian processes to solve these models. In this paper, numerical Gaussian processes for predicting the latent solution of a type of delay PDEs with multi-delays are investigated, and various delay PDEs are studied, including problems governed by variable-order fractional order operators and nonlinear operators, so as to adapt to the needs of practical applications. Numerical Gaussian processes are very good at fitting latent solution of PDEs, when all observation data is noisy and discontinuous. And the methodology can clearly quantify the uncertainty of the predicted solution. For complex boundaries controlled by ODEs, we consider mixed boundary conditions of delay PDEs in this paper. And we also apply Runge-Kutta methods to enhance the prediction accuracy of these problems. Finally, we design seven numerical examples to investigate the efficiency of NGPs and how the noisy data influences the solution of our studied problems.

Suggested Citation

  • Zhang, Wenbo & Gu, Wei, 2024. "Machine learning for a class of partial differential equations with multi-delays based on numerical Gaussian processes," Applied Mathematics and Computation, Elsevier, vol. 467(C).
  • Handle: RePEc:eee:apmaco:v:467:y:2024:i:c:s0096300323006677
    DOI: 10.1016/j.amc.2023.128498
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    References listed on IDEAS

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    1. Zhao Chen & Yang Liu & Hao Sun, 2021. "Physics-informed learning of governing equations from scarce data," Nature Communications, Nature, vol. 12(1), pages 1-13, December.
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