Research article

A machine learning framework for data driven acceleration of computations of differential equations

  • Received: 25 July 2018 Accepted: 03 September 2018 Published: 10 October 2018
  • We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of trainable parameters. These parameters are determined in an o ine training process by (approximately) minimizing suitable (possibly nonconvex) loss functions by (stochastic) gradient descent methods. The proposed algorithm is designed to be always consistent with the underlying di erential equation. Numerical experiments involving both linear and non-linear ODE and PDE model problems demonstrate a significant gain in computational e ciency over standard numerical methods.

    Citation: Siddhartha Mishra. A machine learning framework for data driven acceleration of computations of differential equations[J]. Mathematics in Engineering, 2019, 1(1): 118-146. doi: 10.3934/Mine.2018.1.118

    Related Papers:

  • We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of trainable parameters. These parameters are determined in an o ine training process by (approximately) minimizing suitable (possibly nonconvex) loss functions by (stochastic) gradient descent methods. The proposed algorithm is designed to be always consistent with the underlying di erential equation. Numerical experiments involving both linear and non-linear ODE and PDE model problems demonstrate a significant gain in computational e ciency over standard numerical methods.


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