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

by S. Mishra

(Report number 2018-28)

Abstract
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 offline training process by (approximately) minimizing suitable (possibly non-convex) loss functions by (stochastic) gradient descent methods. The proposed algorithm is designed to be always consistent with the underlying differential equation. Numerical experiments involving both linear and non-linear ODE and PDE model problems demonstrate a significant gain in computational efficiency over standard numerical methods.

Keywords: Numerical methods, Differential equations, Machine learning, deep neural networks

@Techreport{M18_782,
  author = {S. Mishra},
  title = {A machine learning framework for data driven acceleration of computations of differential equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2018-28},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2018/2018-28.pdf },
  year = {2018}
}

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