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

Seminar for Applied Mathematics (SAM), D-Math, ETH Zürich, Rämistrasse 101, Zürich-8092,Switzerland

We propose a machine learning framework to accelerate numerical computations oftime-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existingnumerical methods as artificial neural networks, with a set of trainable parameters. These parametersare 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 tobe always consistent with the underlying differential equation. Numerical experiments involving bothlinear and non-linear ODE and PDE model problems demonstrate a significant gain in computational efficiency over standard numerical methods.
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Keywords machine learning; deep learning; differential equations; non-convex optimization; time-dependent problems

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

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