Personalized Algorithm Generation: A Case Study in Learning ODE Integrators

SIAM Journal on Scientific Computing, vol. 44, pp. A1911–A1933

Abstract

We study the learning of numerical algorithms for scientific computing, which combines mathematically driven, handcrafted design of general algorithm structure with a data-driven adaptation to specific classes of tasks. This represents a departure from the classical approaches in numerical analysis, which typically do not feature such learning-based adaptations. As a case study, we develop a machine learning approach that automatically learns effective solvers for initial value problems in the form of ordinary differential equations (ODEs), based on the Runge--Kutta (RK) integrator architecture. We show that we can learn high-order integrators for targeted families of differential equations without the need for computing integrator coefficients by hand. Moreover, we demonstrate that in certain cases we can obtain superior performance to classical RK methods. This can be attributed to certain properties of the ODE families being identified and exploited by the approach. Overall, this work demonstrates an effective learning-based approach to the design of algorithms for the numerical solution of differential equations. This can be readily extended to other numerical tasks.

Authors 7

1. Y. P. Guo

   Agency for Science, Technology and Research · National University of Singapore · Institute of High Performance Computing

   Affiliation as printed

   Department of Mathematics, National University of Singapore, 117543, Singapore

   Department of Mathematics, National University of Singapore, 117543, Singapore; Institute of High Performance Computing, A*STAR, 138632, Singapore

2. Felix Dietrich

   Technical University of Munich

   Affiliation as printed

   Institut für Informatik, TU München, Boltzmannstr. 3, 85748 Garching

3. Tom Bertalan

   Johns Hopkins University

   Affiliation as printed

   Department of Chemical and Biomolecular Engineering, Whiting School of Engineering, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA

4. Danimir T. Doncevic Aachen

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute of Energy and Climate Research -Energy Systems Engineering (IEK-10), Forschungszen-trum Jülich GmbH, 52425 Jülich, Germany;

   RWTH Aachen University, Aachen 52062, Germany

5. Manuel Dahmen

   Forschungszentrum Jülich

   Affiliation as printed

   Institute of Energy and Climate Research -Energy Systems Engineering (IEK-10), Forschungszen-trum Jülich GmbH, 52425 Jülich, Germany

6. Ioannis G. Kevrekidis

   Agency for Science, Technology and Research · Johns Hopkins University · National University of Singapore · Institute of High Performance Computing

   Affiliation as printed

   Department of Chemical and Biomolecular Engineering and Department of Applied Mathematics and Statistics, Whiting School of Engineering, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA

   Department of Mathematics, National University of Singapore, 117543, Singapore

   Department of Mathematics, National University of Singapore, 117543, Singapore; Institute of High Performance Computing, A*STAR, 138632, Singapore

7. Qianxiao Li corresponding

   Agency for Science, Technology and Research · Johns Hopkins University · National University of Singapore · Institute of High Performance Computing

   Affiliation as printed

   Department of Chemical and Biomolecular Engineering and Department of Applied Mathematics and Statistics, Whiting School of Engineering, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA

   Department of Mathematics, National University of Singapore, 117543, Singapore

   Department of Mathematics, National University of Singapore, 117543, Singapore; Institute of High Performance Computing, A*STAR, 138632, Singapore