Abstract.

Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave suboptimally when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalizes better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML.

Keywords

  1. initial value problems
  2. quadrature
  3. time-stepping
  4. machine learning
  5. reinforcement learning

MSC codes

  1. 34A12
  2. 34A38
  3. 65D32
  4. 65L05
  5. 65L06
  6. 68T05

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Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A579 - A595
ISSN (online): 1095-7197

History

Submitted: 15 April 2021
Accepted: 10 October 2022
Published online: 26 April 2023

Keywords

  1. initial value problems
  2. quadrature
  3. time-stepping
  4. machine learning
  5. reinforcement learning

MSC codes

  1. 34A12
  2. 34A38
  3. 65D32
  4. 65L05
  5. 65L06
  6. 68T05

Authors

Affiliations

Michael Dellnitz
Department of Mathematics, Paderborn University, Paderborn, Germany.
Eyke Hüllermeier
Department of Computer Science, LMU Munich, Munich, Germany.
Modeling and Simulation of Complex Processes, Zuse Institute Berlin, Berlin, Germany.
Sina Ober-Blöbaum
Department of Mathematics, Paderborn University, Paderborn, Germany.
Department of Mathematics, Paderborn University, Paderborn, Germany.
Department of Computer Science, Paderborn University, Paderborn, Germany.
Karlson Pfannschmidt
Department of Computer Science, Paderborn University, Paderborn, Germany.

Notes

*
Submitted to the journal’s Methods and Algorithms for Scientific Computing section April 15, 2021; accepted for publication (in revised form) October 10, 2022; published electronically April 26, 2023.

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