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.


  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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Scipy v1.5.4 Reference Guide: Integration and ODEs (scipy.integrate.quad), https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html.
P. Deuflhard and F. Bornemann, Scientific Computing with Ordinary Differential Equations, Texts Appl. Math. 42, Springer, New York, 2012.
P. Deuflhard and A. Hohmann, Numerical Analysis in Modern Scientific Computing, Springer, New York, 2003.
J. Dormand and P. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6 (1980), pp. 19–26.
J. Fan, Z. Wang, Y. Xie, and Z. Yang, A theoretical analysis of deep Q-learning, in Proceedings of the 2nd Conference on Learning for Dynamics and Control, A. M. Bayen, A. Jadbabaie, G. Pappas, P. A. Parrilo, B. Recht, C. Tomlin, and M. Zeilinger, eds., Proc. Mach. Learn. Res. 120, 2020, pp. 486–489.
X. Glorot and Y. Bengio, Understanding the difficulty of training deep feedforward neural networks, in Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, Y. W. Teh and M. Titterington, eds., Proc. Mach. Learn. Res. 9, PMLR, 2010, pp. 249–256, http://proceedings.mlr.press/v9/glorot10a.html.
E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Ser. Comput. Math. 31, Springer, Berlin, 2013.
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Springer, Berlin, 1996.
E. Hairer, G. Wanner, and S. P. Nørsett, Solving Ordinary Differential Equations I, Springer, Berlin, 1993.
M. J. Kearns and S. P. Singh, Finite-sample convergence rates for Q-learning and indirect algorithms, in Advances in Neural Information Processing Systems, 1999, pp. 996–1002.
D. P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization, https://arxiv.org/abs/1412.6980, 2014.
S. Klus, F. Nüske, S. Peitz, J.-H. Niemann, C. Clementi, and C. Schütte, Data-driven approximation of the Koopman generator: Model reduction, system identification, and control, Phys. D, 406 (2020), 132416.
M. Knöller, A. Ostermann, and K. Schratz, A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data, SIAM J. Numer. Anal., 57 (2019), pp. 1967–1986.
T. P. Lillicrap, J. J. Hunt, A. Pritzel, N. Heess, T. Erez, Y. Tassa, D. Silver, and D. Wierstra, Continuous Control with Deep Reinforcement Learning, poster, International Conference on Learning Representations, 2016.
Y. Liu, J. N. Kutz, and S. L. Brunton, Hierarchical deep learning of multiscale differential equation time-steppers, Philos. Trans. A, 380 (2022), 20210200.
V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski, S. Petersen, C. Beattie, A. Sadik, I. Antonoglou, H. King, D. Kumaran, D. Wierstra, S. Legg, and D. Hassabis, Human-level control through deep reinforcement learning, Nature, 518 (2015), pp. 529–533.
R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack, Springer, Berlin, 1983.
M. L. Piscopo, M. Spannowsky, and P. Waite, Solving differential equations with neural networks: Applications to the calculation of cosmological phase transitions, Phys. Rev. D, 100 (2019), 016002.
M. Raissi, P. Perdikaris, and G. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), pp. 686–707.
F. Regazzoni, L. Dedè, and A. Quarteroni, Machine learning for fast and reliable solution of time-dependent differential equations, J. Comput. Phys., 397 (2019), 108852.
SciPy, v1.5.4 Reference Guide: Integration and ODEs (scipy.integrate.solve_ivp), https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html.
L. Shampine, Vectorized adaptive quadrature in MATLAB, J. Comput. Appl. Math., 211 (2008), pp. 131–140.
L. F. Shampine and M. W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comput., 18 (1997), pp. 1–22.
T. Shinbrot, C. Grebogi, J. Wisdom, and J. A. Yorke, Chaos in a double pendulum, Amer. J. Phys., 60 (1992), pp. 491–499, https://doi.org/10.1119/1.16860.
J. Sirignano and K. Spiliopoulos, DGM: A deep learning algorithm for solving partial differential equations, J. Comput. Phys., 375 (2018), pp. 1339–1364.
R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduction, MIT Press, Cambridge, MA, 2018.
M. Tsatsos, The Van der Pol Equation, Ph.D. thesis, https://arxiv.org/abs/0803.1658, 2008.
P. Virtanen et al., SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods, 17 (2020), pp. 261–272.
C. J. C. H. Watkins and P. Dayan, Q-learning, Mach. Learn., 8 (1992), pp. 279–292.
Wolfram Language & System Documentation Center: The Design of the NDSolve Framework, Wolfram Research, https://reference.wolfram.com/language/tutorial/NDSolveDesign.html.

Information & Authors


Published In

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


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


  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



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.


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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