AI Pontryagin or how neural networks learn to control complex systems
AI & Data Science / 19. Januar 2022
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Assistant Professor of Computational Social Science
Lucas ist Assistenzprofessor für rechnergesützte Sozialwissenschaften an der Frankfurt School of Finance & Management. Seine Forschung nutzt numerische und analytische Methoden, um praxisnahe Probleme aus den Sozialwissenschaften, der Medizin und Biologie zu lösen. Einige angewandte Beispiele seiner Forschungstätigkeit sind epidemische Prozesse, politische Polarisierung und der Anstieg von Antibiotikaresistenz.


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The field of complex systems science is concerned with the study of emergent behaviors associated with the interactions of a system’s constituents. Historically, there is a close connection between complex systems science, applied mathematics, and statistical mechanics (a subfield of physics). Examples of complex systems include a broad range of phenomena such as the Earth’s climate, power grids, and epidemic spreading in populations of humans and animals. For more information on these topics, we refer the reader to the following summary on the Nobel Prize in Physics 2021 that was awarded „for groundbreaking contributions to our understanding of complex systems“. Our ability to understand the behavior of certain systems is closely related to (i) describing and quantifying their dynamics with a suitable theoretical model (e.g., differential equations), (ii) forecasting their evolution, and finally (iii) controlling or steering them from undesired to desired states. Because of factors such as sensitive dependence on initial conditions and nonlinearities that arise in complex dynamical systems, it is, however, often not possible to efficiently control them.

How can we control complex systems?

In two recent works, we show how a class of artificial neural networks that we refer to as AI Pontryagin can easily learn actions (or control signals) that can steer complex systems from a certain initial state to a desired target state within a predefined time. We find that the studied neural networks are not only able to learn to control complex systems, but they also learn to do so using minimal resources. The ability to learn minimal-impact interventions results from an implicit energy regularization mechanism.

AI Pontryagin is named after the Russian mathematician Lev Pontryagin, who made significant contributions to research in pure and applied mathematics, including optimal control theory. Pontryagin’s maximum principle is widely used to derive optimal control signals (i.e., those that minimize the amount of resources or „energy’’) for a given dynamical system. We show in our work that the applicability of AI Pontryagin is not limited to low-dimensional systems with regular interaction patterns; it can be applied to control technical and biological systems that exhibit irregular network structures, various evolution patterns, and adaptive and emergent behaviors.

Possible applications of AI Pontryagin

In addition to providing arguments on the reasons why artificial neural networks are able to efficiently control complex systems, we show in our work that AI Pontryagin offers a new tool that requires less effort to model and calculate control signals for complex systems than other approaches (e.g., Pontryagin’s maximum principle and deep reinforcement learning). Using AI Pontryagin as a tool to calculate interventions may help scientists, engineers, and practitioners to focus more on modeling dynamical systems and studying the effect of interventions on them, instead of devoting much of their time to deriving control signals. The only inputs necessary are (i) a dynamical system, (ii) its initial state, and (iii) a desired target state.

In summary, our work discusses the advantages of a very versatile AI-based control framework over traditional control theory, and it shows that artificial neural networks are capable of representing controls for high-dimensional complex systems.

This article was jointly written by Thomas Asikis, Nino Antulov-Fantulin, and Lucas Böttcher.


(1) T. Asikis, L. Böttcher, N. Antulov-Fantulin, Neural Ordinary Differential Equation Control of Dynamics on Graphs, Physical Review Research (in press)

(2) L. Böttcher, N. Antulov-Fantulin, T. Asikis, AI Pontryagin or how artificial neural networks learn to control dynamical systems, Nature Communications 13, 333 (2022)

(3) L. Böttcher, T. Asikis, I. Fragkos, Solving Inventory Management Problems with Inventory-dynamics-informed Neural Networks, arXiv:2201.06126

(4) GitHub repository

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