Thomas Geert de Jong

Alessandro Corbetta, Thomas Geert de Jong

Neural networks are increasingly employed to model, analyze and control non-linear dynamical systems ranging from physics to biology. Owing to their universal approximation capabilities, they regularly outperform state-of-the-art model-driven methods in terms of accuracy, computational speed, and/or control capabilities. On the other hand, neural networks are very often they are taken as black boxes whose explainability is challenged, among others, by huge amounts of trainable parameters. In this paper, we tackle the outstanding issue of analyzing the inner workings of neural networks trained to classify regular-versus-chaotic time series. This setting, well-studied in dynamical systems, enables thorough formal analyses. We focus specifically on a family of networks dubbed Large Kernel Convolutional Neural Networks (LKCNN), recently introduced by Boullé et al. (2021). These non-recursive networks have been shown to outperform other established architectures (e.g. residual networks, shallow neural networks and fully convolutional networks) at this classification task. Furthermore, they outperform ``manual'' classification approaches based on direct reconstruction of the Lyapunov exponent. We find that LKCNNs use qualitative properties of the input sequence. In particular, we show that the relation between input periodicity and activation periodicity is key for the performance of LKCNN models. Low performing models show, in fact, analogous periodic activations to random untrained models. This could give very general criteria for identifying, a priori, trained models that have poor accuracy.

Thomas de Jong, Valentina Breschi, Maarten Schoukens et al.

In this paper, we consider the design of data-driven predictive controllers for nonlinear systems from input-output data via linear-in-control input Koopman lifted models. Instead of identifying and simulating a Koopman model to predict future outputs, we design a subspace predictive controller in the Koopman space. This allows us to learn the observables minimizing the multi-step output prediction error of the Koopman subspace predictor, preventing the propagation of prediction errors. To avoid losing feasibility of our predictive control scheme due to prediction errors, we compute a terminal cost and terminal set in the Koopman space and we obtain recursive feasibility guarantees through an interpolated initial state. As a third contribution, we introduce a novel regularization cost yielding input-to-state stability guarantees with respect to the prediction error for the resulting closed-loop system. The performance of the developed Koopman data-driven predictive control methodology is illustrated on a nonlinear benchmark example from the literature.

Thomas Geert de Jong, Hirofumi Notsu, Kohei Nakajima

Nature is pervaded with oscillatory dynamics. In networks of coupled oscillators patterns can arise when the system synchronizes to an external input. Hence, these networks provide processing and memory of input. We present a universal framework for harnessing oscillator networks as computational resource. This computing framework is introduced by the ubiquitous model for phase-locking, the Kuramoto model. We force the Kuramoto model by a nonlinear target-system, then after substituting the target-system with a trained feedback-loop it emulates the target-system. Our results are two-fold. Firstly, the trained network inherits performance properties of the Kuramoto model, where all-to-all coupling is performed in linear time with respect to the number of nodes and parameters for synchronization are abundant. The latter implies that the network is generically successful since the system learns via sychronization. Secondly, the learning capabilities of the oscillator network, which describe a type of collective intelligence, can be explained using Kuramoto model's order parameter. In summary, this work provides the foundation for utilizing nature's oscillator networks as a new class of information processing systems.