The prediction of complex nonlinear dynamical systems with the help of machine learning has become increasingly popular in different areas of science. In particular, reservoir computers, also known as echo-state networks, turned out to be a very powerful approach, especially for the reproduction of nonlinear systems. The reservoir, the key component of this method, is usually constructed as a sparse, random network that serves as a memory for the system. In this work, we introduce block-diagonal reservoirs, which implies that a reservoir can be composed of multiple smaller reservoirs, each with its own dynamics. Furthermore, we take out the randomness of the reservoir by using matrices of ones for the individual blocks. This breaks with the widespread interpretation of the reservoir as a single network. In the example of the Lorenz and Halvorsen systems, we analyze the performance of block-diagonal reservoirs and their sensitivity to hyperparameters. We find that the performance is comparable to sparse random networks and discuss the implications with regard to scalability, explainability, and hardware realizations of reservoir computers.
Identifying and describing the dynamics of complex systems is a central challenge in various areas of science, such as physics, finance, or climatology. While machine learning algorithms are increasingly overtaking traditional approaches, their inner workings and, thus, the drivers of causality remain elusive. In this paper, we analyze the causal structure of chaotic systems using Fourier transform surrogates and three different inference techniques: While we confirm that Granger causality is exclusively able to detect linear causality, transfer entropy and convergent cross-mapping indicate that causality is determined to a significant extent by nonlinear properties. For the Lorenz and Halvorsen systems, we find that their contribution is independent of the strength of the nonlinear coupling. Furthermore, we show that a simple rationale and calibration algorithm are sufficient to extract the governing equations directly from the causal structure of the data. Finally, we illustrate the applicability of the framework to real-world dynamical systems using financial data before and after the COVID-19 outbreak. It turns out that the pandemic triggered a fundamental rupture in the world economy, which is reflected in the causal structure and the resulting equations.
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