Systems exhibiting complex dynamics are ubiquitous in nature. Examples include biological systems such as the human heart, traffic models, weather systems and global climate. For all these examples, and many more, there is often a need to predict how dynamical variables of these systems will involve in time. For systems exhibiting complex or chaotic dynamics, this is a challenging task, which is further complicated by the fact that often only partial measured data is available. Along with the ever-growing need for algorithms which are capable of performing accurate predictions, there is also a need for these algorithms to be energy efficient. The goal of the CZS Nexus project “Interpretable surrogates for efficient analog timeseries forecasting” is to develop computational efficient algorithms which can be implemented in physical hardware rather than as software on a traditional computer.

Simplified training via reservoir computing

The starting point for this research is reservoir computing, a machine learning approach in which the input response of a fixed dynamical system is trained via linear regression to approximate a given target. The dynamical system acting as the reservoir could for example be a network of micromechanical oscillators or a semiconductor laser subject to optical self-feedback, which can be operated very energy efficiently compared with implementations in a digital computer. In contrast to other machine learning approaches, in reservoir computing only the final output layer is trained. This has the advantage that it can be easily implemented in hardware, however at the cost of overall accuracy and system size.

training dataDr. Lina Jaurigue
Left: timeseries training data. Center: reservoir trained to predict the timeseries one step ahead. Right: the trained reservoir operated autonomously to predict the evolution of the timeseries.

Computing with small reservoirs

Our work includes learning how parameters of the reservoir influence the dynamics of the system after training. The goal being to reliably obtain the desired characteristics even when the reservoir is small. This is challenging because, for small reservoirs variations in the input can lead to large changes in the final system. An example of this is depicted in Fig.2 where several reservoirs have been trained to emulate the famous Lorenz chaotic attractor, also referred to as the “butterfly” attractor. We use methods from nonlinear dynamics and statistics to understand why these differences in the trained systems arise. In [1] we have worked towards this goal by showing the influence of the coupling topology of small networks and came to the counter-intuitive conclusion that uncoupled nodes lead to more reliable reconstruction of the Lorenz system. This is an exciting result as it allows for a deeper understand of the dynamical systems.

Four examples of a 10-dimensional reservoirLuci Fumagalli
Four examples of a 10-dimensional reservoir trained to reconstruct the Lorenz attractor (green). The true Lorenz trajectory is shown in orange.

The importance of time-delay

The timescales on which the timeseries we want to predict and the timescales on which the reservoir evolves, play a crucial role in optimizing the performance of a reservoir computer. For many tasks it can be advantageous to include a time-delay in the input or the output of the reservoir and to use this as a control parameter. This allows the “memory” of the reservoir to be tailored to the task and the size of the reservoir to be reduced [2,3,4].

Physical nonlinearities

For the energy efficient implementation of machine learning algorithms, it is desirable to utilize the inherent nonlinear properties of physical hardware. To this end, we investigate how suitable hardware can be incorporated in the reservoir computing approach. This includes the study of the optical systems, such as semiconductor quantum dot lasers [5], and electronic systems, such as memristors. These physical systems should then be used instead of the usual network depicted in Fig.1 (center, right).

Summary

By gaining an understanding of the dynamics that can arise in trained systems of physical nonlinearities and tailoring the memory properties of these systems in a task-dependent manner, we work towards energy efficient edge devices for timeseries forecasting in medical and industrial applications.

[1] Chaotic attractor reconstruction using small reservoirs—the influence of topology, L. C. Jaurigue, Mach. Learn.: Sci. Technol. 5(3), 035058, (2024).

[2] Reducing reservoir computer hyperparameter dependence by external timescale tailoring, L. C. Jaurigue and K. Lüdge, Neuromorph. Comput. Eng. 4(1), 014001, (2024).

[3] Post-processing methods for delay embedding and feature scaling of reservoir computers, J. A. Jaurigue, J. Robertson, A. Hurtado, L. C. Jaurigue and K. Lüdge Commun. Eng. 4, 10, (2025).

[4] Efficient Optimisation of Physical Reservoir Computers using only a Delayed Input, E. Picco, L. C. Jaurigue, K. Lüdge and S. Massar Commun. Eng. 4(1), 3, (2025).

[5] Time-Multiplexed Reservoir Computing with Quantum-Dot Lasers: Impact of Charge-Carrier Scattering Timescale, H. Dong, L. C. Jaurigue and K. Lüdge Phys. Status Solidi RRL 2025, 2400433, (2025).

Contact

Dr. Lina Jaurigue
CZS Junior Group Leader
Department of Mathematics and Natural Sciences
Email: lina.jaurigue@tu-ilmenau.de
Phone: +49 3677 69-3649