Topological Entropy Correlates with the Predictive Power of Multiplexed Ensemble Reservoir Computing

Modeling nonlinear, multiscale, and transiently chaotic biological processes remains a major challenge in computational biology. Traditional deep learning models, while powerful, require large datasets and lack mechanistic interpretability, limiting their effectiveness for time-resolved biological systems. Reservoir computing (RC) offers a promising alternative by leveraging the rich transient dynamics of fixed nonlinear systems, yet standard RC architectures struggle with high-dimensional biological data and complex temporal regimes. Here, we introduce Dynamical System Machine Learning (DynML), a multiplexed reservoir framework designed to model gene-expression dynamics in systems such as liver regeneration and Drosophila embryogenesis. DynML encodes biological signals using heterogeneous Lorenz reservoirs and employs a single global readout to capture stage-dependent dynamics with high predictive accuracy. We further show that reservoir topological entropy quantitatively predicts model performance, linking dynamical richness to biological forecasting accuracy. Beyond biological time-series modeling, we demonstrate the generality of DynML on the MNIST handwritten digit classification task using a Rossler-based chaotic reservoir, showing that fixed dynamical cores with linear readouts can also support high-dimensional static classification. Overall, DynML provides a scalable, interpretable, and computationally efficient framework that unifies biological time-series modeling and conventional machine-learning tasks within a single dynamical systems paradigm. Author summaryComplex biological phenomena such as development, regeneration, and disease progression emerge from time-dependent gene-expression programs governed by nonlinear, multiscale dynamics. Capturing these dynamics remains challenging for conventional machine-learning approaches, which typically require large datasets and lack interpretability. In this study, we introduce Dynamical System Machine Learning (DynML), a modeling framework that leverages the transient dynamics of chaotic systems to learn and predict biological time series. DynML transforms gene-expression measurements into high-dimensional dynamical representations using ensembles of nonlinear reservoirs, enabling accurate prediction of future expression states with simple and interpretable linear readouts. We apply DynML to both synthetic dynamical systems and real biological datasets, including Drosophila embryonic development and human liver regeneration, where it achieves high predictive accuracy across multiple temporal transitions. Importantly, we show that the predictive performance of DynML is strongly linked to the topological entropy of the reservoir dynamics, providing a principled and quantitative measure of model expressiveness. Beyond biological time-series prediction, we demonstrate that the same dynamical framework can also classify static data, achieving strong performance on handwritten digit recognition. Together, our results establish DynML as a scalable and interpretable approach for modeling complex biological dynamics, and highlight how concepts from dynamical systems theory can guide the design of effective machine-learning models for biological data.