SIGMa-DS: System identification from the geometric manifold of dynamical synchronization

From predicting critical phase transitions to a global bifurcation structure, recurrent neural networks and, in particular, reservoir computers (RCs), have emerged as a computationally efficient, experimentally realizable, and predictive framework for accurately modeling and extrapolating temporal data. In principle, such extrapolations are only possible if RCs capture and model the generative equations that produce the data, thereby establishing themselves as effective surrogate models and digital twins. However, our inability to directly extract and perturb the internal models learned by RCs limits our ability to understand and test the models' predictions in regimes where we have collected no data, thereby limiting their utility as predictive models. Here, we address these limitations by formulating the process of reservoir computing through the lens of differential geometry-a process we coin "SIGMa-DS"-where we leverage the nonlinear manifold geometry of RC dynamics by developing a theory of dynamical generalized synchronization (DGS). Using DGS, we extract and perturb the internal models learned and generated by RCs and quantitatively explore the nonlinear geometry of the synchronization manifolds. Moreover, this geometric formulation does not require knowledge or identification of the equations of the reservoir dynamics, allowing it to readily generalize to experimental systems. Together, we provide a robust geometric framework that enables understanding and causal hypothesis testing in predictive dynamical models.