Time-delay reservoir for signal demixing using Kalman weight updates in fixed point and limit cycle regimes

In this paper, we study the problem of separating chaotic signals using time-delay reservoir computers with online training via Kalman filtering. Time delay reservoir computers are hardware-efficient and suitable for experimental, high-speed implementation. We demonstrate that incorporating an online training scheme significantly enhances the performance of time-delay reservoirs in challenging signal demixing tasks. In particular, we apply a sliding-window technique to update the readout weights and show that it can improve accuracy compared to the offline ridge regression readout in various scenarios. Here we mainly focus on the separation of two trajectories generated by the Lorenz system with different initial conditions, which is an especially difficult task since both signals share nearly identical statistical properties. We also study mixtures of signals from two different systems, specifically the Lorenz and Mackey-Glass systems, to predict the signal that contributes weakly to the mixture. Furthermore, this approach enables the time-delay reservoir computer to operate effectively in regimes where the nonlinear delay differential equation exhibits a limit cycle attractor in the absence of input, which we find to be less affected by small inaccuracies in the online weight updates than the stable fixed-point regime. This broadens the range of dynamical settings suitable for signal separation. The highest prediction accuracy, regardless of window size, is typically achieved near critical points where the system's qualitative behavior changes.