Parameter-aware reservoir computing for learning parameterized nonlinear dynamical systems with delay and cumulative effects

Complex dynamical systems often exhibit distinct behaviors under varying control parameters and environmental conditions, posing a major challenge for unified modeling and prediction. Conventional reservoir computing (RC) has limited ability to capture parameter-dependent dynamical evolution, particularly when delay and cumulative effects change the effective state representation. To address this limitation, we propose a parameter-aware reservoir computing (PARC) for learning parameterized nonlinear dynamical systems with delay and cumulative effects. In PARC, the physical delay parameter τ remains a prescribed property of the data-generating system. For each prescribed physical delay, validation-based optimization selects a dimensionless delay-condition encoding b τ that is best suited to the current reservoir configuration. The selected encoding is used as a model-level conditioning input through the guide channel. This design allows the reservoir to adapt its parameter conditioning to delay-induced dynamics. We validate PARC on delayed logistic systems under both noiseless and noisy observation settings. The results show that PARC achieves stable long-horizon prediction and accurately reconstructs the attractor geometry and statistical “climate” of the underlying dynamics. The results further show that the proposed framework can handle both single-parameter and multi-parameter variations within a unified framework, indicating that it captures a parameter-dependent family of dynamical evolutions rather than a single trajectory mapping. We further apply PARC to NDVI time series as a real-world example of a spatiotemporal dynamical system with delay and heterogeneity. These results indicate that PARC provides a unified and robust framework for learning and predicting parameterized nonlinear dynamical systems.