Theoretical analysis of coupling-induced inductive bias in PGC-ESN

Develop a rigorous theoretical analysis of how incorporating the target dynamical system’s coupling structure into the reservoir of the physics-guided clustered echo state network (PGC-ESN) functions as an inductive bias that regularizes the model and constrains the hypothesis space to a more optimal region for learning dynamical systems.

Background

The paper introduces the physics-guided clustered echo state network (PGC-ESN), which embeds prior coupling structure from the target dynamical system into the reservoir architecture. Empirical evaluations on Lorenz-96 and Kuramoto–Sivashinsky systems show that leveraging coupling knowledge improves both attractor reconstruction and short-term prediction compared with standard, randomly clustered, and parallelized ESNs.

Based on these findings, the authors argue that coupling knowledge acts as an inductive bias during learning by regularizing the model and constraining the hypothesis space. However, a formal theoretical account of this effect has not been provided and is explicitly identified as an open direction.