Geometry-aware reservoirs: Patch-wise Jacobian lifting with cross-patch couplings for piecewise-linear modelling of chaotic flows

Accurate, long-horizon prediction of chaotic dynamics is notoriously difficult once numerical errors exceed the local Lyapunov time—and becomes even harder when the governing equations are unknown. We introduce the Local Tangent-Space Reservoir (LTSR), a geometry-aware Echo-State Network that embeds empirically estimated Jacobians of a chaotic attractor directly into its recurrent weight matrix. A single training trajectory is first partitioned into P clusters; within each patch we solve a regularised least-squares problem to obtain the one-step Jacobian, truncate it to the d dominant expanding/contracting directions, and store the result as a diagonal block. Off-diagonal blocks are filled with identity couplings scaled by the empirical Markov transition matrix, yielding a block matrix whose spectral radius is subsequently normalised to guarantee the Echo-State Property. A leaky integrator adds tunable memory while a second-order polynomial read-out is learned by ridge regression alone—no recurrent parameters are trained.