Model-free blind reconstruction of dynamical systems using geometry-aware echo state networks

Estimating trajectories from corrupted observations of nonlinear dynamical systems is a challenging blind inverse problem when neither clean target trajectories nor governing equations are available. Conventional echo state network (ESN) readout estimation is supervised and requires a desired target output, corresponding here to access to the ground-truth clean trajectory during readout estimation. We introduce Regularized Maximum Likelihood Phase-Space Volume (R-MLPSV), which extends the MLPSV criterion for target-free ESN readout estimation. R-MLPSV uses trajectory-adaptive annular neighborhoods derived from empirical k-nearest-neighbor distances together with Frobenius-norm regularization. A fixed ESN reservoir is driven by noisy observations, and its inverse-correction readout is selected by minimizing the R-MLPSV geometric criterion rather than a clean-target loss. We establish a local geometric-stability condition under which sufficiently small residual delay-space perturbations relative to the adaptive annular-boundary margin preserve the R-MLPSV neighborhood structure. We further characterize the reservoir correction subspace: exact pointwise recovery within the proposed correction family is possible only when the disturbance lies in the column space of the reservoir-state matrix. Experiments on an AR(2) benchmark, Lorenz and Rössler chaotic systems, and epileptic electroencephalogram (EEG) recordings show that adaptive neighborhoods produce more consistently finite and better-supported geometric evaluations than fixed-radius MLPSV. These results characterize R-MLPSV–ESN as a numerically robust, geometry-constrained framework for target-free ESN readout selection under corrupted observations.