Wasserstein Geometry of Information Loss in Nonlinear Dynamical Systems

Abstract

Time-delay embedding is a powerful technique for reconstructing the dynamics of nonlinear systems. However, the reconstruction map is not always an embedding, a condition rarely verified in practice. When the reconstruction map is non-injective, multiple latent states may map to the same reconstructed state, leading to multi-valued $n$-step evolution. Consequently, the induced system no longer admits a deterministic closure, and the dispersion of future trajectories leads to ambiguity. In this work, we establish a measure-theoretic framework to quantify the ambiguity induced by multi-valued evolution and introduce intrinsic stochasticity to quantify the ambiguity over a finite horizon. For numerical implementation, we use the $k$-nearest-neighbor estimator to approximate intrinsic stochasticity under finite-resolution and finite-sampling settings. Numerical experiments on the synthetic and real-world datasets are consistent with the expectation: reconstructions closer to deterministic closure tend to produce lower scores, and deterministic predictors that take reconstructions with lower empirical closure scores as input are associated with lower rollout errors, suggesting that intrinsic stochasticity provides a new perspective for understanding failures of reconstruction and serves as a diagnostic for selecting reconstruction maps.