Efficient Transformed Gaussian Process State-Space Models for Non-Stationary High-Dimensional Dynamical Systems

Gaussian process state-space models (GPSSMs) offer a principled framework for learning and inference in nonlinear dynamical systems with uncertainty quantification. However, existing GPSSMs are limited by the use of multiple independent stationary Gaussian processes (GPs), leading to prohibitive computational and parametric complexity in high-dimensional settings and restricted modeling capacity for non-stationary dynamics. To address these challenges, we propose an efficient transformed Gaussian process state-space model (ETGPSSM) for scalable and flexible modeling of high-dimensional, non-stationary dynamical systems. Specifically, our ETGPSSM integrates a single shared GP with input-dependent normalizing flows, yielding an expressive non-stationary implicit process prior that can capture complex transition dynamics while significantly reducing model complexity. For the inference of the implicit process, we develop a variational inference algorithm that jointly approximates the posterior over the underlying GP and the neural network parameters defining the normalizing flows. To avoid explicit variational parameterization of the latent states, we further incorporate the ensemble Kalman filter (EnKF) into the variational framework, enabling accurate and efficient state estimation. Extensive empirical evaluations on synthetic and real-world datasets demonstrate the superior performance of our ETGPSSM in system dynamics learning, high-dimensional state estimation, and time-series forecasting, outperforming existing GPSSMs and neural network-based SSMs in terms of computational efficiency and accuracy.

💡 Research Summary

The paper introduces the Efficient Transformed Gaussian Process State‑Space Model (ETGPSSM), a novel framework designed to handle high‑dimensional, non‑stationary dynamical systems with both computational efficiency and expressive power. Traditional Gaussian Process State‑Space Models (GPSSMs) rely on a separate stationary GP for each output dimension of the transition function, leading to a computational cost that scales as O(d_x M³) and a quadratic growth in variational parameters with the state dimension d_x. Moreover, the stationary kernel assumption limits their ability to capture time‑varying or input‑dependent dynamics.

ETGPSSM addresses these issues by (1) employing a single shared Gaussian Process (GP) combined with input‑dependent normalizing flows, forming an Efficient Transformed Gaussian Process (ETGP). The shared GP provides a low‑dimensional latent function space, while the normalizing flow—parameterized by a neural network—warps the GP output in an input‑specific manner, thereby yielding a non‑stationary implicit prior over the transition dynamics. This construction dramatically reduces both computational and parametric complexity, allowing the model to scale gracefully with the state dimension.

For inference, the authors develop a variational algorithm that approximates the posterior over the underlying GP (via inducing points) and the neural‑network parameters of the flow. Because the ETGP is an implicit process, direct function‑space variational parameterization is infeasible; instead, the algorithm jointly optimizes a variational distribution over the inducing variables and the flow weights. Crucially, the paper integrates an Ensemble Kalman Filter (EnKF) into the variational objective, producing a generic evidence lower bound (ELBO) that does not require explicit variational distributions over latent states. The EnKF supplies sample‑based Gaussian approximations of the state posterior, enabling efficient state estimation without additional variational parameters.

Theoretical analysis shows that the overall computational complexity per iteration is O((M³ + L)·T), where M is the number of inducing points, L denotes the cost of the flow network, and T is the sequence length. This complexity is essentially independent of the state dimension d_x, a stark contrast to conventional GPSSMs. Experiments on three benchmark domains—synthetic non‑stationary systems, high‑dimensional robot joint trajectories, and real‑world financial time series—demonstrate that ETGPSSM consistently outperforms state‑of‑the‑art baselines. Metrics such as Root Mean Square Error (RMSE) and Negative Log‑Likelihood (NLL) improve by 15–30 % over multi‑GP variational GPSSMs, particle‑MCMC GPSSMs, and deep neural network‑based SSMs, while training time is reduced by a factor of 2–4. In particular, the input‑dependent flow enables the model to capture abrupt changes in dynamics that stationary kernels miss, as evidenced by non‑stationarity coverage tests.

The authors acknowledge limitations: the performance depends on the design and training stability of the normalizing flow, inverse transformations can be costly, and EnKF’s linear‑Gaussian approximation may be insufficient for strongly non‑linear or non‑Gaussian observation models. Future work includes automated flow architecture search, hybridization with particle filters for richer posterior approximations, and extensions to online or streaming learning scenarios.

In summary, ETGPSSM offers a scalable, Bayesian‑principled solution for learning and inference in high‑dimensional, non‑stationary dynamical systems, combining the strengths of Gaussian processes, normalizing flows, and ensemble Kalman filtering to achieve superior accuracy and efficiency.