Variational Inference and Learning of Koopman Invariant Subspaces
Ramen Ghosh · HAL (Le Centre pour la Communication Scientifique Directe) · 2025
We propose a novel variational inference framework for discovering Koopmaninvariant subspaces from time series data, with applications to forecasting and dimensionality reduction in high-dimensional nonlinear systems. The Koopman operator provides a linear, infinite-dimensional representation of nonlinear dynamics through the evolution of observables. However, identifying finite-dimensional subspaces that are approximately invariant under Koopman evolution remains a core challenge, which our framework addresses. Our approach formulates the discovery of such subspaces as a variational optimization problem: We seek a set of observables that maximize predictive temporal coherence while preserving linear evolution under the Koopman operator. This yields a tractable objective related to the variational approach for Markov processes (VAMP), which we optimize using basis expansions such as polynomials with possible extensions to neural or kernel models. We establish theoretical conditions under which the learned subspaces capture dominant dynamical modes and derive error bounds on the approximation quality with respect to Koopman spectral projections. We validate the method on a metastable stochastic benchmark system, ensuring its reliability. The learned invariant subspaces enable accurate multi-step prediction and interpretable low-rank approximations of the operator spectrum. Our results bridge Koopman's theory with variational learning and offer a rigorous and practical approach to identifying linearizable structures in complex systems, instilling confidence in the method's reliability. Recent work has explored data-driven methods for approximating the Koopman operator, including dynamic mode decomposition (DMD), extended DMD (EDMD), and kernel or neural network variants. These methods typically assume a fixed dictionary of observables or learn an embedding in