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State-dependent polytopic Koopman autoencoder for long-horizon prediction of nonlinear dynamical systems
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- Authors
- Sangjun Park, Hyung-Ok Lee, Eden Kim, Yumin Hwang, Hyun-Young Lee, Seok-Kap Ko
- Issue Date
- 2026-09
- Citation
- Nonlinear Dynamics, v.114, pp.1-27
- ISSN
- 0924-090X
- Publisher
- Springer Nature
- Language
- English
- Type
- Journal Article
- Abstract
- Long-horizon prediction of nonlinear dynamical systems remains challenging because small one-step modeling errors can compound over recursive rollouts and eventually degrade prediction accuracy. This issue is relevant to Koopman-based models when a suitable finite-dimensional Koopman-invariant subspace is not available. To address this problem, we propose a state-dependent polytopic Koopman autoencoder, referred to as spKAE. The idea is to recompose the operator as a convex combination of learned dictionary matrices, with mixing weights obtained from the current rollout latent vector. This recomposition preserves a linear evolution form while it allows the operator to adapt along the rollout. From an analytical perspective, we establish a representational inclusion result relative to KAE and derive a latent-space mismatch-propagation bound relating mismatch to one-step modeling errors and the norms of the learned dictionary matrices. We evaluate spKAE on four nonlinear systems and show that it achieves more accurate long-horizon rollouts than the KAE-based and autoregressive flow-map learning-based models, with clear improvements on the Lorenz-63 system and the 2D incompressible Navier–Stokes dataset. The mixing heatmaps present a diagnostic view of operator recomposition over time, revealing system-dependent recomposition patterns in our experiments.
- Keyword
- Koopman operator, Koopman autoencoder, Nonlinear dynamical systems, State-dependent dynamics, Polytopic operator, Long-horizon prediction
- KSP Keywords
- Invariant subspace, Koopman Operator, Learning-based, Map learning, Modeling errors, Nonlinear dynamical system, Over time, Prediction accuracy, convex combination, learned dictionary, nonlinear systems