A Hybrid Framework for Efficient Koopman Operator Learning

TL;DR

This work tackles the challenge of learning Koopman representations for nonlinear dynamics by blending semidefinite programming with learning-based embeddings. The authors first use an SDP to estimate the observable space, memory, and a provisional Koopman operator, then train an autoencoder to learn efficient forward and inverse mappings that realize a linear propagation in a learned latent space. The proposed two-stage approach reduces the reliance on extensive hyperparameter tuning and improves prediction accuracy and training efficiency, especially in chaotic systems like the Lorenz attractor. The results show that SDP-informed spectral configurations and memory selection significantly enhance one-step predictions and trajectory fidelity, offering a practical pathway to scalable Koopman-based modeling for complex dynamical systems.

Abstract

Koopman analysis of a general dynamics system provides a linear Koopman operator and an embedded eigenfunction space, enabling the application of standard techniques from linear analysis. However, in practice, deriving exact operators and mappings for the observable space is intractable, and deriving an approximation or expressive subset of these functions is challenging. Programmatic methods often rely on system-specific parameters and may scale poorly in both time and space, while learning-based approaches depend heavily on difficult-to-know hyperparameters, such as the dimension of the observable space. To address the limitations of both methods, we propose a hybrid framework that uses semidefinite programming to find a representation of the linear operator, then learns an approximate mapping into and out of the space that the operator propagates. This approach enables efficient learning of the operator and explicit mappings while reducing the need for specifying the unknown structure ahead of time.

Figures (7)

• Figure 1: Our model and training scheme. Red lines show the data path for solving the SDP and determining the latent space size. blue lines show the reconstruction phase, which ensures recoverable states before pretraining. Green lines show the pretraining phase that populates the auxiliary network with the SDP-derived eigenvalues. Black lines refer to the final data path used for fine-tuning and inference.
• Figure 2: MSE Curves During Training: Our W/ Pretraining system was able to outperform both other models on 1-step prediction on the test set, while our W/o Pretraining system still out performed the Lusch model and the eigenvalue sweep.
• Figure 3: Predicted state space trajectories for each system: In \ref{['fig:ffoa_rmse_compare']}, the additional real eigenvalue found by the SDP allows for more accurate prediction of the $z$ component. Because Lorenz was not included as a baseline in Lusch_2018, a parameter sweep was performed to determine its spectral configuration.
• Figure 4: $L_2$ Error On Example Trajectory: We display the $L_2$ error for each example trajectory vs. 1-step prediction shown in \ref{['fig:traj_compare']}. In general, the W/ Pretraining model has a lower $L_2$ error across all environments for most time steps
• Figure 5: Final Test MSE for Fluid Flow on Attractor (order 1) for varying number of eigenvalues:Blue outline marks models with an embedded dimension of 3 or less. The star indicates the SDP-derived spectral configuration, which is the best for total number of eigenvalues up to 3, with diminishing returns until dimension 5.