Koopman AutoEncoder via Singular Value Decomposition for Data-Driven Long-Term Prediction

TL;DR

This paper addresses the challenge of accurate long-term prediction for nonlinear dynamics by enhancing the Koopman Autoencoder (KAE) with an unrolled SVD-based mechanism. By decomposing the Koopman matrix as ${K=U\Sigma V^T}$ and incorporating loss terms that push the singular values toward unity while enforcing unitary structure on $U$ and $V$, the proposed USVD-CKAE directly controls eigenvalues to lie near the unit circle, promoting stable long-horizon forecasts. The method also extends CKAE by aligning forward and backward dynamics in the SVD space, enabling consistent predictions without costly per-iteration eigendecomposition. Empirical results on fluid flow dynamics show USVD-CKAE reduces 1000-step prediction errors by up to $66.43\%$ relative to Vanilla KAE and $91.01\%$ relative to CKAE, with comparable or lower computational cost than iterative SVD baselines, highlighting its practical impact for data-driven long-term forecasting. Overall, the work provides a scalable, effective strategy for enforcing unit-circle eigenstructure in Koopman-based models to improve long-term predictive performance.

Abstract

The Koopman autoencoder, a data-driven technique, has gained traction for modeling nonlinear dynamics using deep learning methods in recent years. Given the linear characteristics inherent to the Koopman operator, controlling its eigenvalues offers an opportunity to enhance long-term prediction performance, a critical task for forecasting future trends in time-series datasets with long-term behaviors. However, controlling eigenvalues is challenging due to high computational complexity and difficulties in managing them during the training process. To tackle this issue, we propose leveraging the singular value decomposition (SVD) of the Koopman matrix to adjust the singular values for better long-term prediction. Experimental results demonstrate that, during training, the loss term for singular values effectively brings the eigenvalues close to the unit circle, and the proposed approach outperforms existing baseline methods for long-term prediction tasks.

Figures (5)

• Figure 1: An illustration of KAE.
• Figure 2: A pair of KAEs with forward and backward prediction for the consistent SVD-based KAE.
• Figure 3: Loss curve evolution during the training of our proposed method USVD-CKAE.
• Figure 4: Prediction error comparison at $200$ epochs.
• Figure 5: Evaluating the predictive performance of our proposed method using performance metrics and trajectory comparison with the ground truth.