Sparse Koopman Autoencoders Identify Local Dynamics in Complex Systems

Koopman autoencoders (KAEs) aim to find a higher-dimensional latent space where complex nonlinear dynamics can be represented linearly. However, systems with multiple basins of attraction, common in many real-world phenomena, pose a significant challenge because they generally cannot be accurately described by a single, finite-dimensional global Koopman embedding. This paper proposes Sparse Koopman Autoencoders (SKAEs) to address this limitation. SKAEs incorporate a sparsity-inducing objective during training, which encourages only a few latent coefficients to be active. This sparsity allows the model to learn distinct "latent supports" that effectively act as interpretable regime variables, identifying different dynamical basins without requiring any pre-existing labels or annotations. Through extensive testing on procedurally generated multibasin systems and chaotic flows, SKAEs consistently show superior forecasting performance compared to traditional dense-latent KAEs. A mechanistic study further confirms that these sparse latent supports are crucial for the quality of the representation and are highly effective in identifying basins on unseen states, whereas dense-latent KAEs tend to collapse into uninformative representations.