Sparse Koopman Autoencoders Identify Local Dynamical Regimes in Multibasin Systems
Sparse Koopman Autoencoders (SKAEs) introduce sparsity-inducing objectives to encoder architectures, enabling latent supports to serve as inspectable basin-modeling principles for systems with multiple attractors Multibasin systems cannot generally admit a single finite-dimensional global Koopman embedding, motivating the need for local regime identification rather than global linearization SKAEs achieve superior forecasting performance compared to dense-latent KAEs across procedurally generated
Analysis
TL;DR
- Sparse Koopman Autoencoders (SKAEs) introduce sparsity-inducing objectives to encoder architectures, enabling latent supports to serve as inspectable basin-modeling principles for systems with multiple attractors
- Multibasin systems cannot generally admit a single finite-dimensional global Koopman embedding, motivating the need for local regime identification rather than global linearization
- SKAEs achieve superior forecasting performance compared to dense-latent KAEs across procedurally generated multibasin systems and chaotic flows
- Mechanistic analysis reveals that latent supports from SKAEs are essential for representation quality and can identify basins on held-out states, while dense KAEs collapse to an uninformative single family
- The approach is fully label-free, requiring no basin annotations or regime labels during training, yet produces interpretable regime variables post-training
Why It Matters
This work addresses a fundamental limitation in Koopman-based learning: the inability of standard autoencoder architectures to capture multiple coexisting dynamical regimes in nonlinear systems. For practitioners working with complex physical systems, climate modeling, or any domain with multistable dynamics, SKAEs offer a principled, unsupervised pathway to discover and exploit local linear structures without manual regime labeling.
Technical Details
- Sparse Koopman Autoencoders (SKAEs): Extends standard Koopman autoencoders by incorporating sparsity-inducing regularization on the encoder, forcing few active latent coefficients per input state and producing discrete latent supports that correspond to dynamical regimes
- Label-free regime discovery: The model learns to partition state space into basins without any basin labels or regime annotations; latent supports are treated as model-produced regime variables after training
- Benchmarking: Evaluated across procedurally generated multibasin systems and chaotic flows, demonstrating that SKAEs outperform dense-latent KAEs in forecasting accuracy
- Mechanistic study: Ablation and analysis confirm that latent supports are both necessary for representation quality and effective at identifying basins on held-out interior states; dense KAEs fail by collapsing all states into a single uninformative family
- Theoretical grounding: Built on the known result that multibasin systems generally lack finite-dimensional global Koopman embeddings, motivating local rather than global linearization
Industry Insight
- Sparse latent representations should be considered a general-purpose inductive bias for any Koopman or linear-dynamics learning task involving systems with multiple coexisting behaviors, not just multibasin dynamics
- The label-free regime discovery capability makes SKAEs particularly valuable for scientific domains where regime annotations are expensive or impossible to obtain, such as fluid dynamics, neuroscience, and climate science
- As multibasin behavior is ubiquitous in real-world nonlinear systems, this approach could become a standard preprocessing step for any pipeline relying on linear approximations of complex dynamics
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