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Finding separatrices of dynamical flows with Deep Koopman Eigenfunctions

NeurIPS 2025 · Probabilistic and scientific modeling › Dynamical systems · approach: Switching and law-discovery dynamics · 0 citations (OpenAlex)

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Learns scalar Koopman eigenfunctions with positive real eigenvalues via a DNN trained on a Koopman PDE residual, so their zero level sets locate separatrices between basins of attraction in high-dimensional multistable systems.

Problem

Characterizing separatrices (basin boundaries) of high-dimensional black-box multistable dynamical systems, such as RNNs and ecological models, where existing tools mainly describe behavior near stable fixed points.

Approach

Approximate a scalar Koopman eigenfunction with positive eigenvalue using a deep network trained on the Koopman PDE residual, normalized by a shuffle-based loss and a balance regularizer, so that its zero set is the separatrix; use multiple sampling distributions to resolve degeneracies, and use the function to design minimal perturbations that cross separatrices.

Key results

• On a 64D GRU 1BFF system, curve-based validation achieved R^2 0.995, and training time stayed roughly 527-718 s across dimensions 32 to 512.
• In the 668D data-trained RNN, the KEF-predicted separatrix points agreed well with ground-truth curve crossings, and the optimized perturbation was smaller than aiming at the target attractor or random separatrix points.
• The 11D fitted gLV model KEF zero level set aligned well with the true separatrix in a 2D projection.

Measured effects reported in the paper

Surprising findings

• Koopman eigenfunctions for multistable systems are generally not square-integrable, so the Hilbert-space framework does not directly apply.
• Without regularization, independently initialized networks converge to spurious solutions that vanish across entire basins or along only some separatrices.

Limitations

• No theoretical guarantees linking KEF approximation accuracy to separatrix location accuracy.
• Requires knowledge of the vector field over the entire phase space, unlike trajectory-based methods.
• Degeneracies of the Koopman PDE require multiple sampling distributions and ensembles, with hand-chosen distributions.

Experimental setup

Datasets: 1-bit flip-flop task 2-bit flip-flop task mouse neural activity PSTH data (data-trained RNN) C. difficile gLV ecological model fit to mouse abundance data

Benchmarks: 1D bistable system damped Duffing oscillator 1BFF GRU 2BFF GRU 11D gLV ecological model 668D data-trained RNN limit cycle system

Metrics: zero level set agreement with separatrix curve-based validation R^2 perturbation norm normalized PDE error

Baselines: kinetic energy minimization finite-time Lyapunov exponent grid search with ODE simulation random points on separatrix

Builds on: Koopman operator theory Deep Ritz Method physics-informed neural networks dynamic mode decomposition

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