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Generative Learning of Separatrices
Ellis R. Crabtree, Dimitris G. Giovanis, Anastasia Georgiou, George Datseris, Ioannis G. Kevrekidis
TL;DR
Separatrices in multistable systems are difficult to characterize because they are complex, sparsely sampled, and costly to resolve in high dimensions. This paper combines uncertainty-aware basin classification with score-based generative modeling to reconstruct these boundaries, successfully matching traditional techniques across three representative systems.
Problem
Separatrices are difficult to characterize because dense basin coverage becomes infeasible in high dimensions and these boundaries typically occupy low-probability, near-zero-volume regions.
Method
The framework uses neural-network basin classification and uncertainty entropy to locate candidate boundaries, then trains generative models on high-uncertainty samples to reconstruct separatrix geometry.
Results
Across three representative dynamical systems, the framework successfully identified and reconstructed basin boundaries with fidelity comparable to traditional techniques.
Takeaways & Limitations
The approach provides a data-driven, density-consistent pathway for extracting separatrix geometry from simulation or experimental time-series data, including in high-dimensional systems.
Takeaways & Limitations
Sparse training data can produce isolated high-uncertainty regions far from true separatrices, causing false-positive identification of spurious boundaries.
Abstract
from arXiv · showhide
The identification and reconstruction of the boundaries separating basins of attraction in multistable, multidimensional dynamical systems presents a fundamental challenge in computational dynamics. These structures govern transition pathways and other important large timescale behavior, yet they remain typically under-sampled since their neighborhood does not get routinely visited during direct simulations. Traditional computational approaches face computational limitations in high-dimensional systems and require a priori knowledge of the dynamical system and its equations. Simplistic sampling methods such as random or uniform sampling of the phase space typically fail to quantitatively approximate separatrices and their structure altogether. We introduce and implement a framework that combines supervised classification with generative modeling to address this challenge. Our approach first trains neural network classifiers on uniformly or randomly sampled initial conditions labeled by their corresponding basins of attraction in the system of interest. Using uncertainty metrics of the trained classifier to quantify decision boundaries, the method then identifies these high uncertainty regions and boundaries of the classifier as preliminary approximate separatrices. Subsequently, score-based generative models are trained specifically on samples from high-uncertainty regions, ultimately generating densities of samples consistent with the empirical density of samples on or close to the manifold that constitutes the separatrix between basins in the sampled region. This approach leverages the complementary strengths of (a) discriminative models for global phase space partitioning and (b) generative models for detailed geometric sampling, resulting in a systematic, iterative, data-driven framework that produces empirically consistent reconstructions of (approximate) separatrix manifolds.
1 Introduction
The paper introduces a framework for reconstructing complex separatrices in multistable dynamical systems by combining supervised basin classification with generative modeling. Classifier uncertainty locates approximate boundaries, while score-based models target detailed sampling near those boundaries.
- Motivation: Separatrices define basin boundaries in multistable systems and may have intricate, complex, or fractal geometric structures.They commonly correspond to stable manifolds of saddle-type invariant sets, unstable source-type invariant sets, or other complex manifolds.
- Challenge: Dense basin coverage can identify separatrices in principle but becomes infeasible as system dimension increases.The computational cost of densely covering basins grows rapidly with dimension.
- Related approaches: Existing approaches include manifold continuation, edge tracking and bisection, and Lyapunov exponent Monte Carlo sampling for high-dimensional separatrix approximation.These methods have been applied across fields including chemical kinetics and reactor design.
- Framework: The proposed framework combines supervised classification of attraction basins with generative-model sampling of densities on separatrix manifolds.It is designed to comprehensively characterize and reconstruct separatrices in multistable dynamical systems.
- Method: Neural classifiers partition phase space and uncertainty metrics identify approximate decision-boundary separatrices, after which score-based models sample high-uncertainty boundary regions for iterative refinement and visualization.The approach combines global boundary localization from discriminative models with detailed geometric sampling from generative models, targeting systems where continuation or bisection is computationally prohibitive.
2 Proposed Mathematical Framework
The framework reconstructs basin separatrices by combining neural-network basin classification with Monte Carlo-dropout uncertainty and score-based generative modeling. It treats high-uncertainty regions as empirical proxies for boundaries between attraction basins, while recognizing that the reconstruction depends on data coverage, model capacity, and classifier geometry.
- Dynamical-system setting: Basin boundaries are generically (n −1)-dimensional stable manifolds of saddle-type invariant sets separating attractor basins in Rn.These invariant sets organize phase-space topology and mediate transitions.
- Classifier and uncertainty: A classification neural network assigns initial conditions to attraction basins and estimates prediction uncertainty using Monte Carlo dropout represented by entropy H.The classifier provides both basin membership predictions and an uncertainty measure for each point.
- Generative sampling: High-uncertainty samples serve as proxy points for local or global separatrices and their surrounding phase-space regions.A generative model is trained with initial conditions labeled by classifier uncertainty to generate new points in these regions.
- Approximation limitations: Finite feedforward networks impose piecewise-smooth decision boundaries, so fractal or fine-scale separatrices are necessarily represented by smoothed approximations.The boundary geometry is constrained by the network’s inductive bias and its depth and width.
- Generative sampling: The score-based generative model retains samples above the 90th-percentile entropy threshold, models their distribution, and samples from the learned high-uncertainty measure.The retained dataset D2 is selected using the Monte Carlo-dropout entropy threshold z.
- Validity conditions: The approximation is most reliable with uniform phase-space coverage, sufficient near-separatrix data, expressive networks, and balanced classes.High-uncertainty regions can be sampled iteratively to improve trajectory coverage near the separatrix.
3 Numerical Examples
The framework is demonstrated on multistable systems with progressively more complex basin boundaries, including the Newton fractal and CSTR model. In the CSTR example, uncertainty-based separatrix identification followed by generative sampling improved geometric reconstruction relative to the classifier alone.
- Threshold sensitivity: A 90th-percentile entropy threshold is used as an illustrative cutoff, while alternative thresholds may be warranted for different accuracy requirements.The sensitivity analysis examines thresholds from the 75th to 99th percentile using point count, spatial coverage, centroid, and coordinate-wise spread.
- Threshold sensitivity: Increasing the entropy threshold contracts the extracted high-entropy region toward a tighter estimate of the separatrix where classification uncertainty is highest.The analysis compares representative 85th, 90th, and 95th percentile thresholds and uses an instability score whose lower values indicate greater robustness.
- CSTR and uncertainty quantification: MC dropout, deep ensemble averaging, and Laplace approximation were compared by evaluating their high-uncertainty regions against a SCIGMA-computed ground-truth separatrix.The comparison assesses different predictive-uncertainty estimates from the trained classifier for separatrix identification.
- CSTR and uncertainty quantification: In the CSTR example, SGM refinement reduced Chamfer distance from 3.05 to 1.58, a 48.2% reduction, while coverage increased from 96.7% to 99%.The generated samples also captured fine-scale geometric features, and Hausdorff distance changed slightly from 20.9 to 20.58.
4 Conclusion
The work introduces a data-driven framework combining supervised classification and generative modeling to approximate separatrices in multistable dynamical systems. Future work will add iterative active learning to address false positives caused by isolated high-uncertainty regions in sparsely sampled areas.
- Conclusion: The framework combines supervised classification with generative modeling to computationally approximate separatrices in multistable dynamical systems.It addresses sampling challenges because basin boundaries govern transitions between stable states but are poorly sampled in standard simulations due to their repelling behavior.
- Conclusion: Future work will augment the method with an iterative active-learning framework to mitigate existing limitations.The stated goal is to improve the method’s handling of uncertainty-driven boundary identification.
- Conclusion: Isolated high-uncertainty regions in sparsely sampled areas can produce false positive identifications of spurious boundaries far from true separatrices.Without local training data, the network may default to uncertain predictions even when far from dynamically significant structures.