Let's do the time-warp-attend: Learning topological invariants of dynamical systems

TL;DR

The paper presents a physics-informed, data-driven framework for classifying dynamical regimes and detecting bifurcations across diverse systems by learning topologically invariant features. It leverages topological augmentations of vector fields, angular representations, and convolutional-attention networks trained on a prototypical Hopf oscillator to achieve out-of-sample generalization. The approach successfully transfers to complex systems, recovers bifurcation boundaries, and applies to high-dimensional models (e.g., the repressilator) and real biological data (pancreatic cell differentiation) with high accuracy. This equation-free, topology-focused method offers a scalable, robust way to analyze long-term dynamical behavior in physical and biological networks without explicit governing equations.

Abstract

Dynamical systems across the sciences, from electrical circuits to ecological networks, undergo qualitative and often catastrophic changes in behavior, called bifurcations, when their underlying parameters cross a threshold. Existing methods predict oncoming catastrophes in individual systems but are primarily time-series-based and struggle both to categorize qualitative dynamical regimes across diverse systems and to generalize to real data. To address this challenge, we propose a data-driven, physically-informed deep-learning framework for classifying dynamical regimes and characterizing bifurcation boundaries based on the extraction of topologically invariant features. We focus on the paradigmatic case of the supercritical Hopf bifurcation, which is used to model periodic dynamics across a wide range of applications. Our convolutional attention method is trained with data augmentations that encourage the learning of topological invariants which can be used to detect bifurcation boundaries in unseen systems and to design models of biological systems like oscillatory gene regulatory networks. We further demonstrate our method's use in analyzing real data by recovering distinct proliferation and differentiation dynamics along pancreatic endocrinogenesis trajectory in gene expression space based on single-cell data. Our method provides valuable insights into the qualitative, long-term behavior of a wide range of dynamical systems, and can detect bifurcations or catastrophic transitions in large-scale physical and biological systems.

Figures (16)

• Figure 1: Overview. Our framework classifies long-term dynamical behavior of point vs periodic attractor across systems and regimes by leveraging topologically-invariant augmentations of a single prototypical system. To encourage generalization, we (1) abstract system-specific vector magnitudes and use angular representations instead, and (2) deploy attention layers which focus on learning essential dynamical cues. Using this framework, we are able to classify and recover the bifurcation boundaries of complex, unseen systems. We apply this method to the problem of recovering cell-cycle scores which distinguish between proliferating and differentiating cell populations. All vector fields are plotted as streamlines.
• Figure 2: Inference of bifurcation boundaries. Each system is plotted at coordinates of its simulated parameter values and colored according to its average cycle prediction across 50 training re-runs. The true parametric bifurcation boundary is plotted in black. Colors are scaled between a perfect cycle classification (red) and perfect point classification (blue) with white at evenly split classification.
• Figure 3: Oscillations in a repressilator system. Example of oscillations in the repressilator system viewed as a time-series (left) or projected onto TetR-LacI protein plain (middle). (right) Predicted oscillatory behavior by our model output upon varying transcription rate without repression ($\alpha$) and the ratio of protein and mRNA degradation rates ($\beta$). The theoretical boundary is marked in black. See details in Appendix \ref{['supp:repressilator']}.
• Figure 4: Proliferation-to-differentiation transition in pancreas development.(left) Gene expression velocity of cells undergoing pancreas endocrinogenesis in UMAP space. Colors distinguish between different cell types. (middle) Cell-cycle score based on the fraction of cells with high S-phase gene expression (with S-score $>$ 0.4). (right) Predicted cell-cycle score computed as the average classification prediction across 50 training re-runs. See details in Appendix \ref{['supp:pancreas']}.
• Figure A5: Framework. Our point-periodic classifier begins by augmenting the vector field data and converting it to an angular representation. Data is then passed through convolutional-attention layers and an MLP classifier.

Theorems & Definitions (6)

• Definition 1: Continuous-time real dynamical system
• Definition 2: Limit cycle
• Definition 3: Asymptotic stability
• Definition 4
• proof
• Definition 5