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Data-driven continuation of patterns and their bifurcations

Wenjun Zhao, Samuel Maffa, Björn Sandstede

TL;DR

The work addresses delineating parameter-domain boundaries between distinct spatial patterns in reaction-diffusion and related systems by building pattern statistics from ensembles of randomized initial data and comparing them with the 2-Wasserstein distance $d_W$. It introduces a probabilistic framework where sublevel-set patterns $A(u)=u^{-1}((- fty,c])$ yield feature functions (e.g., connected components, area distributions, roundness) and a statistic $oldsymbol{ u}_f(p)$ that depends continuously on parameters. Bifurcation curves are traced by maximizing the rate of change of pattern statistics along predictor-corrector arclength continuations, using an objective $G_f(p,q)=d_W(oldsymbol{ u}_f(p),oldsymbol{ u}_f(q))$ and a finite-difference bifurcation surrogate. The approach is demonstrated across numerous PDE models and a stochastic ABM, providing a flexible, automated method to compute pattern boundaries without bespoke boundary-value formulations, with robust behavior under various discretization and sampling choices. It offers a path toward parameter inference and multi-parameter extensions in complex pattern-forming systems.

Abstract

Patterns and nonlinear waves, such as spots, stripes, and rotating spirals, arise prominently in many natural processes and in reaction-diffusion models. Our goal is to compute boundaries between parameter regions with different prevailing patterns and waves. We accomplish this by evolving randomized initial data to full patterns and evaluate feature functions, such as the number of connected components or their area distribution, on their sublevel sets. The resulting probability measure on the feature space, which we refer to as pattern statistics, can then be compared at different parameter values using the Wasserstein distance. We show that arclength predictor-corrector continuation can be used to trace out transition and bifurcation curves in parameter space by maximizing the distance of the pattern statistics. The utility of this approach is demonstrated through a range of examples involving homogeneous states, spots, stripes, and spiral waves.

Data-driven continuation of patterns and their bifurcations

TL;DR

The work addresses delineating parameter-domain boundaries between distinct spatial patterns in reaction-diffusion and related systems by building pattern statistics from ensembles of randomized initial data and comparing them with the 2-Wasserstein distance . It introduces a probabilistic framework where sublevel-set patterns yield feature functions (e.g., connected components, area distributions, roundness) and a statistic that depends continuously on parameters. Bifurcation curves are traced by maximizing the rate of change of pattern statistics along predictor-corrector arclength continuations, using an objective and a finite-difference bifurcation surrogate. The approach is demonstrated across numerous PDE models and a stochastic ABM, providing a flexible, automated method to compute pattern boundaries without bespoke boundary-value formulations, with robust behavior under various discretization and sampling choices. It offers a path toward parameter inference and multi-parameter extensions in complex pattern-forming systems.

Abstract

Patterns and nonlinear waves, such as spots, stripes, and rotating spirals, arise prominently in many natural processes and in reaction-diffusion models. Our goal is to compute boundaries between parameter regions with different prevailing patterns and waves. We accomplish this by evolving randomized initial data to full patterns and evaluate feature functions, such as the number of connected components or their area distribution, on their sublevel sets. The resulting probability measure on the feature space, which we refer to as pattern statistics, can then be compared at different parameter values using the Wasserstein distance. We show that arclength predictor-corrector continuation can be used to trace out transition and bifurcation curves in parameter space by maximizing the distance of the pattern statistics. The utility of this approach is demonstrated through a range of examples involving homogeneous states, spots, stripes, and spiral waves.

Paper Structure

This paper contains 17 sections, 7 theorems, 57 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Pattern statistics: A probabilistic framework
  3. Tracing bifurcations using pattern statistics
  4. Initialization
  5. Predictor-corrector steps
  6. Assessing dependence on algorithmic parameters
  7. Results
  8. Discussion
  9. Models
  10. Barkley and Bär--Eiswirth models:
  11. Brusselator:
  12. Bullara--De Decker:
  13. Gray--Scott:
  14. Rayleigh--Bénard:
  15. Rössler:
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $B$ be a one-dimensional compact $C^2$ submanifold of $D$, then there exists an open neighborhood $V$ of $B$ in $D$ and a $C^2$-diffeomorphism $\theta\colon B\times(-1,1)\to V$, $(b,y)\mapsto\theta(b,y)$ so that $\theta|_{B\times\{0\}}$ is a $C^2$-diffeomorphism onto $B$. The map $\theta$ is ref

Figures (19)

  • Figure 1: The panels illustrate (i) a color plot of the solution of the underlying PDE model at time that is so large that spots and stripes have emerged, (ii) the corresponding sublevel set, (iii) the associated $\alpha$-shape (a polygonal approximation of the boundary of the sublevel set), and (iv) the distribution (histogram) of three different feature functions evaluated on each of ten simulations that start from different random initial data, namely the areas of the connected components of the $\alpha$-shape (top), their perimeters (center), and the roundness score, which is the inverse of the isoperimetric ratio or, equivalently, the fraction of area over perimeter and therefore measures of how elongated each connected component is (bottom).
  • Figure 2: Shown is a segment of the transition curve that separates the regions in parameter space where, respectively, spots and stripes are prevalent in the Brusselator model. The feature function is given by the roundness scores of the connected components. Sample pattern statistics (given by histograms of feature evaluations of direct simulations with an ensemble of randomized initial data) and sample patterns are included in the insets. The colored disks correspond to parameter values where the pattern statistics was computed during continuation, with colors indicating the expectation of the roundness score.
  • Figure 3: We illustrate the definition of $\alpha$-shapes and their dependence on the radius $\alpha$. Panel (i) shows the original data set. The associated $\alpha$-shapes for radii $\alpha_1$ and $\alpha_2$ with $\alpha_1<\alpha_2$ are shown as polygons in panels (ii) and (iii), respectively. The circles $\partial U_{\alpha_{1,2}}$ determine which data points are connected by edges.
  • Figure 4: Panel (i) illustrates the curve $\Gamma$ that separates parameter regions where spots and stripes prevail. The feature function $f_\mathrm{Conn}$ assigns to each pattern the number of its connected components: as indicated in panel (ii), the expectation $E(\mu_f)$ of its pattern statistics should therefore change from low to high values as we cross from the stripe into the spot region.
  • Figure 5: Panel (i) outlines the geometry for the predictor-corrector step, where we omitted the subscript $m$ in the notation. Panel (ii) indicates at which points we evaluate the bifurcation function $\mathfrak{g}(z)$ (labeled by disks) and how this translates into evaluations of the pattern statistics $\mu_f(p)$ (labeled by crosses). The points in green correspond to the initial bisection in step 1., while the points in blue correspond to the refinement in step 2. for the case $\sigma=1$.
  • ...and 14 more figures

Theorems & Definitions (7)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7