Coarsening dynamics of fingerprint labyrinthine patterns: Machine learning assisted characterization

TL;DR

This paper shows that fingerprint labyrinthine patterns formed under the Turing–Swift–Hohenberg framework coarsen primarily through localized defects (junctions and terminals) whose dynamics are constrained by the surrounding stripe network, leading to slow, arrested relaxation rather than conventional diffusive coarsening. A template-matching CNN (TM-CNN) is used to detect and classify defects in real space, enabling defect statistics, interconversions, and spatial correlations to be quantified via defect densities and pair distribution functions. The analysis reveals strong short-range defect correlations, defect clustering, and interconversion pathways that drive coarsening toward a frozen state with a finite residual defect density. The study advances phase-ordering theory by extending it to pattern-forming systems with geometric frustration and demonstrates a practical, physics-informed ML approach for decoding complex nonequilibrium morphologies across disciplines.

Abstract

Fingerprint labyrinthine patterns exhibit a level of structural complexity beyond simple stripe phases, combining local stripe order with a dense network of point-like defects. Unlike symmetry-breaking phases, where coarsening proceeds via diffusive defect annihilation, or conventional stripe phases, where curvature-driven motion of extended grain boundaries dominates, the coarsening of fingerprint labyrinths is governed primarily by localized junction and terminal defects. Using the Turing-Swift-Hohenberg equation, we study the nonequilibrium relaxation of fingerprint labyrinthine patterns following a quench. To go beyond conventional Fourier-based diagnostics, we employ a template-matching convolutional neural network (TM-CNN) to identify and track junctions and terminals directly in real space, enabling a quantitative characterization of defect statistics and spatial correlations. We show that, although these point-like defects drive coarsening, their motion is strongly constrained by the surrounding stripe geometry, leading to slow, nondiffusive dynamics that are qualitatively distinct from both conventional phase ordering and stripe coarsening. Together, these results establish defect-mediated dynamics as the central organizing principle of fingerprint labyrinthine coarsening and demonstrate the effectiveness of machine-learning-assisted approaches for complex pattern-forming systems.

Figures (10)

• Figure 1: Coarsening dynamics of the fingerprint labyrinthine phase of the TSH equation at increasing times. Coarsening is dominated by stripe breaking and reconnection events, corresponding to the annihilation of junctions and stripe terminals, while the motion of extended grain boundaries is strongly suppressed. The evolution is governed by local topological rearrangements within the stripe network rather than curvature-driven domain growth.
• Figure 2: Time evolution of the global structure factor for the fingerprint labyrinthine phase obtained from simulations of the TSH equation. The top panels show the two-dimensional structure factor $S(\mathbf{q})$ at representative times, displaying a ring-like distribution centered at a well-defined wave number. The bottom panels show the corresponding angularly averaged structure factor $I(q) \sim \exp[-(q - q_c)^2/2\sigma^2]$. Solid lines indicate Gaussian fits to $I(q)$, and $\sigma$ denotes the standard deviation of the fit, which characterizes the radial width of the ring.
• Figure 3: (a) Time evolution of the ring width $\sigma$ in the angular-averaged structure factor of the fingerprint labyrinthine pattern, shown as the normalized deviation $(\sigma-\sigma_\infty)/\sigma_\infty$, where $\sigma_\infty$ is the long-time limit. The relaxation is well described by a power-law decay (solid line). (b) Corresponding evolution of the correlation length $L \sim 1/\sigma$, normalized by its asymptotic value $L_\infty$, showing slow growth associated with the narrowing of the structure-factor ring.
• Figure 4: Schematic of point-like defects in stripe patterns. (a) Junction defect (red-stripe viewpoint), corresponding to a $-1/2$ disclination in an effective orientational description. (b) Terminal defect, corresponding to a $+1/2$ disclination, with the sign set by the winding of the local stripe orientation. (c) Bound pair of terminal (blue-stripe) and junction (red-stripe) forming a composite defect with zero net disclination charge, behaving as an effective dislocation of the stripe pattern.
• Figure 5: Schematic illustration of the template-matching convolutional neural network (TM-CNN) workflow used to identify point-like defects in labyrinthine patterns. Starting from the input image, a set of predefined templates is used to perform template matching, generating correlation maps that highlight candidate defect locations. A low detection threshold is employed to ensure that all potential junctions and terminals are captured, followed by non-maximum suppression to remove duplicate detections. Image patches centered on these candidate locations are then extracted and passed to a convolutional neural network, which classifies each patch as a junction (J), a terminal (T), or a false detection (F). This two-stage procedure combines the sensitivity of template matching with the selectivity of a CNN, enabling efficient and accurate detection of point defects.