Early-warning the compact-to-dendritic transition via spatiotemporal learning of two-dimensional growth images

TL;DR

This work tackles forecasting incipient interfacial instabilities during nonequilibrium electrodeposition by formulating a horizon-based CDT early-warning task on spatiotemporal growth images. It demonstrates that reliable prediction requires end-to-end learning of joint spatial and temporal representations, with CNN--GRU (and CNN--TCN) outperforming fixed-feature baselines. A low-dimensional latent state extracted from the learned dynamics acts as a surrogate for progressive morphological destabilization, offering a mechanistic interpretation of pre-transition signals. Transferability across reaction-rate conditions is limited but systematically improvable via fine-tuning, underscoring the need to adapt models to operating conditions. The framework provides a general approach for predictive monitoring and potential closed-loop control of pattern-forming nonequilibrium growth systems.

Abstract

Transitions between distinct dynamical regimes are ubiquitous in nonequilibrium systems. As a prototypical example, deposition growth is often accompanied by irreversible morphological instabilities. Forecasting such transitions from pre-transition configurations remains fundamentally challenging, as early precursors are weak, spatially heterogeneous, and masked by inherent fluctuations. Here, we investigate compact-to-dendritic transitions (CDTs) in a two-dimensional particle-based electrodeposition model and formulate a horizon-based early-warning task using trajectory-resolved transition points. We demonstrate that anticipating the CDT is intrinsically a spatiotemporal problem: neither static morphological descriptors nor temporal learning applied to predefined features alone yields reliable predictive signals. In contrast, end-to-end learning of jointly optimized spatial and temporal representations from growth images enables robust anticipation across a wide range of prediction horizons. Analysis of the learned latent dynamics reveals the emergence of a low-dimensional surrogate variable that tracks progressive morphological destabilization and undergoes reorganization near the transition. We further show that the learned spatiotemporal representation exhibits limited but systematic transferability across reaction-rate conditions, with predictive performance degrading as the inference condition departs from the training condition, consistent with changes in the latent-state dynamics. Overall, our results establish a general formulation for forecasting incipient instabilities in nonequilibrium interfacial growth, with implications for the predictive monitoring and control of pattern-forming driven systems.

Figures (17)

• Figure 1: Setup for early-warning learning. (A) Schematic illustration of temporal sequence construction and early-warning labeling. At a reference temporal index $t=T$, the model observes an input sequence of $L_s$ past growth images, $\{I_{T-L_s+1}, \ldots, I_T\}$. The compact-to-dendritic transition (CDT) occurs at $T_c$, and the critical distance is defined as $D_c = T_c - T$. The label $y$ is assigned as alarm-positive if $T_c \in [T,\, T+E)$, i.e., if the transition occurs within a future prediction horizon of length $E$, and as alarm-negative otherwise. (B) Architecture of the CNN--GRU model for end-to-end spatiotemporal learning. CNN extracts spatial feature representations $e_t$ from each image, GRU models their temporal evolution into a hidden state $h_T$, and an MLP outputs the predicted label $\hat{y}$ (alarm-positive or alarm-negative).
• Figure 2: Morphological characterization across the compact-to-dendritic transition (CDT). (A) Representative electrodeposition morphologies during growth at $\log_{10} k = -1.69$. (B) Fractal dimension $d_f$ (Eq. \ref{['eq:df']}) as a function of growth progression. (C) Dimensionless interfacial length $\tilde{L}_{\mathrm{int}}$ (Eq. \ref{['eq:inter_norm']}) as a function of growth progression. Panels (B) and (C) present results for three reaction rates, $\log_{10} k = -1.69$ (green), $-1.90$ (orange), and $-2.12$ (blue). As a visual reference, red dashed lines indicate the CDT at $R_g = R_c$. Here, the ensemble-averaged value of $R_c$ is adopted, reported in Ref. jacobsonContinuumLimitDendritic2025b. (D) Two-dimensional t-SNE embedding of 100 growth trajectories at $\log_{10} k = -1.69$ constructed from the $e_{\mathrm{phys}}$ representations. Markers with black edges denote the starting and end points of each trajectory, and marker color indicates the morphological state: green for compact growth and red for dendritic growth. Grey lines connect each trajectory. Thick colored lines show five representative growth trajectories in the latent space.
• Figure 3: F1-score performance of early-warning models as a function of $E - D_c$ at $L_s = 5$. Circle and triangle markers denote the F1-positive and F1-negative scores, respectively (Eq. \ref{['eq:f1_both']}). Here, $D_c = T_c - T$ is the temporal distance between the prediction time and the trajectory-specific transition point, and $E$ is the prediction horizon. $E \leq D_c$ corresponds to the alarm-negative (compact) regime, whereas $E > D_c$ corresponds to the alarm-positive regime. $D_c$ is determined using the sequence-specific $R_c$. Panels A, G, H, and I correspond to end-to-end learning models. Red vertical lines denote the boundaries between the two early-warning prediction regimes. The grid resolution ($H \times W = 32 \times 32$), CNN feature dimension ($S_F = 16$), and global spatial pooling size ($H_F \times W_F = 1 \times 1$) are fixed across all models.
• Figure 4: Normalized activation of the five most important spatial features $e_t$ for $E=5$ during growth at $\log_{10} k=-1.69$: (A) CNN--GRU, (B) CNN$^{cl}$, (C) Rad$^{phys}$, and (D) CNN--TCN. As a visual reference, red vertical lines indicate the CDT at $R_g = R_c$, adopting the ensemble-averaged value of $R_c$ reported in Ref. jacobsonContinuumLimitDendritic2025b. Bluer colors indicate features with higher importance score.
• Figure 5: Latent-space speed during growth at $\log_{10} k = -1.69$. (A--C) Hidden-state velocity $v_t^h = \lVert h_t - h_{t-1} \rVert$ for the GRU-based models: (A) CNN--GRU, (B) CNN$^{cl}$--GRU, and (C) Rad$^{phys}$--GRU. (D) Latent speed $v_t^C = \lVert c_t - c_{t-1} \rVert$ for the CNN--TCN model, computed from the feature vector $c_t$ of the final hidden layer. (E--H) Magnified views of the regions near the transition point $T_c$ for panels (A--D), respectively. Different colors indicate different values of $E$, consistent with the color coding in Fig. \ref{['fig:main_results']}. Red vertical lines denote the boundaries between the two early-warning prediction regimes, and black horizontal lines serve as guides to the eye.