Neural surrogates for crystal growth dynamics with variable supersaturation: explicit vs. implicit conditioning

1. Problem Statement

Crystal growth simulations, particularly those involving interface motion and faceting, are computationally expensive. Traditional methods like Phase-Field (PF) models require solving stiff nonlinear partial differential equations (PDEs) on fine meshes, which limits the ability to screen materials or simulate long timescales and large domains.

While Machine Learning (ML) surrogates, specifically Convolutional Recurrent Neural Networks (CRNNs), have shown promise in accelerating these simulations, a critical gap exists in how these models handle variable driving forces. In crystal growth, the supersaturation parameter ($\Delta\mu$) dictates both the growth rate and the fine details of the crystal morphology (e.g., sharpness of facets).

• The Challenge: How should a neural network be conditioned to predict evolution under variable supersaturation?
  • Implicit Conditioning: The model infers $\Delta\mu$ from a short sequence of initial frames (treating it as a hidden variable).
  • Explicit Conditioning: The model is provided the numerical value of $\Delta\mu$ as an input alongside a single initial frame.
• Goal: To quantitatively compare these two approaches to determine which offers better accuracy, data efficiency, and generalization for crystal growth dynamics.

2. Methodology

A. Physical Model (Ground Truth)

• Governing Equation: The authors use a 2D Allen-Cahn phase-field model under isothermal conditions.
• Order Parameter: $\phi$ distinguishes the crystal phase ($\phi=1$) from the liquid/gas phase ($\phi=0$).
• Anisotropy: To mimic realistic faceting, an anisotropic kinetic coefficient $k(\alpha) = 1 + \beta \cos(N\alpha)$ is introduced ($N=6$ for hexagonal shapes).
• Variable: The supersaturation $\Delta\mu$ is varied uniformly in the range $[0.2, 0.8]$.
• Data Generation: 7,500 simulation sequences were generated on a $128 \times 128$ grid. Each sequence consists of 200 time-frames (spanning $200\tau$) starting from random elliptical seeds.

B. Neural Network Architectures

Two CRNN architectures based on Convolutional Gated Recurrent Units (ConvGRU) were developed, both with ~140,000 parameters and 3 stacked layers:

1. NNseq (Implicit Conditioning):

   • Input: A "mini-sequence" of $s$ frames (e.g., $s=3, 5, 7$).
   • Mechanism: The recurrent layers act as a memory to implicitly infer the driving force ($\Delta\mu$) and the evolution rate from the temporal changes in the input frames.
   • Task: Sequence completion/prediction.

2. NNpar (Explicit Conditioning):

   • Input: A single initial frame ($\phi_t$) concatenated with a spatial tensor representing the constant value of $\Delta\mu$.
   • Mechanism: The network is explicitly told the driving force.
   • Task: Predict the full sequence from a single snapshot.

C. Training Strategy

• Loss Function: Mean Squared Error (MSE) between the predicted and ground-truth phase fields.
• Curriculum Learning: Training started by predicting only the last frame of a sequence, gradually increasing complexity to predict longer horizons.
• Datasets: Models were trained on datasets of varying sizes ($N_{set} = 500, 1500, 5000, 7500$) to test data efficiency.

3. Key Contributions

1. Systematic Comparison of Conditioning Strategies: The paper provides the first direct, quantitative comparison between implicit (sequence-based) and explicit (parameter-based) conditioning for crystal growth surrogates.
2. Quantification of Data Efficiency: It demonstrates that explicit conditioning is significantly more data-efficient. To achieve the same accuracy as an explicitly conditioned model, the implicitly conditioned model requires a dataset approximately 15 times larger.
3. Generalization Capabilities: The study validates that these surrogates can generalize to:
   • Larger Domains: Successfully scaling to $2048 \times 2048$ grids (16x the training size) without accuracy loss.
   • Longer Timescales: Extrapolating to sequences 10x longer than the training data.
   • Variable Initial Conditions: Handling initial seed coverages significantly lower than those in the training set.
4. Morphological Fidelity: The models successfully learn not just the growth rate scaling but also the subtle morphological changes (corner sharpness) induced by varying $\Delta\mu$.

4. Results

A. Prediction Accuracy

• Explicit vs. Implicit: The NNpar (explicit) models consistently outperformed NNseq (implicit) models.
  • NNpar ($5000$ samples): Achieved a median Maximum Mean Absolute Error (MAE) of 0.011, with 90% of predictions having errors $< 0.018$.
  • NNseq ($7500$ samples): Even with a 50% larger dataset, the best implicit model achieved a median MAE of 0.043, with 90% of predictions having errors $< 0.075$.
• Data Efficiency: The explicit model trained on only 500 samples performed comparably to the implicit model trained on 7,500 samples.

B. Sensitivity to Supersaturation ($\Delta\mu$)

• NNpar: Maintained high fidelity across the entire training range ($\Delta\mu \in [0.2, 0.8]$) and could extrapolate up to $\Delta\mu = 1.0$ with manageable error increases.
• NNseq: Struggled significantly at low supersaturation ($\Delta\mu < 0.3$). The model failed to infer the slow growth rate from short mini-sequences, leading to large errors. It is only reliable for $\Delta\mu \in [0.3, 0.8]$.

C. Scalability and Extrapolation

• Domain Size: Both models scaled perfectly to $2048 \times 2048$ domains. The error distribution remained consistent with the $128 \times 128$ training results.
• Time Extrapolation: Models successfully predicted sequences up to $2000\tau$ (10x training length). Error accumulation was minimal for NNpar but more pronounced for NNseq.
• Initial Coverage: Both models generalized well to initial seed coverages as low as $\theta_0 = 0.03$ (compared to training $\approx 0.11$), though NNpar remained more robust.

5. Significance and Conclusion

• Best Practice: The study concludes that explicit conditioning is the superior strategy whenever the governing parameters (like supersaturation) are known. It yields higher accuracy, requires less training data, and offers better robustness against extrapolation.
• When to use Implicit: Implicit conditioning (NNseq) remains a viable option only when the driving forces are unknown (e.g., analyzing experimental data where $\Delta\mu$ cannot be measured), but it demands significantly larger datasets to compensate.
• Computational Impact: While the current Allen-Cahn solver is fast, the authors note that these surrogates offer massive speedups (potentially $10\times$ on GPU) for more complex, computationally intensive PDEs (e.g., Cahn-Hilliard or finite element methods).
• Scientific Value: The work proves that ML surrogates can capture complex physical phenomena (faceting anisotropy) driven by variable parameters, paving the way for high-throughput screening of crystal growth processes in materials science.

Availability: The datasets and code are open-source on Materials Cloud and GitHub, respectively.