Technical Summary: Thermally Conditioned Time-Series Forecasting of Grain Growth in Polycrystalline Materials

Problem Statement

Grain growth during heat treatment significantly dictates the microstructure and mechanical properties of polycrystalline materials. While high-fidelity numerical methods (e.g., front-tracking, phase-field) can accurately simulate these processes, their reliance on solving Partial Differential Equations (PDEs) incrementally over time results in prohibitive computational costs for long or non-isothermal thermal cycles. Previous data-driven approaches successfully predicted grain growth under isothermal conditions using recurrent neural networks (RNNs) to forecast grain size distributions as time series. However, these models failed to capture the strong influence of thermal history on grain boundary kinetics, as grain boundary mobility is explicitly temperature-dependent (following Arrhenius-type relations). Consequently, models trained solely on isothermal data cannot accurately predict evolution under realistic industrial heat treatments, which involve complex heating, holding, and cooling stages.

Methodology

The authors propose a reformulated time-series forecasting framework that explicitly incorporates thermal information to handle non-isothermal conditions.

Data Generation and Preprocessing:

• Simulations: Grain growth evolutions were simulated using the validated front-tracking framework (TRM) for 304L stainless steel. Initial microstructures were generated via the LaVoGen algorithm.
• Thermal Cycles: Two datasets were created: one-hour (3600 s) and four-hour (14,400 s) cycles, covering a wide range of heating/cooling rates and peak temperatures.
• Representation: Microstructural states were represented as normalized grain size histograms (30 bins). Thermal states were characterized by instantaneous temperature ($T$) and thermal rate ($\dot{T}$).
• Dataset: The data was downsampled to a 1-minute resolution. A sliding window strategy ($W=5$) was used to construct samples, with thermal descriptors standardized via z-score normalization.

Model Architecture:
The proposed surrogate model is a hybrid deep learning architecture designed to jointly process microstructural and thermal data:

1. Microstructural Encoding: A 1D Convolutional Neural Network (CNN) encodes the grain size histograms to capture short-range morphological dependencies between adjacent bins.
2. Thermal Encoding: A dedicated Multi-Layer Perceptron (MLP) embeds the thermal descriptors ($T$ and $\dot{T}$).
3. Fusion and Temporal Prediction: The latent vectors from the CNN and MLP are concatenated and fed into a recurrent network (evaluating Simple RNN, GRU, and LSTM).
4. Prediction Strategy: The model predicts the variation of the distribution relative to the previous state (residual formulation) rather than the absolute state, using a scaling factor ($\alpha=0.1$) to stabilize auto-regressive rollout. Predictions are made in square-root space to balance bin weights, then squared back.
5. Training: The model uses a "pushforward" strategy, unrolling over three steps using its own predictions to reduce the training-inference mismatch. The loss function is Mean Absolute Error (MAE) on square-rooted frequencies.

Conditioning Strategies:
The study evaluated four strategies to isolate the impact of thermal inputs:

• C0: No thermal input (baseline).
• C1: Temperature ($T$) only.
• C2: Thermal rate ($\dot{T}$) only.
• C3: Combined $T$ and $\dot{T}$.

Key Results

1. Importance of Thermal Conditioning:
Thermal information is indispensable for non-isothermal prediction. The unconditioned model (C0) failed with a Mean Relative Error (MRE) of 45.26% ($R^2 = 0.8055$). Introducing thermal descriptors drastically improved performance:

• C3 (Combined $T + \dot{T}$) achieved the best results: MRE = 2.99% and $R^2 = 0.9970$ for one-hour cycles.
• C1 ($T$ only) and C2 ($\dot{T}$ only) also performed well (MRE ~3.1% and 4.5% respectively), but the combined approach yielded the highest accuracy.

2. Recurrent Architecture Comparison:
Among the recurrent blocks, LSTM outperformed both GRU and Simple RNN.

• LSTM: MRE = 2.99%, $R^2 = 0.9970$.
• GRU: MRE = 4.52%, $R^2 = 0.9935$.
• Simple RNN: MRE = 10.33%, $R^2 = 0.9812$.
  The LSTM's ability to retain information over long sequences was critical for minimizing error accumulation during auto-regressive rollout.

3. Architectural Components:
The hybrid approach (CNN + MLP + LSTM) outperformed a direct LSTM input. Encoding features separately before fusion reduced the MRE from 3.80% to 2.99% and eliminated simulations with errors exceeding 30%.

4. Generalization to Unseen Histories:
The model was tested on four thermal histories excluded from training, including multi-stage and non-monotonic cycles.

• The model successfully generalized to unseen cycles, with MREs ranging from 1.91% to 14.66% and $R^2$ values generally exceeding 0.94.
• Errors were lowest during heating phases and highest during cooling, particularly for complex, multi-stage cycles where grain growth kinetics change abruptly.

5. Computational Efficiency:
While not explicitly quantified in seconds in the results section, the paper notes that the surrogate generates complete trajectories in seconds, compared to the tens of minutes required for the underlying TRM simulations, representing a speedup of over two orders of magnitude.

Significance and Claims

The paper claims to provide a practical step toward surrogate models applicable to realistic, non-isothermal heat treatment scenarios. By extending time-series forecasting from isothermal to non-isothermal regimes, the work demonstrates that:

• Explicitly conditioning the model with both instantaneous temperature and thermal rate is necessary to capture thermo-kinetic effects.
• A hybrid architecture (CNN for microstructure, MLP for thermal context, LSTM for temporal dynamics) effectively preserves physical consistency while maintaining computational efficiency.
• The framework can generalize to complex, unseen thermal paths (including multi-stage heating/cooling), offering a computationally efficient alternative to high-fidelity PDE-based simulations for large-scale studies and process-level applications.

The authors maintain a modest tone, acknowledging that while the model reproduces main trends with high accuracy, errors do accumulate over time, particularly in cooling phases and for complex thermal histories, though they remain within physically admissible bounds.