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

Eliane Younes; Elie Hachem; Marc Bernacki

cond-mat.mtrl-sciNov 4, 2026

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

Authors: Eliane Younes, Elie Hachem, Marc Bernacki

Organizations: Mines Paris, PSL University Centre for material forming (CEMEF), UMR CNRS 06904 Sophia Antipolis, France

Abstract

Grain growth during heat treatment strongly affects the microstructure of polycrystalline materials and, consequently, their mechanical properties. Although numerical methods such as front-tracking or front-capturing simulations can accurately describe grain growth, their high computational cost limits their use for long or non-isothermal thermal cycles. In previous work, grain growth was successfully predicted under isothermal conditions by treating grain size distributions as time series and forecasting their evolution using recurrent neural networks, allowing complete grain growth trajectories to be generated in seconds rather than through costly PDE-based simulations. In the present work, this forecasting framework is reformulated to handle non-isothermal conditions by explicitly incorporating thermal information into the model architecture. Grain size distributions are encoded using a convolutional neural network, thermal descriptors are embedded through a dedicated multilayer perceptron, and temporal evolution is predicted using a recurrent network. Instantaneous temperature and thermal rate are provided at each time step, allowing the model to adapt its predictions to changing thermal conditions. The framework is trained and evaluated on datasets generated from validated front-tracking simulations across a wide range of non-isothermal thermal paths. The results show that the thermally conditioned model accurately reproduces grain growth dynamics under complex thermal histories, achieving, in k-fold cross-validation over all simulations, mean relative errors of 2.99% (one-hour cycles) and 4.51% (four-hour cycles) with coefficients of determination exceeding 0.99, and mean relative errors between 1.91% and 14.66% on ten simulations following four unseen thermal histories of increasing complexity.

Figures & tables

Parameter	Value(s)	Description
$a$ (mm)	2, 3, 4, 5	Side length of the square domain
Distribution	$\mathcal{N}(\bar{R},\sigma^{2})$	Initial ECR distribution (LaVoGen)
$\bar{R}$ ( $\mu$ m)	20	Mean equivalent circle radius (ECR)
$\sigma$ ( $\mu$ m)	2, 8, 16, 32	Standard deviation of the ECR distribution
$T$ (K)	293.15–1323.15	Temperature range
$dT/dt$ heating (K s -1 )	0.215–1.0	Heating rate range

Table 1: Main parameters used for LaVoGen microstructure generation and TRM grain growth simulations under non-isothermal thermal cycles.

Panel of Figure 1 — Figure 1: Non-isothermal thermal histories considered in this study. Short-duration cycles (1 h) are used to investigate fast thermo-kinetic regimes, whereas long-duration cycles (4 h) are introduced to examine slower grain growth conditions. In each case, the thermal evolution is characterized by the instantaneous temperature $T(t)$ and the corresponding thermal rate $\dot{T}(t)$ .

Figure 2: Schematic representation of the thermally conditioned surrogate model for grain size distribution forecasting. A temporal window of histograms is encoded using a CNN, while the thermal descriptors $T(t)$ and $\dot{T}(t)$ are processed through a dedicated MLP. The resulting feature vectors are concatenated and passed to a recurrent temporal model (RNN, GRU, or LSTM). The output projection predicts the next grain size distribution, which is reused during auto-regressive rollout with known future thermal conditions.

Panel of Figure 3 — Figure 3: Thermal histories of the four generalization cases excluded from training. For each case, the temperature evolution (left) and the corresponding thermal rate (right) are reported throughout the cycle.

Configuration	MRE (%) $\downarrow$	$R^{2}\uparrow$
C0 – No thermal input	45.26	0.8055
C1 – Temperature only ( $T$ )	3.10	0.9969
C2 – Thermal rate only ( $\dot{T}$ )	4.53	0.9905
C3 – Temperature + thermal rate ( $T+\dot{T}$ )	2.99	0.9970

Table 2: Global prediction performance averaged over all 86 one-hour simulations (

k

-fold cross-validation)

Recurrent block	MRE (%) $\downarrow$	$R^{2}\uparrow$
Simple RNN	10.33	0.9812
GRU	4.52	0.9935
LSTM	2.99	0.9970

Table 3: Comparison of recurrent formulations within the hybrid architecture (CNN + MLP + recurrent).

Panel of Figure 4 — Figure 4: Comparison of the recurrent formulations within the hybrid architecture (CNN + MLP + recurrent block) over the 86 one-hour simulations: mean (a) MAE, (b) MRE and (c) $R^{2}$ over time.

Configuration	CNN	MLP	MRE (%) $\downarrow$	Max. MRE (%) $\downarrow$	$R^{2}\uparrow$
LSTM only	–	–	3.80	10.75	0.9942
CNN + MLP + LSTM	✓	✓	2.99	8.17	0.9970

Table 4: Impact of architectural components under identical thermal conditioning (one-hour dataset,

k

-fold cross-validation).

Configuration	Heating $\downarrow$	Plateau $\downarrow$	Cooling $\downarrow$
C0 – No thermal input	41.26	39.78	56.50
C1 – $T$ only	0.48	3.76	5.61
C2 – $\dot{T}$ only	1.44	5.34	7.48
C3 – $T+\dot{T}$	0.44	3.59	5.49

Table 5: Phase-wise Mean Relative Error (averages over the 86 one-hour simulations).

Panel of Figure 5 — Figure 5: Temporal evolution of forecasting accuracy and physical consistency for the selected hybrid CNN–MLP–LSTM model (C3) over the 86 one-hour simulations: mean (line) and minimum–maximum envelope (shaded area). (a) Sum of predicted relative frequencies. (b) Mean Absolute Error (MAE) over time. (c) Mean Relative Error (MRE) over time. (d) Coefficient of determination ( $R^{2}$ ) over time.

Panel of Figure 6 — Figure 6: Comparison of predicted (CNN–MLP–LSTM) and TRM normalized grain-size distributions for a representative test sequence at selected prediction steps: (a) the initial state, (b) the first predicted time step, (c) an intermediate stage of the thermal cycle, (d) a later stage of grain growth, and (e) the final state of the sequence.

Figure 7: MRE (top) and $R^{2}$ (bottom) of each of the 86 one-hour simulations ( $k$ -fold cross-validation), sorted by increasing MRE and coloured by the standard deviation $\sigma_{0}$ of the initial distribution. The dashed lines give the mean values.

Panel of Figure 8 — Figure 8: Four-hour thermal cycles: (a) MRE and (b) $R^{2}$ over time for the 208 simulations ( $k$ -fold cross-validation): mean (line) and minimum–maximum envelope (shaded area).

Panel of Figure 9 — Figure 9: Four-hour thermal cycles: comparison of predicted (CNN–MLP–LSTM) and TRM normalized grain-size distributions for a representative simulation at (a) the initial state, (b) the first predicted time step, (c) $t=60$ min, (d) $t=120$ min, (e) $t=180$ min, and (f) the final state ( $t=240$ min).

Figure 10: MRE (top) and $R^{2}$ (bottom) of each of the 208 four-hour simulations ( $k$ -fold cross-validation), sorted by increasing MRE and coloured by the standard deviation $\sigma_{0}$ of the initial distribution. The dashed lines give the mean values.

Case	Model	$a$ (mm)	$\sigma_{0}$ ( $\mu$ m)	MRE (%) $\downarrow$	$R^{2}\uparrow$
Case 1 – one-hour cycle	1 h	3	16 ∗	1.91	0.9988
Case 2 – four-hour cycle	4 h	5	8 ∗	10.40	0.9922
			16	14.66	0.9825
			32	12.70	0.9705
Case 3 – multi-stage cycle	1 h	3	2	8.84	0.9919
			8 ∗	6.82	0.9916

Table 6: Prediction performance on the ten simulations of the four unseen thermal cycles. The simulation shown in the corresponding figure is marked with an asterisk; † : intermediate holding stage at 1190.15 K instead of 1123.15 K.

Panel of Figure 11 — Figure 11: Case 1: Comparison of reference (TRM) and predicted (CNN–MLP–LSTM) normalized grain-size distributions for a representative one-hour unseen thermal cycle at (a) the initial state, (b) the first predicted time step, (c) the end of heating, (d) the end of the isothermal holding stage, and (e) the final state after cooling.

Panel of Figure 12 — Figure 12: Case 2: Comparison of reference (TRM) and predicted (CNN–MLP–LSTM) normalized grain-size distributions for a representative four-hour unseen thermal cycle at (a) the initial state, (b) the first predicted time step, (c) the end of heating, (d) the end of the extended isothermal holding stage, and (e) the final state after cooling.

Panel of Figure 13 — Figure 13: Case 3: Comparison of reference (TRM) and predicted (CNN–MLP–LSTM) normalized grain-size distributions for a representative multi-stage unseen thermal cycle at (a) the initial state, (b) after the first isothermal plateau, (c) after partial cooling, (d) following reheating and holding, (e) at the end of the second heating stage, and (f) after the final plateau.

Panel of Figure 14 — Figure 14: Case 4: Comparison of reference (TRM) and predicted (CNN–MLP–LSTM) normalized grain-size distributions for a representative multi-stage non-monotonic thermal cycle at (a) the initial state, (b) the first predicted time step, (c) the end of the first heating stage, (d) after the first holding stage, (e) after reheating and holding at high temperature, (f) following the subsequent holding stage, (g) at the end of the final heating stage, and (h) after the final holding stage.

Materials Science

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Abstract

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