Retrainable physics-integrated neural differentiable modeling of sintering across material systems

Zeping Chen; Ani Aprahamian; Khachatur V. Manukyan; Tengfei Luo

cond-mat.mtrl-sciSep 25, 2026

Retrainable physics-integrated neural differentiable modeling of sintering across material systems

Authors: Zeping Chen, Ani Aprahamian, Khachatur V. Manukyan, Tengfei Luo

Organizations: Department of Chemical and Biomolecular Engineering, University of Notre Dame, Notre Dame, IN, USA · Nuclear Science Laboratory, Department of Physics & Astronomy, University of Notre Dame, Notre Dame, IN, USA · Department of Chemistry & Biochemistry, University of Notre Dame, Notre Dame, IN, USA · Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA

Abstract

Sintering is widely used to manufacture ceramics, but coupled densification and grain growth, material-dependent kinetics, and sparse measurements complicate predictive modeling and process design. We present Sinter-PiNDiff, a retrainable physics-integrated neural differentiable framework for predicting density and grain-size evolution. Two neural networks learn densification and grain-growth coefficients within coupled rate equations, while a smooth saturation factor attenuates densification near theoretical density. The same governing structure, network architecture, and training procedure were fitted independently to published data for MgO, Al-doped ZnO, and CaO-doped ThO2. Tests at held-out temperatures and compositions yielded the lowest mean error in all twelve material-metric comparisons against multilayer perceptron and residual network baselines. For MgO, Al-doped ZnO, and CaO-doped ThO2, respectively, density normalized root-mean-square errors were 14.6%, 10.8%, and 14.4%, and grain-size errors using the same metric were 8.6%, 12.1%, and 19.3%. Removing evolving density from both neural-network inputs increased density and grain-size trajectory errors in all three systems and ten of twelve aggregate errors, supporting density-dependent kinetic feedback. Deep ensembles estimated model disagreement, but empirical coverage showed that the uncertainty bands were not calibrated and did not capture all model-data discrepancies. These results establish Sinter-PiNDiff as a retrainable framework for sparse-data prediction and uncertainty-informed selection of sintering conditions.

Figures & tables

Figure 1: Overview of the Sinter-PiNDiff framework. Neural networks learn material-dependent densification and grain-growth coefficients, which are embedded within sintering evolution equations to predict density and grain-size histories. The common governing structure and network architecture are independently retrained for MgO, Al-doped ZnO, and CaO-doped ThO 2 , with separate weights learned for each material system. Schematic output curves and shaded bands illustrate the predicted evolution and ensemble uncertainty, respectively.

Figure 2: Differentiable autoregressive architecture of Sinter-PiNDiff. (a) Density and grain size evolve through recurrent Sinter-PiNDiff cells with neural-network parameters shared across time steps. Predictions at observation times contribute to the training objective, and gradients propagate through the complete rollout. (b) Within each cell, two neural networks receive the temperature $T$ , current density $\rho_{n}$ , and composition $(\chi_{\mathrm{d}},\chi_{\mathrm{b}})$ to predict the effective coefficients $k_{\mathrm{nn},n}$ and $D_{\mathrm{nn},n}$ . The density-saturation factor $\psi(\rho_{n})$ attenuates the densification rate as the relative density approaches one. The resulting rates are integrated using an explicit Euler update to obtain $G_{n+1}$ and $\rho_{n+1}$ .

Figure 3: Experimental design and data partitioning for the three material systems. Filled and open circles denote training and testing conditions, respectively; $\rho$ and $G$ indicate the availability of density and grain-size measurements. Shaded cells identify the testing conditions used for the architecture rollout comparison: $1600\,^{\circ}\mathrm{C}$ for MgO, $0.60$ mol% Al at $1200\,^{\circ}\mathrm{C}$ for Al-doped ZnO, and $8.8$ mol% CaO at $1550\,^{\circ}\mathrm{C}$ for CaO-doped ThO 2 . These conditions were selected independently of model performance as the testing conditions containing time-resolved measurements of both outputs that maximized the normalized distance from the corresponding training design. For MgO, the higher-temperature condition was selected to resolve the equal-distance case and evaluate rapid coupled kinetics beyond the highest grain-size-supervised training temperature.

Model	Learned mapping	Parameters
MLP	$[t,T,\chi_{\mathrm{d}},\chi_{\mathrm{b}}]\mapsto[\rho(t),G(t)]$	55,014
ResNet	$[\rho_{n},T,\chi_{\mathrm{d}},\chi_{\mathrm{b}},t_{n}]\mapsto[\dot{\rho}_{n},\dot{G}_{n}]$	55,174
Sinter-PiNDiff	$[T,\rho_{n},\chi_{\mathrm{d}},\chi_{\mathrm{b}}]\mapsto[D_{\mathrm{nn},n},k_{\mathrm{nn},n}]$ , followed by the retained sintering equations	5,538

Table 1: Learned mappings and trainable parameter counts for the architectures included in the model comparison. Parameter counts correspond to one independently trained model and are identical across the three material systems.

Figure 4: Temporal predictions for MgO across the investigated sintering temperatures. The upper and lower rows show density and grain-size evolution, respectively. Filled and open circles denote training and testing experimental data. Solid curves show the five-model ensemble mean, and shaded bands indicate $\pm 3$ ensemble standard deviations. No experimental grain-size measurements were available at $1650\,^{\circ}\mathrm{C}$ .

Figure 5: Composition-dependent temporal predictions for Al-doped ZnO at $1200\,^{\circ}\mathrm{C}$ . Density and grain-size evolution are shown in the upper and lower rows, respectively, for Al contents from $0$ to $1.20$ mol%. The intermediate $0.08$ and $0.60$ mol% compositions were withheld for testing. Filled and open circles denote training and testing measurements. Solid curves show the five-model ensemble mean, and shaded bands indicate $\pm 3$ ensemble standard deviations.

Figure 6: Temperature-dependent density predictions for Al-doped ZnO after $120$ min of sintering. Results are shown from $1100$ to $1400\,^{\circ}\mathrm{C}$ for Al contents from $0$ to $1.20$ mol%. Filled and open circles denote training and testing measurements. Solid curves show the five-model ensemble mean, and shaded bands indicate $\pm 3$ ensemble standard deviations.

Figure 7: Temporal density predictions for CaO-doped ThO 2 . Columns correspond to CaO contents of $0$ , $2.31$ , $4.25$ , and $8.8$ mol%, and rows correspond to temperatures from $1300$ to $1600\,^{\circ}\mathrm{C}$ . Filled and open circles denote training and testing measurements. Solid curves show the five-model ensemble mean, and shaded bands indicate $\pm 3$ ensemble standard deviations.

Figure 8: Temporal grain-size predictions for CaO-doped ThO 2 . Columns correspond to CaO contents of $0$ , $2.31$ , $4.25$ , and $8.8$ mol%, and rows correspond to temperatures of $1500$ , $1550$ , and $1600\,^{\circ}\mathrm{C}$ . Filled and open circles denote training and testing measurements. Solid curves show the five-model ensemble mean, and shaded bands indicate $\pm 3$ ensemble standard deviations.

Material system	Observable	$N_{\mathrm{test}}$	Within $\pm 2\sigma$	Within $\pm 3\sigma$
MgO	Density	13	10/13 (76.9%)	11/13 (84.6%)
	Grain size	11	2/11 (18.2%)	6/11 (54.5%)
Al-doped ZnO	Density	19	9/19 (47.4%)	12/19 (63.2%)
	Grain size	10	8/10 (80.0%)	8/10 (80.0%)
CaO-doped ThO 2	Density	84	66/84 (78.6%)	76/84 (90.5%)
	Grain size	25	13/25 (52.0%)	16/25 (64.0%)

Table 2: Empirical coverage of the Sinter-PiNDiff ensemble bands over the testing observations. Entries report the number of covered observations divided by the total number of test observations, with the corresponding percentage in parentheses. Here,

\sigma

denotes the standard deviation across five independently trained ensemble members.

Table 3: Predictive performance of Sinter-PiNDiff and data-driven baseline models. Values are reported as the mean $\pm$ one sample standard deviation across five independent model initializations evaluated on the testing dataset. Each normalized root-mean-square error (NRMSE) pools all available observations of the corresponding output and is expressed as a percentage of its test-output range. Conditions with endpoint-only measurements contribute only those observations. Final-density mean absolute error (MAE) is reported in percentage points (p.p.), and final-grain-size mean absolute percentage error (MAPE) is reported as a percentage. Lower values indicate better performance; the lowest mean for each material system and metric is shown in bold.

Figure 9: Comparison of temporal predictions for MgO at $1600\,^{\circ}\mathrm{C}$ . Predictions from the MLP, ResNet, and Sinter-PiNDiff models are shown from left to right. The upper and lower rows show density and grain-size evolution, respectively. Open circles denote testing measurements, solid curves show the five-model ensemble means, and shaded bands indicate $\pm 3$ ensemble standard deviations.

Figure 10: Comparison of temporal predictions for $0.60$ mol% Al-doped ZnO at $1200\,^{\circ}\mathrm{C}$ . Predictions from the MLP, ResNet, and Sinter-PiNDiff models are shown from left to right. The upper and lower rows show density and grain-size evolution, respectively. Open circles denote testing measurements, solid curves show the five-model ensemble means, and shaded bands indicate $\pm 3$ ensemble standard deviations.

Figure 11: Comparison of temporal predictions for $8.8$ mol% CaO-doped ThO 2 at $1550\,^{\circ}\mathrm{C}$ . Predictions from the MLP, ResNet, and Sinter-PiNDiff models are shown from left to right. The upper and lower rows show density and grain-size evolution, respectively. Open circles denote testing measurements, solid curves show the five-model ensemble means, and shaded bands indicate $\pm 3$ ensemble standard deviations.

Figure 12: Ablation of density feedback in the learned kinetic closures. (a–c) Density and (d–f) grain-size predictions for representative testing conditions. Panels (a,d) show MgO at $1500\,^{\circ}\mathrm{C}$ , panels (b,e) show $0.08$ mol% Al-doped ZnO at $1200\,^{\circ}\mathrm{C}$ , and panels (c,f) show $2.31$ mol% CaO-doped ThO 2 at $1550\,^{\circ}\mathrm{C}$ . Within each material system, the representative condition was selected as the testing condition whose density NRMSE ratio was closest to the median. Solid black curves denote the full Sinter-PiNDiff model, dashed orange curves denote the ablated model in which $\rho_{n}$ was removed from both neural-network inputs, and open circles denote experimental measurements. (g) Aggregate ablation error ratios over the full testing dataset for density and grain-size trajectory NRMSE, final-density MAE, and final-grain-size MAPE. The ratio is defined as the error of the model without the density input divided by the error of the full model; values greater than one therefore favor the full model, whereas values below one favor the ablated model.

Materials Property Prediction

Materials property prediction uses data and physical models to estimate how composition, structure, and processing determine a material’s behavior.

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Physics-Informed Attention Mechanism and Generalization Capability of Deep Learning-Based Grain Growth Evolution Prediction

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Mines Paris, PSL University Centre for Material Forming (CEMEF), UMR CNRS 06904 Sophia Antipolis, France

OOD Generalization Physics-Informed ML

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Machine Learning (ML) models for grain growth prediction are typically trained on idealized synthetic data, yet practical applications require generalization to conditions outside the training distribution. This study evaluated the Out-Of-Distribution (OOD) generalization capability of the trained model from our previous study across three test cases, including experimental microstructures, microstructures characterized by a bimodal grain size distribution, and abnormal grain growth. To further probe whether physics-informed architectural design could improve robustness under these different conditions, a boundary-masked attention mechanism was proposed specifically for grain growth, constraining attention to grain boundary pixels. Both the baseline and the proposed physics-informed attention model were evaluated without retraining or fine-tuning on the OOD data. Both models successfully generalized to all three test cases, yet the boundary-masked attention mechanism provided substantial improvements, with the most notable gains for microstructures characterized by a bimodal grain size distribution, where Structural Similarity Index Measure (SSIM) improved from \num{0.6221} to \num{0.7609} and mean grain size (

\overline{R}

) error decreased from \SI{8.75}{\percent} to \SI{3.57}{\percent}. The attention heatmap analysis revealed that the boundary-masked attention model learned to concentrate attention on large grain boundaries in a manner consistent with curvature-driven grain growth physics, emerging from training without being explicitly encoded into the architecture. These results indicate that models trained on synthetic data can generalize to diverse OOD conditions without retraining, and that physics-informed attention may improve accuracy when the boundary morphology matches the training domain.

Abstract

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