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

2026-09-25

AI Summary

This study addresses the prediction challenges in sintering arising from the coupling of densification and grain growth, material-specific kinetic variations, and data sparsity by proposing the Sinter-PiNDiff framework. This method employs physics-integrated neural differential equations with a dual-network architecture, where neural networks learn rate equation coefficients under embedded physical constraints. A smooth saturation factor is introduced to capture near-theoretical-density behavior, while deep ensembles enable uncertainty quantification and retrainable modeling across materials under small-sample conditions. Evaluated across twelve comparative experiments on three materials including MgO, Sinter-PiNDiff achieves the lowest prediction errors, yielding density normalized root-mean-square errors (NRMSE) of only 10.8%–14.6%. These results establish it as an efficient and robust predictive tool for sintering processes.

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.

Problem

• sintering
• predictive modeling
• densification
• grain growth
• sparse data

Innovation

• Physics-integrated neural differentiable modeling
• Retrainable framework
• Sintering kinetics
• Coupled rate equations
• Deep ensembles