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

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.