Machine learning nested in multiphysics finite element analysis: application to flash sintering

Finite element modelling is a powerful tool for predicting temperature field and deformation in flash sintering. However, conventional finite element modelling approaches often rely on empirical constitutive models, which struggle to capture the highly complex, nonlinear behaviour of materials under flash sintering conditions. To overcome this limitation, this study develops a framework that can train artificial neural networks (ANNs) when nested in multiphysics finite element analysis. By replacing empirical rules with ANNs, the finite element model can learn directly from data and, thus, operate as a digital twin of the flash sintering process. A two-step training strategy is adopted: pretraining of ANNs using synthetic data generated from empirical rules, followed by nested retraining within a commercial finite element package using experimental data. To enable the second-step training, a novel backpropagation algorithm integrated with the Adam optimiser is developed. The results demonstrate a significant improvement in predictive accuracy: the mean squared error between predicted and measured strains decreased from 0.0316 (modified Olevsky's constitutive law) to 0.0034 (ANNs). Beyond improving model fidelity, the approach enables continual adaptation as new data become available and is extensible to all engineering problems governed by differential equations.