Gradient Enhanced-Expert Informed Neural Network (GE-EINN) for forming depth prediction from a small-scale metal stamping dataset

Abstract

Machine learning (ML) prediction tools are becoming an ever-growing topic of interest across various disciplines but are often perceived as probabilistic black boxes with good interpolation and lesser extrapolation performances. To this end, this contribution investigates the modeling and performance of a Gradient Enhanced-Expert Informed Neural Network (GE-EINN) consisting of a neural network where the backpropagation was upgraded by user-defined constraints in terms of partial differential equations (PDEs) aimed at improving the prediction accuracy outside the latent space. Experiments and finite element analysis (FEA) results under cold and warm forming conditions of pure Titanium sheet employed in Li-battery housing were carried out by recording the maximum filling depth before failure onset, subsequently correlated to 17 material and process features. Given the absence of physics-based PDEs, three numerical solutions based on the three features with the highest importance were developed for the GE-EINN model and associated with three residuals' functions based on firstand second-order PDEs. As benchmarks for the developed GE-EINN model, a Gradient Boosting (GB) ensemble and a Deep Neural Network (DNN) algorithm were considered. First, no significant differences were observed between GE-EINN, GB, and DNN during the cross-validation process, with average deviations equal to 3.9 %, 5.6 %, and 5.7 %. However, when tested on additional points within and outside the latent space, the GE-EINN model showed average improvements equal to 25.6 % and 114.2 % for the GB model, and 30.7 % and 67.3 % for the DNN model, respectively. All three models were further tested on a metallic bipolar plate (MBP) for proton exchange membrane fuel cell (PEMFC), resulting in 46.8 % and 14.5 % average improvements to GB and DNN formulations, when the GE-EINN residuals include first and second-order derivatives, but no PDEs crossproducts among multiple features. Overall, the results show the improvements provided by the GE-EINN approach in extrapolation scenarios but also the need for additional research on the equation discovery phase to improve the generality of the solution.