A Hybrid Shallow–Deep Neural Network-Based Constitutive Model for High-Temperature Flow Stress Prediction of Ti-6Al-4V Titanium Alloy and Its Integration into MATLAB-Based Engineering Mathematics Instruction

Ti-6Al-4V titanium alloy is widely used in aerospace, space exploration, and automotive manufacturing. Characterizing its high-temperature flow stress is essential for engineering design and numerical simulation. To address the limited prediction accuracy of the conventional Johnson–Cook constitutive model and the generalization challenges of neural networks trained on small datasets, this study develops a hybrid shallow–deep neural network workflow. Mechanical tests at 25, 100, 150, 200, and 300 °C characterize the constitutive behavior and fracture features of large Ti-6Al-4V castings. A shallow neural network learns the response within the sampled domain, while a long short-term memory (LSTM) network extends temperature-dependent stress predictions beyond this domain. The generated data support subsequent nonlinear regression to determine the parameters of a modified Johnson–Cook constitutive model, which is compared with the available experimental responses. The workflow is further incorporated into a project-based engineering mathematics software course using the Deep Learning Toolbox in MATLAB R2021a. Through experimental data processing, constitutive fitting, neural-network-assisted data generation, subinterval error evaluation, and failure diagnosis, the instructional case develops students’ capabilities in data-driven modeling, reproducible computation, and cautious extrapolation.