Constrained Neural Networks Approach of the Cahn-hilliard Equation With u-dependent Mobility

This paper presents a comparative study between the finite element method and neural networks for the numerical resolution of the Cahn-Hilliard equation with concentration-dependent mobility. The neural approach introduced here is based on c-PINNs (constrained Physics-Informed Neural Networks), which incorporate physical constraints directly into their architecture. This integration enables them to effectively capture the complex dynamics of phase separation. Compared to the classical finite element method, c-PINNs adapt flexibly to local variations and resolve phase interfaces with high precision. Their ability to dynamically adjust to spatial gradients reduces the need for fine discretization and extensive mathematical analysis, making them particularly well-suited for complex and high-order problems. Error analysis confirms their robustness and predic-tive strength, highlighting their convergence toward the desired solution. This approach thus paves the way for more accurate and efficient simulations in the field of materials science and engineering.