Structure-preserving generalized transferable neural networks for the Cahn-Hilliard equation

Abstract

This paper is concerned with a structure-preserving neural network-based framework for the Cahn-Hilliard equation in mixed form. We employ generalized transferable neural networks (GTransNet) for spatial approximation and stabilized backward differentiation formulas (BDF) for temporal discretization. The resulting first- and second-order in time GTransNet-BDF schemes are shown to conserve mass and satisfy energy stability at the time-discrete level. The schemes are implemented by a collocation-based method, in which a least-squares system with constant coefficient matrix needs to be solved at each time step. The solution of this system, which determines the output-layer weights of the network, violates mass conservation due to the expected nonzero least-squares residual. To overcome this issue, we introduce a novel post-processing mass-conserving projection that enforces the mass constraint through a minimization problem, whose solution can be computed at negligible computational cost. A key advantage of the proposed method lies in its predetermined hidden layers and mesh-free nature, making the method applicable to complex domains, variable mobility, and long-time simulations. Extensive numerical experiments in two and three dimensions verify convergence, mass conservation, and energy dissipation as well as demonstrate the accuracy and robustness of the proposed GTransNet-BDF schemes.