End-to-end deep learning method for solving nonlocal phase-field models
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We present an efficient end-to-end deep learning approach for solving nonlocal phase-field models of Allen-Cahn and Cahn-Hilliard type. Traditional PDE-based phase-filed models often require severe grid refinement near diffuse interfaces. In contrast, nonlocal phase-filed models can generate truly sharp interfaces, sometimes only one grid cell wide, but introduce increased computational complexity due to long-range interactions. Our method addresses this challenge by learning the fully discrete time-stepping operator associated with a Fourier collocation discretization in space and a semi-implicit scheme in time. We introduce convolutional neural network architectures whose loss functions are defined using the residual of the fully discrete systems, enabling training without ground-truth solution data. To account for nonlocal effects, the network inputs include the nonlocal interaction term as an additional input channel. Extensive numerical experiments demonstrate accurate and stable prediction of the model dynamics across a range of scenarios, with significant reductions in computational cost at comparable fidelity. The proposed framework offers a scalable alternative for large-scale nonlocal phase-field simulations and can be readily extended to other nonlocal or multiscale evolution problems.
