Interfacial dynamics underlie a wide range of phenomena, including phase transitions, microstructure coarsening, pattern formation, and thin-film growth, and are typically described by stiff, time-dependent nonlinear partial differential equations (PDEs). Traditional numerical methods, including finite difference, finite element, and spectral techniques, often become computationally prohibitive when dealing with high-dimensional problems or systems with multiple scales. Neural operators (NOs), a class of deep learning models, have emerged as a promising alternative by learning mappings between function spaces and efficiently approximating solution operators. In this work, we introduce the Multi-Head Neural Operator (MHNO), an extended neural operator framework specifically designed to address the temporal challenges associated with solving time-dependent PDEs. Unlike existing neural operators, which either struggle with error accumulation or require substantial computational resources for high-dimensional tensor representations, MHNO employs a novel architecture with time-step-specific projection operators and explicit temporal connections inspired by message-passing mechanisms. This design allows MHNO to predict all time steps after a single forward pass, while effectively capturing long-term dependencies and avoiding parameter overgrowth. We apply MHNO to solve various phase field equations, including antiphase boundary motion, spinodal decomposition, pattern formation, atomic scale modeling, and molecular beam epitaxy growth model, and compare its performance with existing NO-based methods. Our results show that MHNO achieves superior accuracy, scalability, and efficiency, demonstrating its potential as a next-generation computational tool for phase field modeling. The code and data supporting this work is publicly available at https://github.com/eshaghi-ms/MHNO.
Eshaghi et al. (2025) studied this question.