High‐Order Accurate Taylor Correction‐Based Backward Differentiation Formulas With Spatial Transferable Neural Networks for Moving Interface Problems

Parabolic partial differential equations with moving interfaces arise in a wide range of applications, including materials science, fluid dynamics, and environmental engineering. The primary challenge in their numerical simulation lies in the evolving position of the interface, which introduces temporal discontinuities and necessitates modifications to standard numerical methods. To address this issue, we propose in this paper a class of Taylor correction‐based backward differentiation formula (BDF‐TC) schemes, specifically designed for the time discretization of moving interface problems. For spatial approximation, we employ the so‐called transferable neural network (TransNet), a special class of two‐layer fully connected neural networks in which the hidden‐layer parameters are predetermined and fixed according to a uniform neuron distribution. In particular, at each time step, two TransNets are constructed to approximate the solutions in the subdomains on both sides of the interface and are connected through the interface conditions. The resulting fully discrete system can then be solved efficiently using linear least‐squares techniques, without resorting to stochastic optimization algorithms. Finally, we present numerical experiments and comparisons in both two and three dimensions to demonstrate the accuracy, stability, and efficiency of the proposed method.