AI 中文总结
本文提出保结构GTransNet-BDF格式求解混合形式Cahn-Hilliard方程，可保质量守恒与能量稳定，适用于复杂场景，数值实验验证其准确性与鲁棒性。
AI 中文摘要
本文研究混合形式Cahn-Hilliard方程的保结构神经网络框架。我们采用广义可迁移神经网络（GTransNet）进行空间近似，采用稳定后向差分公式（BDF）进行时间离散化。所得时间一阶和二阶GTransNet-BDF格式在时间离散层面可保持质量守恒并满足能量稳定性。该格式通过配点法实现，每时间步需求解系数矩阵为常数的最小二乘方程组，该方程组的解（决定网络输出层权重）因最小二乘残差非零会破坏质量守恒。为解决此问题，我们提出一种新颖的后处理质量守恒投影，通过极小化问题强制质量约束，其解的计算成本可忽略不计。该方法的核心优势在于预设隐藏层及无网格特性，使其适用于复杂区域、可变迁移率及长时间模拟。大量二维和三维数值实验验证了其收敛性、质量守恒性、能量耗散性，同时证明了所提GTransNet-BDF格式的准确性与鲁棒性。
英文摘要
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.
