高阶变尺度能量变分神经网络用于相场梯度流

中文摘要 AI 辅助

本文针对相场梯度流提出了一种高阶变尺度能量变分神经网络（VS-EVNN）方法。每个离散阶段使用一个更新状态网络，其空间输入以界面尺度为中心并进行归一化，而采样能量和移动惩罚相对于基线EVNN公式保持不变。这种重定标源于能量中的坐标变换，该变换使变换后泛函的梯度系数归一化。我们将尺度化表示与高阶多阶段变分外推格式以及启发式自适应时间步控制器相结合。每个阶段被求解为一个网络最小化问题，其中较早阶段在求积网格上作为固定锚点，从而实现了这些多阶段变分格式的神经实现，包括用于Cahn-Hilliard流的迁移率加权、质量守恒版本。该方法在一维和二维Allen-Cahn和Cahn-Hilliard流上进行了测试，后者采用质量守恒度量。在报告的优化设置下，尺度化网络比未尺度化的EVNN基线产生更低的解误差，且高阶格式在测试示例中提高了精度。

英文摘要

In this paper, we develop a high-order variable-scaled energetic variational neural network (VS-EVNN) method for phase-field gradient flows. Each discretization stage uses an updated-state network whose spatial inputs are centered and divided by the interface scale, while the sampled energy and movement penalties remain unchanged relative to the baseline EVNN formulation. This rescaling is motivated by a coordinate change in the energy that normalizes the gradient coefficient of the transformed functional. We combine the scaled representation with high-order, multi-stage variational extrapolation schemes and a heuristic adaptive time-stepping controller. Each stage is solved as a network minimization problem, in which the earlier stages serve as fixed anchors on the quadrature grid, yielding a neural realization of these multi-stage variational schemes, including a mobility-weighted, mass-conserving version for Cahn--Hilliard flows. The method is tested on one- and two-dimensional Allen--Cahn and Cahn--Hilliard flows, the latter in the mass-conserving metric. Under the reported optimization settings, the scaled networks yield lower solution errors than the unscaled EVNN baseline, and the higher-order schemes improve the accuracy in the test examples.

发表机构

• School of Mathematics, Southeast University（东南大学数学学院）
• Department of Mathematics, University of Alabama（阿拉巴马大学数学系）

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