Allen–Cahn equation is a reaction–diffusion equation and is widely used for modeling phase separation. Machine learning methods for solving the Allen–Cahn equation in its strong form suffer from inaccuracies in collocation techniques, errors in computing higher-order spatial derivatives, and the large system size required by the space–time approach. To overcome these challenges, we propose solving the L2 gradient flow of the Ginzburg–Landau free energy functional, which is equivalent to the Allen–Cahn equation, thereby avoiding the second-order spatial derivatives associated with the Allen–Cahn equation. A minimizing movement scheme is employed to solve the gradient flow problem, eliminating the complexities of a space–time approach. We utilize a separable neural network that efficiently represents the phase field through low-rank tensor decomposition. As we use the minimizing movement scheme to numerically solve the gradient flow problem, we thus, refer to the proposed method as the Separable Deep Minimizing Movement (SDMM) method. The evaluation of the functional in the minimizing movement scheme using the Gauss quadrature technique bypasses the inaccuracies associated with collocation techniques traditionally used to solve partial differential equations. A hyperbolic tangent transformation is introduced on the phase field prior to the evaluation of the functional to ensure that it remains strictly bounded within the values of the two phases. For this transformation, theoretical guarantee for energy stability of the minimizing movement scheme is established. Our results suggest that this transformation helps to improve the accuracy and efficiency significantly. The proposed method resolves the challenges faced by state-of-the-art machine learning techniques, outperforming them in both accuracy and efficiency. It is also the first machine learning method to achieve an order of magnitude speed improvement over the finite element method. In addition to its formulation and computational implementation, several case studies illustrate the applicability of the proposed method.11The source code is available on GitHub. - https://github.com/vmattey/SDMM.

中文翻译：

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使用可分离神经网络进行基于梯度流的相场建模


Allen-Cahn 方程是一种反应-扩散方程，广泛用于模拟相分离。用于求解强形式的 Allen-Cahn 方程的机器学习方法存在搭配技术不准确、计算高阶空间导数错误以及时空方法所需的大系统大小等问题。为了克服这些挑战，我们建议求解 Ginzburg-Landau 自由能泛函的 L2 梯度流，它相当于 Allen-Cahn 方程，从而避免了与 Allen-Cahn 方程相关的二阶空间导数。采用最小化运动方案来解决梯度流问题，消除了时空方法的复杂性。我们利用了一个可分离的神经网络，它通过低秩张量分解有效地表示相场。当我们使用最小化运动方案对梯度流问题进行数值求解时，我们将所提出的方法称为可分离深度最小化运动 （SDMM） 方法。使用高斯正交技术对最小化运动方案中的泛函进行评估，绕过了传统上用于求解偏微分方程的搭配技术所带来的不准确性。在评估泛函之前，在相场上引入双曲切变换，以确保它严格限制在两个相的值范围内。针对这种变换，建立了最小化运动方案能量稳定性的理论保证。我们的结果表明，这种转换有助于显著提高准确性和效率。 所提出的方法解决了最先进的机器学习技术所面临的挑战，在准确性和效率方面都优于它们。它也是第一个比有限元方法速度提高一个数量级的机器学习方法。除了公式和计算实现之外，几个案例研究还说明了所提出的方法的适用性。11源代码可在 GitHub 上找到。- https://github.com/vmattey/SDMM。