We propose a mesh-free method to solve nonconvex energy minimization problems for martensitic phase transitions and twinning in crystals, using the deep learning approach. These problems pose multiple challenges to both analysis and computation, as they involve multiwell gradient energies with large numbers of local minima, each involving a topologically complex microstructure of free boundaries with gradient jumps. We use the Deep Ritz method, whereby candidates for minimizers are represented by parameter-dependent deep neural networks, and the energy is minimized with respect to network parameters. The new essential ingredient is a novel activation function proposed here, which is a smoothened rectified linear unit we call SmReLU; this captures the structure of minimizers where usual activation functions fail. The method is mesh-free and thus can approximate free boundaries essential to this problem without any special treatment, and is extremely simple to implement. We show the results of many numerical computations demonstrating the success of our method.

翻译：我们提出一种无网状方法,用深层学习方法解决非碳化物能源最小化问题,解决水晶中非碳化物级转换和结对的最小化问题。这些问题对分析和计算都提出了多重挑战,因为涉及多孔梯度能量,当地微型能源数量众多,每个系统都涉及一个具有梯度跳跃的自由边界的地形复杂微结构。我们使用深里兹方法,即最小化物的候选体由依赖参数的深神经网络代表,而网络参数的能量则被最小化。新的基本成分是在这里提出的一种新型激活功能,这是一个我们称之为SmReLU的平滑修正线性单元;它捕捉到通常活性功能失效的最小化器的结构。这种方法是无网状的,因此可以在没有特殊处理的情况下接近这个问题所必需的自由边界,并且非常简单。我们展示了许多数字计算的结果,显示了我们的方法的成功。