人工智能固体力学教学：非均匀固体的快速求解器
Teaching Solid Mechanics to Artificial Intelligence: a fast solver for heterogeneous solids
摘要：
我们提出了一种深度神经网络（DNN）作为非均匀非线性材料系统中局部应力（原则上是应变）计算的快速代理模型。我们表明，DNN预测的局部应力与约3.8%的平均绝对百分比误差（MAPE）的情况下，非均匀弹性介质和相邻域之间的机械相位差高达1.5倍，同时执行103倍的速度比频谱求解器。速度的提高源于这样一个事实，即在训练之后，DNN无需任何迭代就可以预测压力，而不是标准非线性求解器的迭代性质。新的DNN代理模型也被证明适用于一般用途：它能够在拓扑结构上与用于训练的几何结构截然不同的几何结构中再现应力分布，这意味着有效地学习底层偏微分方程描述的场景。即使在相邻区域之间的弹性刚度具有高达2倍的机械对比度和4倍的屈服应力的弹塑性材料的情况下，其中常规求解器通常需要大量的迭代来达到应力预测，经训练的模型在网络的一个单个前向评估步骤中以6.4%的MAPE模拟微观力学，即，即使对于这种非线性问题的情况也没有任何迭代计算。结果揭示了一个全新的和高效的方法来解决非线性力学边界值问题和/或增加现有的解决方案的方法在相应的混合变量，与目前最快的解决方案相比，加速到8300倍的异质弹塑性材料。
英文摘要：
We propose a deep neural network (DNN) as a fast surrogate model for local stress (and in principle strain) calculation in inhomogeneous non-linear material systems. We show that the DNN predicts the local stresses with about 3.8% mean absolute percentage error (MAPE) for the case of heterogeneous elastic media and a mechanical phase contrast of up to factor of 1.5 among neighboring domains, while performing 103 times faster than spectral solvers. The speed-up arises from the fact that after training, the DNN predicts the stress without any iterations, as opposed to the iterative nature of standard non-linear solvers. The new DNN surrogate model also proves suited for general purposes: it is capable to reproduce the stress distribution in geometries topologically far different from those used for training, implying effective learning of scenarios described by the underlying partial differential equations. Even in the case of elasto-plastic materials with up to 2 times mechanical contrast in elastic stiffness and 4 times in yield stress among adjacent regions, where conventional solvers typically require a substantial number of iterations to arrive at stress predictions, the trained model simulates the micromechanics with a MAPE of 6.4% in one single forward evaluation step of the network, i.e. without any iterative calculations even for the case of such a non-linear problem. The results reveal a completely new and highly efficient approach to solve non-linear mechanical boundary value problems and/or augment existing solution methods in corresponding hybrid variants, with an acceleration up to factor of 8300 for heterogeneous elastic-plastic materials in comparison to the currently fastest available solvers.
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