论文标题
用于近似线性椭圆PDE的近似解决方案的混合MGA-MSGD ANN训练方法
A hybrid MGA-MSGD ANN training approach for approximate solution of linear elliptic PDEs
论文作者
论文摘要
我们介绍了一种混合“修改的遗传算法 - 属性随机梯度下降”（MGA-MSGD）训练算法，该算法可大大提高PDES通过ANN（人工神经网络）的PDES中描述的3D机械问题的求解的准确性和效率。这种提出的方法允许选择许多感兴趣的位置，在这些位置期望状态变量满足与物理问题相关的管理方程。与经典的PDE近似方法（例如有限差异或有限元方法）不同，无需在整个计算域中建立和重建物理场数量，以预测在特定关注的特定位置的机械响应。 MGA-MSGD的基本思想是操纵可学习参数的组件，负责误差爆炸，以便我们可以以相对较大的学习速率训练网络，从而避免捕获本地最小值。所提出的训练方法对学习率的率，训练点密度和分布以及随机初始参数的敏感性不太敏感。最小化的距离函数是我们介绍PDE的位置，包括任何物理定律和条件（所谓的，物理学通知ANN）。遗传算法被修改为适用于这种类型的ANN，其中利用了粗级随机梯度下降（CSGD）以做出后代资格的决定。与标准培训算法（如经典SGD和Adam Optimiser）相比，采用呈现的方法，准确性和效率都有很大的提高。通过在足够细的网格中引入有限元方法（FEM）作为参考位移，研究并确保局部位移精度。解决了一个稍微复杂的问题，以确保其可行性。
We introduce a hybrid "Modified Genetic Algorithm-Multilevel Stochastic Gradient Descent" (MGA-MSGD) training algorithm that considerably improves accuracy and efficiency of solving 3D mechanical problems described, in strong-form, by PDEs via ANNs (Artificial Neural Networks). This presented approach allows the selection of a number of locations of interest at which the state variables are expected to fulfil the governing equations associated with a physical problem. Unlike classical PDE approximation methods such as finite differences or the finite element method, there is no need to establish and reconstruct the physical field quantity throughout the computational domain in order to predict the mechanical response at specific locations of interest. The basic idea of MGA-MSGD is the manipulation of the learnable parameters' components responsible for the error explosion so that we can train the network with relatively larger learning rates which avoids trapping in local minima. The proposed training approach is less sensitive to the learning rate value, training points density and distribution, and the random initial parameters. The distance function to minimise is where we introduce the PDEs including any physical laws and conditions (so-called, Physics Informed ANN). The Genetic algorithm is modified to be suitable for this type of ANN in which a Coarse-level Stochastic Gradient Descent (CSGD) is exploited to make the decision of the offspring qualification. Employing the presented approach, a considerable improvement in both accuracy and efficiency, compared with standard training algorithms such as classical SGD and Adam optimiser, is observed. The local displacement accuracy is studied and ensured by introducing the results of Finite Element Method (FEM) at sufficiently fine mesh as the reference displacements. A slightly more complex problem is solved ensuring its feasibility.
