We propose the use of machine learning techniques to find optimal quadrature rules for the construction of stiffness and mass matrices in isogeometric analysis (IGA). We initially consider 1D spline spaces of arbitrary degree spanned over uniform and non-uniform knot sequences, and then the generated optimal rules are used for integration over higher-dimensional spaces using tensor product sense. The quadrature rule search is posed as an optimization problem and solved by a machine learning strategy based on gradient-descent. However, since the optimization space is highly non-convex, the success of the search strongly depends on the number of quadrature points and the parameter initialization. Thus, we use a dynamic programming strategy that initializes the parameters from the optimal solution over the spline space with a lower number of knots. With this method, we found optimal quadrature rules for spline spaces when using IGA discretizations with up to 50 uniform elements and polynomial degrees up to 8, showing the generality of the approach in this scenario. For non-uniform partitions, the method also finds an optimal rule in a reasonable number of test cases. We also assess the generated optimal rules in two practical case studies, namely, the eigenvalue problem of the Laplace operator and the eigenfrequency analysis of freeform curved beams, where the latter problem shows the applicability of the method to curved geometries. In particular, the proposed method results in savings with respect to traditional Gaussian integration of up to 44% in 1D, 68% in 2D, and 82% in 3D spaces.

翻译：我们提出利用机器学习技术寻找等几何分析（IGA）中刚度矩阵和质量矩阵构造的最优求积规则。首先考虑定义在均匀与非均匀节点序列上的任意阶一维样条空间，进而通过张量积形式将生成的最优规则推广至高维空间积分。求积规则的搜索被表述为优化问题，并采用基于梯度下降的机器学习策略求解。然而，由于优化空间高度非凸，搜索成功与否强烈依赖于求积点数量与参数初始化。为此，我们引入动态规划策略，从较低节点数的样条空间最优解初始化参数。通过该方法，我们在包含最多50个均匀单元、多项式阶数最高达8次的IGA离散化样条空间中找到了最优求积规则，展示了该方法在此场景下的普适性。对于非均匀划分，该方法同样在相当数量的测试案例中找到了最优规则。我们还在两个实际案例中评估了生成的最优规则，即拉普拉斯算子的特征值问题和自由形态曲梁的固有频率分析，后者展示了该方法对曲线几何结构的适用性。特别地，与传统高斯积分相比，所提方法在一维、二维和三维空间中分别可节省高达44%、68%和82%的计算量。