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.

该研究使用机器学习技术提出了一种优化静态和动态刚度和质量矩阵构造的四面体数值积分规则方法，实现对不规则网格几何建模的精度处理。

基于机器学习的等几何分析最优求积规则探索

Machine Learning Discovery of Optimal Quadrature Rules for Isogeometric  Analysis

We present the application of a class of deep learning, known as Physics
Informed Neural Networks (PINN), to learning and discovery in solid mechanics.
We explain how to incorporate the momentum balance and constitutive relations
into PINN, and explore in detail the application to linear elasticity, and
illustrate its extension to nonlinear problems through an example that
showcases von~Mises elastoplasticity. While common PINN algorithms are based on
training one deep neural network (DNN), we propose a multi-network model that
results in more accurate representation of the field variables. To validate the
model, we test the framework on synthetic data generated from analytical and
numerical reference solutions. We study convergence of the PINN model, and show
that Isogeometric Analysis (IGA) results in superior accuracy and convergence
characteristics compared with classic low-order Finite Element Method (FEM). We
also show the applicability of the framework for transfer learning, and find
vastly accelerated convergence during network re-training. Finally, we find
that honoring the physics leads to improved robustness: when trained only on a
few parameters, we find that the PINN model can accurately predict the solution
for a wide range of parameters new to the network---thus pointing to an
important application of this framework to sensitivity analysis and surrogate
modeling.

本文介绍了物理信息神经网络在固体力学中应用的方法，展示了通过使用多网络模型，结合动量平衡和本构关系，可以更准确地呈现一些场量变量。同时，通过测试合成数据并和解析解和数值解进行比较，验证了模型的有效性和精度，并指出了等几何分析在准确性和收敛性方面的优于有限元法的特点。我们还探索了该框架在机器学习中的应用，并发现物理信息对于提高模型的鲁棒性有很大作用。