Convolutional neural networks are now seeing widespread use in a variety of
fields, including image classification, facial and object recognition, medical
imaging analysis, and many more. In addition, there are applications such as
physics-informed simulators in which accurate forecasts in real time with a
minimal lag are required. The present neural network designs include millions
of parameters, which makes it difficult to install such complex models on
devices that have limited memory. Compression techniques might be able to
resolve these issues by decreasing the size of CNN models that are created by
reducing the number of parameters that contribute to the complexity of the
models. We propose a compressed tensor format of convolutional layer, a priori,
before the training of the neural network. 3-way kernels or 2-way kernels in
convolutional layers are replaced by one-way fiters. The overfitting phenomena
will be reduced also. The time needed to make predictions or time required for
training using the original Convolutional Neural Networks model would be cut
significantly if there were fewer parameters to deal with. In this paper we
present a method of a priori compressing convolutional neural networks for
finite element (FE) predictions of physical data. Afterwards we validate our a
priori compressed models on physical data from a FE model solving a 2D wave
equation. We show that the proposed convolutinal compression technique achieves
equivalent performance as classical convolutional layers with fewer trainable
parameters and lower memory footprint.

本文提出使用压缩张量格式的卷积层、使用单向滤波器替代卷积层的三向 / 二向滤波器等方式压缩卷积神经网络模型，减少参数数量，从而降低内存占用，在预测时间和训练时间上表现良好。同时，文章将此技术应用于预测有限元物理数据。

卷积神经网络的先验压缩在波模拟器中的应用

A priori compression of convolutional neural networks for wave  simulators

Numerical methods such as finite element have been flourishing in the past
decades for modeling solid mechanics problems via solving governing partial
differential equations (PDEs). A salient aspect that distinguishes these
numerical methods is how they approximate the physical fields of interest.
Physics-informed deep learning is a novel approach recently developed for
modeling PDE solutions and shows promise to solve computational mechanics
problems without using any labeled data. The philosophy behind it is to
approximate the quantity of interest (e.g., PDE solution variables) by a deep
neural network (DNN) and embed the physical law to regularize the network. To
this end, training the network is equivalent to minimization of a well-designed
loss function that contains the PDE residuals and initial/boundary conditions
(I/BCs). In this paper, we present a physics-informed neural network (PINN)
with mixed-variable output to model elastodynamics problems without resort to
labeled data, in which the I/BCs are hardly imposed. In particular, both the
displacement and stress components are taken as the DNN output, inspired by the
hybrid finite element analysis, which largely improves the accuracy and
trainability of the network. Since the conventional PINN framework augments all
the residual loss components in a "soft" manner with Lagrange multipliers, the
weakly imposed I/BCs cannot not be well satisfied especially when complex I/BCs
are present. To overcome this issue, a composite scheme of DNNs is established
based on multiple single DNNs such that the I/BCs can be satisfied forcibly in
a "hard" manner. The propose PINN framework is demonstrated on several
numerical elasticity examples with different I/BCs, including both static and
dynamic problems as well as wave propagation in truncated domains. Results show
the promise of PINN in the context of computational mechanics applications.

本文提出了一种基于物理学信息神经网络（PINN）的方法来解决在没有标注数据的情况下建模弹性动力学问题的挑战，进一步解决了复杂的 I/BCs 在弱正则化 PINN 框架下无法很好满足的问题，在多个数值弹性例子中展示了该方法的可行性。

基于物理学信息的无标签数据计算弹性动力学的深度学习

Physics informed deep learning for computational elastodynamics without  labeled data

We introduce a novel approach to automatic unstructured mesh generation using
machine learning to predict an optimal finite element mesh for a previously
unseen problem. The framework that we have developed is based around training
an artificial neural network (ANN) to guide standard mesh generation software,
based upon a prediction of the required local mesh density throughout the
domain. We describe the training regime that is proposed, based upon the use of
\emph{a posteriori} error estimation, and discuss the topologies of the ANNs
that we have considered. We then illustrate performance using two standard test
problems, a single elliptic partial differential equation (PDE) and a system of
PDEs associated with linear elasticity. We demonstrate the effective generation
of high quality meshes for arbitrary polygonal geometries and a range of
material parameters, using a variety of user-selected error norms.

使用机器学习为出现的问题预测最佳有限元网格，从而自动生成了高质量的网格，该方法基于神经网络对标准网格生成软件进行指导。