论文信息

题目：Transfer learning enhanced physics informed neural network for phase-field modeling of fracture

作者：Somdatta Goswami

期刊会议： Machine Learning (stat.ML); Machine Learning (cs.LG)

年份：2019

论文地址：https://arxiv.org/abs/1907.02531

代码：

基础补充

内容

动机

动机：
The proposed PINN has to be trained at each load/displacement step, which can potentially make the algorithm computationally inefficient.

• To address this issue, we propose to use the concept ‘transfer learning’ wherein, instead of re-training the complete network, we only retrain the network partially while keeping the weights and the biases corresponding to the other portions fixed. With this setup, the computational efficiency of the proposed approach is significantly enhanced.

However, despite its excellent performance in the domain of image processing and computer science, there are two major issues when it comes to the application of ANN in the engineering fields.

• First, in the course of analyzing complex engineering systems, data acquisition is often computationally expensive. Consequently, we may have access to a limited amount of training samples (i.e., we work in the small data regime).
• Secondly, and perhaps more importantly, ANN trained from training data cannot ensure that the physics of the problem will be satisfied。

问题定义

创新：

Compared to conventional residual based PINN, the proposed approach has two major advantages.

• First, the imposition of boundary conditions is relatively simpler and more robust.
• Second, the order of derivatives present in the functional form of the variational energy is of lower order than in the residual form used in conventional PINN and hence, training the network is faster.

In this paper, we propose a new PINN algorithm for studying the growth and propagation of fracture in brittle materials.The proposed approach differs from the existing PINNs on several aspects ：

• we do not minimize the residual of the governing differential equations; instead, we propose to minimize the variational energy of the system. One major advantage of the proposed variational energy formulation resides in the fact that it requires derivatives one order lowerthan in the conventional residual minimization approach
• Secondly, in almost all the available PINN methods, either trapezoidal rule or Monte Carlo integration is used for computing the integral by sampling the domain with either randomly or uniformly spaced points. In this setup, a large number of integration pointsare required to obtain accurate results. This, in turn, increases the computational cost of the approach. To address this issue, we utilize the Gauss-Legendre quadrature rules(正交规则). However, directly generating Gauss points within the whole domain is not efficient for integrating non-smooth functions, which are common in modeling fracture. Therefore, motivated from finite element analysis and isogeometric analysis, we divide the computational domain intoa number of elementsand then, the Gauss points are generated within each element.

方法

To strike a balance between the boundary-loss and the residual loss function, a penalty parameter has to be introduced with the boundary-loss term.This approach has two major disadvantages:

• First, the boundary terms and the energy/residual component for the interior are often conflicting in nature (as one increases the other decreases). This makes the optimization problem difficult to solve.
• Secondly, the penalty parameter in this approach has to be modulated manually. This also complicates the optimization problem as the selection of proper penalty parameters is tedious and time-consuming.
  To address this issue, we propose to modify the neural network output so that the boundary conditions are exactly satisfied. As a consequence, no component corresponding to the boundary loss is needed in the loss function of the proposed approach. This significantly simplifies the objective function to be minimized

实验

结论

不足

不懂

可借鉴地方

最后

以上就是害怕电话为你收集整理的Transfer learning enhanced physics informed neural network for phase-field modeling 论文笔记论文信息基础补充内容的全部内容，希望文章能够帮你解决Transfer learning enhanced physics informed neural network for phase-field modeling 论文笔记论文信息基础补充内容所遇到的程序开发问题。

如果觉得靠谱客网站的内容还不错，欢迎将靠谱客网站推荐给程序员好友。