论文标题
物理集成的机器学习：将神经网络嵌入Navier-Stokes方程中。第一部分
Physics-integrated machine learning: embedding a neural network in the Navier-Stokes equations. Part I
论文作者
论文摘要
在本文中，研究了物理学 - （或PDE-）集成机器学习（ML）框架。 Navier-Stokes（NS）方程是使用Chorin的投影方法使用Python的Tensorflow库来求解的。提供了解决方案的方法，该方法与在Fortran中实现的经典解决方案进行了比较。该解决方案与神经网络（NN）集成。这样的集成使人们可以在NN的直接输出中训练嵌入在NS方程中的NN；取而代之的是，对NN进行了对现场数据（感兴趣量）的培训，该数据是NS方程的解决方案。为了证明框架的性能，制定了一个案例研究：考虑了具有非恒定速度依赖性动态粘度的2D盖驱动腔。对NN进行了训练，可以预测速度场的动态粘度。当直接在可用数据（动态粘度的字段）上训练NN时，将物理集成的ML的性能与经典ML框架进行了比较。这两个框架都表现出相似的精度。但是，尽管具有其复杂性和计算成本，但物理综合的ML提供了主要优势，即：（i）NN的目标输出（标记的培训数据）可能是未知的，并且可以使用PDE恢复； （ii）不必从大数据中提取和预处理信息（培训目标），而是可以由PDE提取； （iii）无需采用物理或规模分离假设来构建闭合模型。本文证明了优势（i），而优势（ii）和（iii）是未来工作的主题。 PDE与ML的集成为更紧密的数据知识连接打开了一扇门，这可能会影响基于物理的模型的进一步开发，用于数据驱动的热流体模型。
In this paper the physics- (or PDE-) integrated machine learning (ML) framework is investigated. The Navier-Stokes (NS) equations are solved using Tensorflow library for Python via Chorin's projection method. The methodology for the solution is provided, which is compared with a classical solution implemented in Fortran. This solution is integrated with a neural network (NN). Such integration allows one to train a NN embedded in the NS equations without having the target (labeled training) data for the direct outputs from the NN; instead, the NN is trained on the field data (quantities of interest), which are the solutions for the NS equations. To demonstrate the performance of the framework, a case study is formulated: the 2D lid-driven cavity with non-constant velocity-dependent dynamic viscosity is considered. A NN is trained to predict the dynamic viscosity from the velocity fields. The performance of the physics-integrated ML is compared with classical ML framework, when a NN is directly trained on the available data (fields of the dynamic viscosity). Both frameworks showed similar accuracy; however, despite its complexity and computational cost, the physics-integrated ML offers principal advantages, namely: (i) the target outputs (labeled training data) for a NN might be unknown and can be recovered using PDEs; (ii) it is not necessary to extract and preprocess information (training targets) from big data, instead it can be extracted by PDEs; (iii) there is no need to employ a physics- or scale-separation assumptions to build a closure model. The advantage (i) is demonstrated in this paper, while the advantages (ii) and (iii) are the subjects for future work. Such integration of PDEs with ML opens a door for a tighter data-knowledge connection, which may potentially influence the further development of the physics-based modelling with ML for data-driven thermal fluid models.
