论文标题
积极培训物理知识的神经网络，汇总和插值参数解决方案
Active Training of Physics-Informed Neural Networks to Aggregate and Interpolate Parametric Solutions to the Navier-Stokes Equations
论文作者
论文摘要
这项工作的目的是训练一个神经网络，该神经网络近似于跨参数空间区域的Navier-Stokes方程，其中参数定义了物理属性，例如域形和边界条件。这项工作的贡献是三重的： 1）为了证明神经网络可以是使用传统的，可信赖的数值方法（例如有限元素）创建的数据培训的全部参数解决方案家庭的有效聚合器。优点包括对物理和参数空间中任何点的压力和速度的快速评估（渐近，〜3 $μs$ /查询）和数据压缩（与其自己的培训数据相比，网络所需的存储空间少99 \％）。 2）证明，神经网络可以在参数空间中准确插入有限元解决方案之间，从而使它们可以立即查询压力和速度场解决方案，以解决从未执行过传统模拟的问题。 3）要引入一种主动学习算法，以便在训练中可以自动查询有限元求解器，以在神经网络的预测最需要改进的位置获得其他培训数据，从而自主获取并有效地分配培训数据在整个参数空间中。 除了上面的项目2的明显效用外，我们还证明了网络在快速参数中的应用，非常准确地预测了管中的狭窄程度，这将导致在给定流量下端到端压力差增加50 \％。该能力在动脉疾病的医学诊断和计算机辅助设计中都可以应用。
The goal of this work is to train a neural network which approximates solutions to the Navier-Stokes equations across a region of parameter space, in which the parameters define physical properties such as domain shape and boundary conditions. The contributions of this work are threefold: 1) To demonstrate that neural networks can be efficient aggregators of whole families of parameteric solutions to physical problems, trained using data created with traditional, trusted numerical methods such as finite elements. Advantages include extremely fast evaluation of pressure and velocity at any point in physical and parameter space (asymptotically, ~3 $μs$ / query), and data compression (the network requires 99\% less storage space compared to its own training data). 2) To demonstrate that the neural networks can accurately interpolate between finite element solutions in parameter space, allowing them to be instantly queried for pressure and velocity field solutions to problems for which traditional simulations have never been performed. 3) To introduce an active learning algorithm, so that during training, a finite element solver can automatically be queried to obtain additional training data in locations where the neural network's predictions are in most need of improvement, thus autonomously acquiring and efficiently distributing training data throughout parameter space. In addition to the obvious utility of Item 2, above, we demonstrate an application of the network in rapid parameter sweeping, very precisely predicting the degree of narrowing in a tube which would result in a 50\% increase in end-to-end pressure difference at a given flow rate. This capability could have applications in both medical diagnosis of arterial disease, and in computer-aided design.
