NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations
NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations
复制标题
NSFnets（纳维-斯托克斯流网）：用于不可压缩纳维-斯托克斯方程的物理神经网络
DOI：
10.1016/j.jcp.2020.109951
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发表时间：
2021-01-05
影响因子：
4.1
通讯作者：
Karniadakis, George Em
中科院分区：
文献类型：
--
作者：
Jin, Xiaowei;Cai, Shengze;Karniadakis, George Em
在过去的50年里，数值求解Navier-Stokes方程的方法有了很大的发展，包括有限差分法、有限元法、谱法，甚至无网格法。然而，在许多真实的情况下，我们仍然不能将无缝（多保真度）数据合并到现有算法中，并且对于工业复杂性应用，网格生成是耗时的并且仍然是一门艺术。缺乏边界条件）或逆问题通常过于昂贵，并且需要不同的公式和新的计算机代码。在这里，我们采用物理信息神经网络（PINN），通过自动微分将控制方程直接编码到深度神经网络中，以克服上述模拟不可压缩层流和湍流的一些限制。我们开发的Navier-Stokes流网络（NSFnets），考虑两种不同的数学公式的Navier-Stokes方程：速度-压力（VP）制定和涡度-速度（VV）制定。由于这是一种新的方法，我们首先选择一些标准的基准问题来评估NSFnet的精度，收敛速度，计算成本和灵活性;解析解和直接数值模拟（DNS）数据库为NSFnet模拟提供了适当的初始和边界条件。空间和时间坐标是NSFnet的输入，而瞬时速度和压力场是VP-NSFnet的输出，瞬时速度和涡度场是VV-NSFnet的输出。这是无监督学习，因此，除了边界和初始条件以及流体属性之外，不需要标记数据。VP或VV控制方程的残差，连同初始和边界条件，被嵌入到NSFnets的损失函数。VP-NSFnet没有提供压力数据，这是一种隐藏状态，通过不可压缩性约束获得，无需额外的计算成本。与传统的数值方法不同，NSFnets继承了神经网络（NN）的特性，因此总误差由逼近误差、优化误差和推广误差组成。在这里，我们根据经验尝试通过改变采样（“残差”）点、迭代求解器和NN架构的大小来量化这些误差。对于层流的解决方案，我们表明，VP和VV配方的准确性相当，但其最佳性能对应于不同的NN架构。初始收敛速度很快，但由于优化误差的主导地位，误差最终饱和到一个平台。对于湍流槽道流，我们表明，NSFnets可以维持在Re τ类似于1000的湍流，但由于昂贵的训练，我们只考虑部分的通道域和DNS数据库提供的子域边界上的速度边界条件。我们还对损失函数中用于平衡数据和物理分量的权重进行了系统的研究，并研究了一种动态计算权重的新方法，以加速训练并提高准确性。在最后一部分中，我们演示了如何在实践中使用NSFnets，即不完整或噪声边界条件的不适定问题以及反问题。我们获得了合理的精确解，以及在这种情况下，不需要改变NSFnets和相同的计算成本在向前适定的问题。我们还提出了一个简单的迁移学习的例子，这将有助于加速不同参数设置的NSFnets的训练。(C)2020爱思唯尔公司All rights reserved.
In the last 50 years there has been a tremendous progress in solving numerically the Navier-Stokes equations using finite differences, finite elements, spectral, and even meshless methods. Yet, in many real cases, we still cannot incorporate seamlessly (multi-fidelity) data into existing algorithms, and for industrial-complexity applications the mesh generation is time consuming and still an art. Moreover, solving ill-posed problems (e.g., lacking boundary conditions) or inverse problems is often prohibitively expensive and requires different formulations and new computer codes. Here, we employ physics-informed neural networks (PINNs), encoding the governing equations directly into the deep neural network via automatic differentiation, to overcome some of the aforementioned limitations for simulating incompressible laminar and turbulent flows. We develop the Navier-Stokes flow nets (NSFnets) by considering two different mathematical formulations of the Navier-Stokes equations: the velocity-pressure (VP) formulation and the vorticity-velocity (VV) formulation. Since this is a new approach, we first select some standard benchmark problems to assess the accuracy, convergence rate, computational cost and flexibility of NSFnets; analytical solutions and direct numerical simulation (DNS) databases provide proper initial and boundary conditions for the NSFnet simulations. The spatial and temporal coordinates are the inputs of the NSFnets, while the instantaneous velocity and pressure fields are the outputs for the VP-NSFnet, and the instantaneous velocity and vorticity fields are the outputs for the VV-NSFnet. This is unsupervised learning and, hence, no labeled data are required beyond boundary and initial conditions and the fluid properties. The residuals of the VP or VV governing equations, together with the initial and boundary conditions, are embedded into the loss function of the NSFnets. No data is provided for the pressure to the VP-NSFnet, which is a hidden state and is obtained via the incompressibility constraint without extra computational cost. Unlike the traditional numerical methods, NSFnets inherit the properties of neural networks (NNs), hence the total error is composed of the approximation, the optimization, and the generalization errors. Here, we empirically attempt to quantify these errors by varying the sampling ("residual") points, the iterative solvers, and the size of the NN architecture. For the laminar flow solutions, we show that both the VP and the VV formulations are comparable in accuracy but their best performance corresponds to different NN architectures. The initial convergence rate is fast but the error eventually saturates to a plateau due to the dominance of the optimization error. For the turbulent channel flow, we show that NSFnets can sustain turbulence at Re-tau similar to 1, 000, but due to expensive training we only consider part of the channel domain and enforce velocity boundary conditions on the subdomain boundaries provided by the DNS data base. We also perform a systematic study on the weights used in the loss function for balancing the data and physics components, and investigate a new way of computing the weights dynamically to accelerate training and enhance accuracy. In the last part, we demonstrate how NSFnets should be used in practice, namely for ill-posed problems with incomplete or noisy boundary conditions as well as for inverse problems.We obtain reasonably accurate solutions for such cases as well without the need to change the NSFnets and at the same computational cost as in the forward well-posed problems. We also present a simple example of transfer learning that will aid in accelerating the training of NSFnets for different parameter settings. (C) 2020 Elsevier Inc. All rights reserved.