用深度神经网络求解定常Navier-斯托克斯方程反问题
Solving Inverse Problems in Steady-State Navier-Stokes Equations using Deep Neural Networks
摘要：
流体动力学反问题在科学和工程中无处不在，其应用范围从电子冷却系统设计到海洋建模。本文提出了一种基于深度神经网络和数值偏微分方程（PDE）的求解稳态Navier-斯托克斯方程反问题的通用方法。我们的方法表示数值模拟作为一个计算图与可微算子。然后，我们通过约束优化来解决逆问题，使用从具有反向模式自动微分的计算图计算的梯度。这种技术使我们能够使用深度神经网络对未知的物理特性进行建模，并将其嵌入到PDE模型中。我们通过使用深度神经网络（DNN）计算空间变化的粘度和电导率场并使用速度场的部分观测值训练DNN来证明我们方法的有效性。我们证明了DNN能够用稀疏和噪声数据对复杂的空间变化的物理场进行建模。我们的实现利用开放访问ADCME，一个库，用于解决逆建模问题，在科学计算中使用自动微分。
英文摘要：
Inverse problems in fluid dynamics are ubiquitous in science and engineering, with applications ranging from electronic cooling system design to ocean modeling. We propose a general and robust approach for solving inverse problems in the steady-state Navier-Stokes equations by combining deep neural networks and numerical partial differential equation (PDE) schemes. Our approach expresses numerical simulation as a computational graph with differentiable operators. We then solve inverse problems by constrained optimization, using gradients calculated from the computational graph with reverse-mode automatic differentiation. This technique enables us to model unknown physical properties using deep neural networks and embed them into the PDE model. We demonstrate the effectiveness of our method by computing spatially-varying viscosity and conductivity fields with deep neural networks (DNNs) and training the DNNs using partial observations of velocity fields. We show that the DNNs are capable of modeling complex spatially-varying physical fields with sparse and noisy data. Our implementation leverages the open access ADCME, a library for solving inverse modeling problems in scientific computing using automatic differentiation.
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