概率神经网络在流体流动代理建模和数据恢复中的应用
Probabilistic neural networks for fluid flow surrogate modeling and data recovery
摘要：
我们考虑使用概率神经网络的流体流动{代理建模}和数据恢复。这个框架是通过假设目标变量是从以输入为条件的高斯分布中采样而构建的。因此，整个公式建立了一个过程来预测这个分布的超参数，然后使用这些超参数来计算给定训练数据的目标函数。我们证明，这个框架有能力提供预测置信区间的概率后验假设的基础上，给出了适当的模型架构和足够的训练数据。本框架的适用性与嘈杂的测量和有限的观察情况下，也进行了评估。为了证明这个框架的能力，我们认为典型的回归问题的流体动力学从降阶建模和空间数据恢复的角度来看，四个规范的数据集。本研究中考虑的例子来自（1）浅水方程，（2）二维圆柱流，（3）带有Gurney襟翼的NACA 0012翼型的尾流，（4）NOAA海表面温度数据集。目前的研究结果表明，概率神经网络不仅产生基于机器学习的流体流{代理}模型，但也系统地量化其中的不确定性，以协助模型的可解释性。
英文摘要：
We consider the use of probabilistic neural networks for fluid flow {surrogate modeling} and data recovery. This framework is constructed by assuming that the target variables are sampled from a Gaussian distribution conditioned on the inputs. Consequently, the overall formulation sets up a procedure to predict the hyperparameters of this distribution which are then used to compute an objective function given training data. We demonstrate that this framework has the ability to provide for prediction confidence intervals based on the assumption of a probabilistic posterior, given an appropriate model architecture and adequate training data. The applicability of the present framework to cases with noisy measurements and limited observations is also assessed. To demonstrate the capabilities of this framework, we consider canonical regression problems of fluid dynamics from the viewpoint of reduced-order modeling and spatial data recovery for four canonical data sets. The examples considered in this study arise from (1) the shallow water equations, (2) a two-dimensional cylinder flow, (3) the wake of NACA0012 airfoil with a Gurney flap, and (4) the NOAA sea surface temperature data set. The present results indicate that the probabilistic neural network not only produces a machine-learning-based fluid flow {surrogate} model but also systematically quantifies the uncertainty therein to assist with model interpretability.
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