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.

中文翻译：

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概率神经网络用于流体代理建模和数据恢复

我们考虑将概率神经网络用于流体流量{代理建模}和数据恢复。通过假设目标变量是从以输入为条件的高斯分布中采样来构造此框架的。因此，总体公式设置了一个程序来预测此分布的超参数，然后将其用于在给定训练数据的情况下计算目标函数。我们证明，在适当的模型架构和足够的训练数据的基础上，该框架具有基于概率后验假设提供预测置信区间的能力。还评估了本框架对具有嘈杂测量值和有限观察值的情况的适用性。为了演示此框架的功能，我们从四个规范数据集的降阶建模和空间数据恢复的角度考虑流体动力学的规范回归问题。本研究中考虑的示例来自（1）浅水方程，（2）二维圆柱流，（3）带格尼襟翼的NACA0012翼型的尾流以及（4）NOAA海面温度数据集。目前的结果表明，概率神经网络不仅产生了基于机器学习的流体流量{替代}模型，而且系统地量化了其中的不确定性以帮助模型的可解释性。（3）带有格尼襟翼的NACA0012机翼的尾流，以及（4）NOAA海面温度数据集。目前的结果表明，概率神经网络不仅产生了基于机器学习的流体流量{替代}模型，而且还系统地量化了其中的不确定性以帮助模型的可解释性。（3）带有格尼襟翼的NACA0012机翼的尾流，以及（4）NOAA海面温度数据集。目前的结果表明，概率神经网络不仅产生了基于机器学习的流体流量{替代}模型，而且系统地量化了其中的不确定性以帮助模型的可解释性。