Probabilistic neural networks for fluid flow surrogate modeling and data recovery

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 (i) the shallow-water equations, (ii) a two-dimensional cylinder flow, (iii) the wake of a NACA0012 airfoil with a Gurney flap, and (iv) 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.

Figure 1

Representative schematic of the probabilistic neural network. We consider the flow reconstruction from sensor measurements of a two-dimensional cylinder wake (Sec. 3c1) as an example problem.

Figure 2

Different canonical test cases considered in the present work.

Figure 3

Information of the data set for the shallow-water equation. (a) Schematic for the generation of training and testing data. Given inputs of $\mathit{r}$ (i.e., the location of an initial Gaussian perturbation to $\eta$), the machine learning ROM is tasked with predicting the evolution of 40 POD coefficients over ten snapshots in time. The prediction is one shot. (b) Convergence of reduced representations to the true field with increasing $M$. The values underneath each field indicate the normalized $L_1$ error norm $\epsilon =|{\eta }_{\mathrm{true}}-{\eta }_{\mathrm{POD}}|/|{\eta }_{\mathrm{true}}-1|$.

Figure 4

Coefficient estimations of two representative test simulations by the present framework with the shallow-water equation showing error bars for two standard deviations around the mean as parametrized by a Gaussian distribution. Final time field reconstructions using estimated POD coefficient means are shown on the right-hand side of each coefficient evolution. True indicates the reference field with 40 POD modes.

Figure 5

POD coefficients and entire field of NOAA sea surface temperature. The value underneath the contour is the $L_2$ error norm ${\epsilon }_{\mathrm{POD}}=\parallel T_{\mathrm{ref}}-T_{\mathrm{POD}}{\parallel }_2/{\parallel T_{\mathrm{ref}}\parallel }_2$.

Figure 6

Prediction of POD coefficient evolution from sensors at the first snapshot. Representative temporal evolution and the reconstructed field with POD eigenvectors at the first, 40th, and 80th weeks are shown.

Figure 7

Dependence of the POD coefficient prediction for sea surface temperature on the number of input sensors. (a) Temporal evolution of four POD coefficients. (b) Reconstructed temperature field with POD eigenvectors. (c) $L_2$ error norm of predicted POD coefficients and reconstructed field. (d) Ensemble average of estimated standard deviation taken over all POD coefficients.

Figure 8

Effect of the number of sensors for coefficient estimation of the shallow-water equation on the accuracy. (a) Field reconstructions with 10, 20, 30, and 35 sensors. Also shown is the relationship between the number of sensors and (b) $L_2$ errors of the estimated POD coefficient and reconstructed field, (c) the normalized $L_1$ error of the reconstructed field, and (d) the ensemble average of the estimated standard deviation taken over all POD coefficients. For reference, the average range of the POD coefficient magnitude is 3.289.

Figure 9

Comparison between the PNN and Gappy POD for POD coefficient reconstruction of two representative testing solutions. For all our testing simulations, the normalized $L_1$ error norms in field reconstruction are 0.155 for the PNN using 30 sensors, 3.18 for gappy POD using 30 sensors, and 0.277 for gappy POD using 3800 sensors. Sensor locations (shown in red) are not provided for the validation with 3800 sensors for visual clarity.

Figure 10

Dependence of the accuracy for coefficient estimation of the shallow-water equation on the magnitude of noisy input. (a) Field reconstructions with 35 random sensors. Also shown is the relationship between the number of sensors and (b) $L_2$ errors of the estimated POD coefficient and reconstructed field, (c) the normalized $L_1$ error of the reconstructed field, and (d) the ensemble average of the estimated standard deviation taken over all POD coefficients. For reference, the average range of the POD coefficient magnitude is 3.289.

Figure 11

Sensor estimation of the cylinder wake using the PNN. The left-hand side shows both input and output sensor locations. The $L_2$ error norm $\epsilon =\parallel s_{6,\mathrm{DNS}}-s_{6,\mathrm{ML}}{\parallel }_2/{\parallel s_{6,\mathrm{DNS}}\parallel }_2$ is 0.0137. Note here that the $\sigma$ can be hardly seen due to the excellent agreement of the estimation with the reference.

Figure 12

Wake reconstruction of the cylinder flow using the PNN. The value below the estimated mean field $\mu$ indicates the $L_2$ error norm $\epsilon =\parallel {\omega }_{\mathrm{DNS}}-{\omega }_{\mathrm{ML}}{\parallel }_2/{\parallel {\omega }_{\mathrm{DNS}}\parallel }_2$.

Figure 13

Sensor estimation of a NACA0012 wake with a Gurney flap using the PNN. The left-hand side shows both input and output sensor locations. The $L_2$ error norm $\epsilon =\parallel s_{6,\mathrm{DNS}}-s_{6,\mathrm{ML}}{\parallel }_2/{\parallel s_{6,\mathrm{DNS}}\parallel }_2$ is 0.0429.

Figure 14

Wake reconstruction of a NACA0012 airfoil with a Gurney flap depending on the number of snapshots for training. The values inside the estimated mean field indicate the $L_2$ error norm $\epsilon =\parallel {\omega }_{\mathrm{DNS}}-{\omega }_{\mathrm{ML}}{\parallel }_2/{\parallel {\omega }_{\mathrm{DNS}}\parallel }_2$. The values inside the estimated standard deviation express the ensemble-averaged standard deviation over the field.

Figure 15

Dependence of estimation accuracy for the whole wake field of a NACA0012 airfoil with a Gurney flap on the magnitude of noisy input using the machine-learned model with (a) $n_{\mathrm{snapshot}}=250$ and (b) $n_{\mathrm{snapshot}}=500$. The values inside the estimated mean field indicate the $L_2$ error norm. The values inside the estimated standard deviation express the ensemble-averaged standard deviation over the field.

Figure 16

Instantaneous temperature field on the sea surface. Green circles are sensors used as input data.

Figure 17

Sensor estimation of sea surface temperature using the PNN. The $L_2$ error norm $\epsilon =\parallel s_{H,\mathrm{ref}}-s_{H,\mathrm{ML}}{\parallel }_2/{\parallel s_{H,\mathrm{ref}}\parallel }_2$ is 0.0163.

Figure 18

Whole field reconstruction of sea surface temperature. Note that the estimated standard deviation is the time-ensemble average value. The value below the estimated mean field indicates the $L_2$ error norm $\epsilon =\parallel T_{\mathrm{ref}}-T_{\mathrm{ML}}{\parallel }_2/{\parallel T_{\mathrm{ref}}\parallel }_2$.

Figure 19

Influence of estimation accuracy for the whole field of sea surface temperature on additional sensors. Five sensors are added (a) randomly and (b) based on the estimated standard deviation in Fig. 18. The values underneath the estimated mean fields indicate the $L_2$ error norm.