| DNN architecture | Parameters |
| Input layer size | 12 (s \in \mathbb{R}^{9}, b \in \mathbb{R}^{3}) |
| Output layer size | 78 (a \in \mathbb{R}^{78}) |
| Hidden layer sizes | 256, 256, 256 |
| Activation function | ReLU (Agarap 2018) |
| Initialization | Xavier (Glorot & Bengio 2010) |
| Input–output scaling | Min–max → [0,1] |
| Batch size | 64 |
| Optimizer | Adam (Kingma & Ba 2014) |
| Loss | MSE (L_{\varepsilon}) |
| Number of epochs | 500 |
| Number of repetitions | 20 |
+
+
+Table 1. The set-up of the DNN applied for the fluidic pinball plant.
+
+
+The output of the neural network contains the truncated POD modal coefficients $a _ { 1 }$ to $a _ { N }$ Both the combined input vector q and output vector a are re-scaled into the range [0, 1] to facilitate convergence. Note that in kNN and LSE, the input vector is non-dimensionalized to an amplitude of O(1) by the freestream velocity $U _ { \infty }$ , which is in accord with the recommendation in Burkov (2019).
+
+A brief summary of the DNN architecture applied in this study can be found in table 1. The architecture of the neural network is determined empirically by balancing the estimation accuracy and the computational cost. Based on the dimension of the combined input vector and the POD modal truncation (see § 4.1), the sizes of the input and the output layers are 12 and 78, respectively. Three 256-node hidden layers are fully connected in between, for each neuron for which the rectified linear activation function (ReLU; Agarap 2018) is specified as the activation function.
+
+To minimize the loss function that indicates the performance of the neural network, all tunable weights and biases are optimized with gradient-based back-propagation via an iterative training approach (LeCun, Bengio & Hinton 2015). In this study, the Adam optimization method (Kingma & Ba 2014) is adopted, and the corresponding loss function is defined as $L _ { \varepsilon } = \langle \| \pmb { a } - \mathbf { \hat { a } } \| _ { 2 } ^ { 2 } \rangle$ , which represents the mean squared error between the desired output a and the actual output aˆ over the training samples. To accelerate the training phase and avoid the convergence to local minimum, the iterative back-propagation is achieved based on the evaluation of mini-batches that are partitioned randomly from the training data. In the present work, the batch size is set to 64, and a total number of 500 epochs were carried out to train the model. Other key settings related to the fluidic pinball dataset applied in this work, including the dataset partition and the iterative training process, will be detailed in § 3.4.
+
+Before the iterative training, all weights are initialized randomly following the suggestion in Glorot & Bengio (2010). In this method, the randomly generated initial weights yield Gaussian distributions, and the initial biases are set to zero. To examine the influence of randomness from the network initialization to the estimation outcome, we trained the network 20 times, and for each time the initial weights are randomly generated. We calculate the averaged outputs and their variances based on the test set, and present these results in $\ S 4$
+
+## 2.5. Linear stochastic estimation
+
+Besides the nonlinear, machine-learning-based methods, we apply LSE (Adrian 1994) as a reference point in this study. LSE represents the most general linear fit between fluctuating sensor signals and POD modal coefficients. The solution of LSE is equivalent to the least-squared estimation error for the estimation of POD modal coefficients. As presented in figure 6, this method analytically optimizes a linear input–output model, such that
+
+$$
+a _ {i} (\boldsymbol {b}, t) = \sum_ {j = 1} ^ {N} \boldsymbol {\tau} _ {i j} q _ {j} (t).\tag{2.7}
+$$
+
+Multiplying by $q _ { k }$ on both sides of (2.7) and taking the ensemble average over b and t, we get
+
+$$
+\sum_ {j = 1} ^ {N} \boldsymbol {\mathsf {T}} _ {i j} \langle q _ {j} q _ {k} \rangle_ {\boldsymbol {b}, t} = \langle a _ {i} q _ {k} \rangle_ {\boldsymbol {b}, t}.\tag{2.8}
+$$
+
+Here, $\langle q _ { j } q _ { k } \rangle _ { b , t }$ denotes the cross-correlation between inputs, and $\langle a _ { i } q _ { k } \rangle _ { b , t }$ represents the autocorrelation between inputs and outputs. Both matrices can be computed directly from data, and a least-squares solution of $\tau _ { i j }$ can be obtained by solving (2.8).
+
+So far, we have described the control-oriented, sensor-based flow estimation problem, as well as a spectrum of machine learning flow estimation methods that are capable to tackle this problem. In the next section, we will present the numerical plant, i.e. the fluidic pinball, that will be utilized in this study to evaluate the performance of these estimation methods.
+
+## 3. Numerical plant: the fluidic pinball
+
+In this section, we introduce the fluidic pinball as a benchmark plant to evaluate control-oriented flow estimation methods described in § 2. An overview of the fluidic pinball is provided in § 3.1, followed by the description of the flow configuration and the numerical solver in § 3.2. The establishment of the fluidic pinball database that will be used in this study is detailed in § 3.3, including control commands generation and data sampling, as well as sensor placement. The partition of the database for the purpose of establishing data-driven mappings, as well as the criteria to evaluate the estimation performance, will be concluded in § 3.4.
+
+
+
+
+
+Figure 6. A schematic of the LSE algorithm for the control-oriented flow estimation. The yellow column and light yellow column represent the sensor signal s and control command b, respectively. The orange column represents the estimation of the POD modal coefficients a. Dots in the transfer matrix correspond to the weights of T , which is visualized under a blue (negative)–white (0)–red (positive) colour scheme.
+
+
+## 3.1. An overview of the fluidic pinball plant
+
+The fluidic pinball plant is a classical control problem that contains rich dynamical behaviours thanks to different flow control inputs (Ishar et al. 2019). This plant is composed of a two-dimensional, uniform flow passing a set of three cylinders of the same diameter D. The centres of the three cylinders form an equilateral triangle pointing in the upstream direction, and the side length of the triangle is 1.5D. The unforced flow exhibits six different Navier–Stokes solutions, including two stable statistically asymmetric periodic vortex sheddings (Deng et al. 2020, 2021a). The independent rotation of three cylinders allows the control of the wake formed downstream, and is capable of replicating most known control mechanisms, namely, phasor control, stagnation point control, boat tailing and base bleeding, as well as high- and low-frequency forcing (Ishar et al. 2019; Cornejo Maceda et al. 2021). In this study, only steady, open-loop control commands are considered. Apart from the uncontrolled case with periodic vortex shedding, steady rotations of cylinders enables the realization of stagnation point control, boat tailing and base bleeding. The schematics of these control mechanisms can be found in figure 7. With the stagnation point control, a unified rotation is exerted on the three cylinders, from which the incoming flow is deflected following the cylinder rotation direction thus attenuated periodic vortex shedding (Seifert 2012). Both boat tailing and base bleeding mechanisms starts from the counter-rotation of the two rearward cylinders. With boat tailing, the top cylinder rotates clockwise and the bottom rotates counter-clockwise. This will guide the boundary layer flowing towards the centre region downstream of the cylinders, which delays the flow separation and leads to significant drag reduction (Geropp & Odenthal 2000; Evrard et al. 2016). When the top cylinder rotates counter-clockwise and the bottom rotates clockwise, the base bleeding mechanism takes place, which generates a fluid jet on the centreline. The jet actuation will limit the interaction between upper and lower shear layers, hence impairing the development of large-scale vortices downstream. The ability of the base bleeding mechanism to reduce bluff-body aerodynamic drag has been reported in Wood (1964), Bearman (1967) and Howell, Sheppard & Blakemore (2003), among many others. With the deployment of sparse sensors in the flow, the fluidic pinball forms an MIMO system. Here, we focus
+
+S. Li, W. Li and B.R. Noack
+
+
+
+
+
+Figure 7. A graphical illustration of three flow control mechanisms in the fluidic pinball plant. Figures from Lin (2021). (a) Stagnation point control. (b) Boat tailing. (c) Base bleeding.
+
+
+on the application of sensor signal, along with the control command, to estimate the instantaneous flow field under steady control.
+
+## 3.2. Flow configuration and numerical solver
+
+The flow is depicted in a two-dimensional Cartesian coordinate system, where the origin is located on the midpoint of the two rear cylinders. The x axis represents the streamwise direction, and the y axis points perpendicular to the freestream direction. Both axes are scaled by the diameter of the cylinders. The velocity vector is scaled by the freestream velocity $U _ { \infty }$ , and is denoted by $\boldsymbol { u } = ( u , v )$ , where u represents the streamwise velocity, and v the transverse velocity. In this study, we consider an incompressible Newtonian fluid with constant density $\rho$ and kinematic viscosity ν. The Reynolds number based on the cylinder diameter D is defined as $R e _ { D } = U _ { \infty } D / \nu$ . Throughout this study, the Reynolds number is set to 100, which corresponds to the asymmetric periodic vortex shedding according to Deng et al. (2020). The computational domain Ω is bounded by a rectangular box $[ - 6 , 2 0 ] \times [ - 6 , 6 ]$ externally and the three cylinders internally:
+
+$$
+\begin{array}{c} \Omega = \{[ x, y ] ^ {\mathrm{T}} \in \mathcal {R} ^ {2}: [ x, y ] ^ {\mathrm{T}} \in [ - 6, 2 0 ] \times [ - 6, 6 ] \cap (x - x _ {i}) ^ {2} \\ + (y - y _ {i}) ^ {2} \geqslant 1 / 4, i = 1, 2, 3 \}. \end{array}\tag{3.1}
+$$
+
+Here, $x _ { i }$ and $y _ { i }$ represent the centres of the three cylinders positioned in the flow, where
+
+$$
+\left. \begin{array}{l l} x _ {1} = - 3 / 2 \cos (3 0 ^ {\circ}), & y _ {1} = 0, \\ x _ {2} = 0, & y _ {2} = - 3 / 4, \\ x _ {3} = 0, & y _ {3} = 3 / 4. \end{array} \right\}\tag{3.2}
+$$
+
+The computational domain Ω is discretized by the unstructured mesh shown in figure 8. This mesh contains 4225 triangles and 8633 vertices. The grid independence of the direct Navier–Stokes solutions has been established by Deng et al. (2020).
+
+The boundary conditions for the inflow, upper and lower boundaries are specified as the freestream velocity $U _ { \infty }$ and a stress-free condition is assumed at the outflow boundary. To model the steady rotation of the cylinders, a no-slip condition is enforced on the cylinder boundaries. The flow adopts the circumferential velocities on the boundary of the front, bottom and top cylinders, which is specified by $b _ { 1 } = U _ { F } , b _ { 2 } = U _ { B }$ and $b _ { 3 } = U _ { T }$ . Here, $b _ { 1 }$ to $b _ { 3 }$ are scaled with $U _ { \infty }$ . For each cylinder, the rotation speed on the boundary is within the range [−2, 2]. A positive value corresponds to counter-clockwise rotation of the cylinders, and vice versa. The control command can then be written as $\pmb { b } = [ b _ { 1 } , b _ { 2 } , b _ { 3 } ] ^ { \mathrm { T } } \in$ $[ - 2 , 2 ] ^ { 3 }$ . The initial condition for the numerical simulations is the symmetric steady solution calculated from the steady Navier–Stokes equations. We use an in-house implicit finite-element method solver UNS3 (Noack & Morzynski´ 2017) that employs an implicit finite element method to perform numerical integration of Navier–Stokes equations with third-order accuracy in space and time. In this study, only the post-transient data are collected for the purpose of constructing the fluidic pinball database that will be used to model the input–output relationship between the sparse sensors and the flow state. We non-dimensionalize the time with $U _ { \infty }$ and $D .$ . For every control command, 100 snapshots are recorded with a time interval $\Delta t = 1$ . The numerical simulation under a single control command takes up around 1 core hour, and the data storage for 100 snapshots is approximately 78 MB. For the unforced flow, the vortex shedding period is approximately 8.5 times the non-dimensionalized time unit. We choose $\Delta t = 1$ in an attempt to balance the computational cost and the overall sampling range in time. The generation of control commands, as well as the placement of virtual sensors, will be detailed in the next subsection.
+
+
+
+
+
+Figure 8. Computational grid for the fluidic pinball simulation.
+
+
+## 3.3. Control command generation and sensor placement
+
+To establish control-oriented flow estimators, a set of 1000 different control commands $B = \{ b _ { j } \} _ { j = 1 } ^ { 1 0 0 0 }$ are generated randomly inside the control parameter space $[ - 2 , 2 ] ^ { 3 }$ . As mentioned in § 3.2, the control parameter $\pmb { b }$ represents the circumferential rotation speed of three cylinders in the flow. Here, we employ the Latin hypercube sampling (McKay, Beckman & Conover 1979) to obtain a set of randomly distributed control commands . As presented in figure 9, this sampling method ensures a nearly uniform distribution of samples inside the multi-dimensional control parameter space. For each control command, we categorize the flow state into periodic and chaotic types after the flow snapshots are obtained from the numerical solver. The classification criteria are based on the relationship between two variables, $\gamma _ { 1 }$ and $\gamma _ { 2 }$ , obtained from the sampled snapshots. Here, $\gamma _ { 1 } = C _ { L } ( t )$ and $\gamma _ { 2 } = ( C _ { L } ( t + \Delta t ) - C _ { L } ( t ) ) / \Delta t .$ , and $C _ { L }$ represents the lift coefficient. The lift coefficient is computed by integrating pressure over the surfaces of three cylinders. From these plots, different characteristics between periodic and chaotic flows can be distinguished clearly. Representative periodic and chaotic phase pattern approximations are displayed in figure 9. For periodic flows, the discrete points in the plots will form circular or periodic trajectories. In contrast, the patterns in chaotic flows are dispersed and disordered. The classification of the flow state will enable us to evaluate the performance of the flow estimators introduced in § 2 under different types of the flow states.
+
+
+
+
+
+Figure 9. Randomly sampled control commands in the control space $[ - 2 , 2 ] ^ { 3 }$ and their projections on the three orthogonal coordinate planes. Periodic and chaotic cases are rendered by red and blue dots, respectively. Representative patterns of $\gamma _ { 1 }$ versus $\gamma _ { 2 }$ in each category are also presented in this figure. See the text for the definitions of $\gamma _ { 1 } , \gamma _ { 2 }$
+
+
+