Deep Learning Forecasts a Strained Turbulent Flow Velocity Field in Temporal Lagrangian Framework: Comparison of LSTM and GRU

: The subject of this study presents an employed method in deep learning to create a model and predict the following period of turbulent ﬂow velocity. The applied data in this study are extracted datasets from simulated turbulent ﬂow in the laboratory with the Taylor microscale Reynolds numbers in the range of 90 < R λ < 110. The ﬂow has been seeded with tracer particles. The turbulent intensity of the ﬂow is created and controlled by eight impellers placed in a turbulence facility. The ﬂow deformation has been conducted via two circular ﬂat plates moving toward each other in the center of the tank. The Lagrangian particle-tracking method has been applied to measure the ﬂow features. The data have been processed to extract the ﬂow properties. Since the dataset is sequential, it is used to train long short-term memory and gated recurrent unit model. The parallel computing machine DEEP-DAM module from Juelich supercomputer center has been applied to accelerate the model. The predicted output was assessed and validated by the rest of the data from the experiment for the following period. The results from this approach display accurate prediction outcomes that could be developed further for more extensive data documentation and used to assist in similar applications. The mean average error and R 2 score range from 0.001–0.002 and 0.9839–0.9873, respectively, for both models with two distinct training data ratios. Using GPUs increases the LSTM performance speed more than applications with no GPUs.


Introduction
Turbulent flow is a nonlinear and random phenomenon [1][2][3]. Water flow in a river, waterfall, airflow passing a wind turbine blade, flow in an engine mixing chamber, smoke from a chimney, and two working flows inside the heat exchanger are examples of turbulent flow in natural events and artificial applications [1][2][3][4][5]. The complexity and multiscale features of turbulent flows make the forecasting of the fluid flow a considerable problem. There are many previous works using experiments and/or numerical methods of turbulent flow to investigate and make efforts to forecast flow periods with specified conditions. However, experiments are costly and, for many applications, could not be performed in a laboratory environment. Computational methods based on partial differential equations, i.e., applying a full-order model, are capable of predicting fluid flow accurately but are computationally costly. High-performance computing is, therefore, essential in those computational methods, yet we are still far from having computing capability to solve even moderately sized problems. Thus, there are limitations in computing costs [6].
These restrictions determine a reliable tool is required to overcome the above-mentioned obstacles. Machine learning based on artificial intelligence has become a pivotal approach

Fluid Flow in Lagrangian Framework
In turbulent flow, it is well-known what statistical aspects of the flow features are applicable [1][2][3]. A Lagrangian framework is an exploration of fluid motion that keeps track of the velocity vector and displacement vector of each flow point, called a fluid particle [1,16]. A fluid particle is a point that moves with the local fluid velocity, and, therefore, it identifies the position at the time t of a fluid particle [16]. The definition of a fluid particle arithmetically is [1]: U i = U i (t, x 1 (t, x 1,0 ), x 2 (t, x 2,0 ), x 3 (t, x 3,0 )) (2) where the fluid particle position and velocity in 3D coordinates is determined by (1) and (2), respectively, x is the position, U is the velocity, t is the time, and i specifies the vector component.
Based on the Lagrangian definition, there is time series data for fluid particles that determine a position and velocity at a specific time. In particular, in turbulent flow, which has no known equation and is investigated using statistics, a sequential dataset could be used from the Lagrangian view for the forecasting model. It is a crucial challenge to be able to have accurate prediction for turbulent flow velocity via an approach that does not need preprocessing to extract hidden features or reduced order methods [6][7][8][17][18][19]. Some numerical methods due have drawbacks, such as missing features because of dimension reduction [6][7][8][17][18][19]. Based on the above description for the velocity in the Lagrangian framework, in this study we will apply velocity denotation (2) to train LSTM or GRU model via velocity component as an input. Then, the model will predict the velocity component for next period as an output.
This study used the dataset for a strained turbulent flow that has been generated in a laboratory. Turbulence intensity has been created with the action of impeller rotors in the corners of a box turbulence facility. The turbulent flow has been strained in the vertical direction (see Figure 1) by the motion of flat circular plates, as shown in the sketch. Equation (3) defines the mean velocity field of the flow: where 2S is the primary strain rate in the Y-dir, S is the mean strain rate for the other two directions, and x, y and z are the particle location. In this work, the flow was considered in 2D, therefore, Z-dir is not addressed. track of the velocity vector and displacement vector of each flow point, called a fluid particle [1,16]. A fluid particle is a point that moves with the local fluid velocity, and, therefore, it identifies the position at the time t of a fluid particle [16]. The definition of a fluid particle arithmetically is [1]: , , , , , , , , , where the fluid particle position and velocity in 3D coordinates is determined by (1) and (2), respectively, x is the position, U is the velocity, t is the time, and i specifies the vector component.
Based on the Lagrangian definition, there is time series data for fluid particles that determine a position and velocity at a specific time. In particular, in turbulent flow, which has no known equation and is investigated using statistics, a sequential dataset could be used from the Lagrangian view for the forecasting model. It is a crucial challenge to be able to have accurate prediction for turbulent flow velocity via an approach that does not need preprocessing to extract hidden features or reduced order methods [6][7][8][17][18][19]. Some numerical methods due have drawbacks, such as missing features because of dimension reduction [6][7][8][17][18][19]. Based on the above description for the velocity in the Lagrangian framework, in this study we will apply velocity denotation (2) to train LSTM or GRU model via velocity component as an input. Then, the model will predict the velocity component for next period as an output.
This study used the dataset for a strained turbulent flow that has been generated in a laboratory. Turbulence intensity has been created with the action of impeller rotors in the corners of a box turbulence facility. The turbulent flow has been strained in the vertical direction (see Figure 1) by the motion of flat circular plates, as shown in the sketch. Equation (3) defines the mean velocity field of the flow: where 2S is the primary strain rate in the Y-dir, S is the mean strain rate for the other two directions, and x, y and z are the particle location. In this work, the flow was considered in 2D, therefore, Z-dir is not addressed. The straining flow case has been created in the experiment with mean strain rate, 2S = 8 s −1 . Equation (3) is based on laminar flow; however, we know that velocity fluctuates in turbulent flow. Further details on the experimental setup are described in Section 3.
The Stokes number for a seeded particle is calculated to ensure a tracer particle meets the requirements and specifies whether a particle introduced to the flow will follow the flow streamline or not. This identification is defined by Equation (4): The straining flow case has been created in the experiment with mean strain rate, 2S = 8 s −1 . Equation (3) is based on laminar flow; however, we know that velocity fluctuates in turbulent flow. Further details on the experimental setup are described in Section 3.
The Stokes number for a seeded particle is calculated to ensure a tracer particle meets the requirements and specifies whether a particle introduced to the flow will follow the flow streamline or not. This identification is defined by Equation (4): where τ p is Stokes' relaxation time. Kolmogorov scale (τ η , defined in Equation (6)) is based on the flow quantities before applying the strain. Stokes' relaxation time τ p is, in turn, calculated by Equation (5): where ρ p is particle density, d p is a spherical particle diameter, and µ is the dynamic fluid viscosity that, in the conducted experiment, was water. The Stokes number significantly greater than 1 (St 1) describes particles that are unaffected by a fluid velocity change and continue their original trajectory; if St 1, the particle will follow the fluid's local velocity. To extract the flow properties, particle image velocimetry method has been employed and once dissipation rate specified via second-order longitudinal velocity structure function, Kolmogorov scales were calculated by: where ν is kinematic viscosity of the fluid, τ η is the Kolmogorov's time scale, and ε is dissipation rate evaluated via second-order longitudinal velocity structure function. The Stokes number for the tracer particles used in the performed experiment are in the range of 0.0063-0.0094.

LSTM and GRU Architecture
The subject of this study is using the velocity components U x and U y , according to definition (2) as sequential input training data for LSTM or GRU model. The model for each component is separated. Even though the velocity is a profound feature in the flow field description and is based on the Lagrangian perspective, it is spatiotemporal; therefore, it carries many flow effects, such as fluctuation, strain, turbulence intensity, and geometry boundary [1][2][3][4][5]. This is the capital concept of this novel proposed approach.
Since the flow velocity is spatial-temporal and is affected via the above-nominated effects, the LSTM or GRU model is trained and learns how to forecast the next period according to received effects. The predicted flow velocity via LSTM or GRU model is validated by test data in the x and y components separately.
Recurrent neural networks are deep network models that can extract sequential data dynamics through recurrent connections. They can be considered as cycles in the network of nodes. Although gradient contraction in recurrent neural networks seems to help exploding gradients, handling vanishing gradients requires a more precise solution [20][21][22]. One of the first and most successful techniques to solve the vanishing problems was presented in the long short-term memory model [9].
In simple recurrent neural networks, long-term memory is in the form of weights, and the weights change gradually during the training and encode general knowledge about the data. Additionally, these networks have short-term memory, which is in the form of fast transient activations and is continuously transferred from one node to another. In the LSTM model, an intermediate storage type is defined through a memory cell (see Figure 2). A memory cell is a composite unit that consists of simpler nodes and acts through a specific connectivity pattern by imbedding new multiplicative nodes.
Each memory cell is composed of an internal state and several multiplicative gates, which control the data as follows: (i) a given input should affect the internal state (the input gate) or not, (ii) the internal state should drive to 0 (the forget gate), (iii) a given neuron's internal state should be able to impact the cell output (the output gate).
A significant distinction between standard RNNs and LSTM is a hidden state gate determined in LSTM. This state provides an appropriate mechanism for when a hidden state should be updated and when it should be reset. These mechanisms are learned, and they resolve the known concerns from standard RNNs. For example, if the first token has a major significance, it will learn not to update the hidden state after the first perception. In addition, it will learn to omit incoherent temporary perceptions. Eventually, it will learn to reset the hidden state whenever it is essential.
Fluids 2022, 7, x FOR PEER REVIEW 5 of 13 Figure 2. Long short-term memory cell; ℎ is hidden state from previous step, is current input, ℎ is new hidden state, is memory cell internal state, is new memory cell internal state, is forget gate, is input gate, is input node, is output gate, is sigmoid function, tanh is hyperbolic tangent function; descriptions are based on [9].
Each memory cell is composed of an internal state and several multiplicative gates, which control the data as follows: (i) a given input should affect the internal state (the input gate) or not, (ii) the internal state should drive to 0 (the forget gate), (iii) a given neuron's internal state should be able to impact the cell output (the output gate).
A significant distinction between standard RNNs and LSTM is a hidden state gate determined in LSTM. This state provides an appropriate mechanism for when a hidden state should be updated and when it should be reset. These mechanisms are learned, and they resolve the known concerns from standard RNNs. For example, if the first token has a major significance, it will learn not to update the hidden state after the first perception. In addition, it will learn to omit incoherent temporary perceptions. Eventually, it will learn to reset the hidden state whenever it is essential.
GRU is a next-generation determination from LSTM with a bit distinction in the model architecture [23]. Literature reports that GRU is comparable in performance is considerably faster to compute than LSTM and has a streamlined model [17,19,24]. The GRU cell that is displayed in Figure 3 is composed of a hidden state, reset gate, and update gate. We can control how much of the previously hidden state might be remembered from the reset gate. is hidden state from previous step, is current input, ℎ is new hidden state, is output, is reset gate, is update gate, is candidate hidden state, is sigmoid function, tanh is hyperbolic tangent function [18].
GRU is a next-generation determination from LSTM with a bit distinction in the model architecture [23]. Literature reports that GRU is comparable in performance is considerably faster to compute than LSTM and has a streamlined model [17,19,24]. The GRU cell that is displayed in Figure 3 is composed of a hidden state, reset gate, and update gate. We can control how much of the previously hidden state might be remembered from the reset gate. is hidden state from previous step, is current input, ℎ is new hidden state, is memory cell internal state, is new memory cell internal state, is forget gate, is input gate, is input node, is output gate, is sigmoid function, tanh is hyperbolic tangent function; descriptions are based on [9].
Each memory cell is composed of an internal state and several multiplicative gates, which control the data as follows: (i) a given input should affect the internal state (the input gate) or not, (ii) the internal state should drive to 0 (the forget gate), (iii) a given neuron's internal state should be able to impact the cell output (the output gate).
A significant distinction between standard RNNs and LSTM is a hidden state gate determined in LSTM. This state provides an appropriate mechanism for when a hidden state should be updated and when it should be reset. These mechanisms are learned, and they resolve the known concerns from standard RNNs. For example, if the first token has a major significance, it will learn not to update the hidden state after the first perception. In addition, it will learn to omit incoherent temporary perceptions. Eventually, it will learn to reset the hidden state whenever it is essential.
GRU is a next-generation determination from LSTM with a bit distinction in the model architecture [23]. Literature reports that GRU is comparable in performance is considerably faster to compute than LSTM and has a streamlined model [17,19,24]. The GRU cell that is displayed in Figure 3 is composed of a hidden state, reset gate, and update gate. We can control how much of the previously hidden state might be remembered from the reset gate. is hidden state from previous step, is current input, ℎ is new hidden state, is output, is reset gate, is update gate, is candidate hidden state, is sigmoid function, tanh is hyperbolic tangent function [18].
is candidate hidden state, σ is sigmoid function, tanh is hyperbolic tangent function [18].
On the other hand, via the update gate, we can understand how much of the new hidden state is just a copy of the old hidden state. This architecture in the GRU establishes two significant features: the reset gate captures short-term dependencies in sequences and the update gate receives long-term dependencies in sequences [23].

Apparatus and Experiment Setup
The experiment has been performed at the Laboratory of Fundamental Turbulence Research (LFTR) at Reykjavik University and the applied facility is shown in Figure 4. The water tank (60 cm × 60 cm × 60 cm) has 20 mm thick acrylic walls (transparent Plexiglas XT) that enables optical access to the data. The eight corners of the box have a triangular shape, while the top and the bottom are nearly circular. An aluminum frame holds the components of the turbulence box together. The turbulence has been generated by eight impellers driven by independently controlled servo motors (Lenz-model: MCS), which were mounted at the eight corners of the cube and point to the center of the tank. The rotation rate of each servo motor is adjustable over a range of 100-4500 rpm at a gearing rate of 0.075. For the used dataset in this study the speed is 1000 rpm. The motion-view filtering software that came with these motors was used to monitor and set up the suited speed of each impeller. The degassing system was used to remove bubbles and solid dust from the water before starting the experiment. The tank has been specifically designed for studying turbulence (Lagrangian and Eulerian motion at moderate Reynolds numbers). The flow facility produces a nearly stationary homogeneous isotropic turbulence near the center of the tank, where measurements have been performed.
the update gate receives long-term dependencies in sequences [23].

Apparatus and Experiment Setup
The experiment has been performed at the Laboratory of Fundamental Turbulence Research (LFTR) at Reykjavik University and the applied facility is shown in Figure 4. The water tank (60 cm × 60 cm × 60 cm) has 20 mm thick acrylic walls (transparent Plexiglas XT) that enables optical access to the data. The eight corners of the box have a triangular shape, while the top and the bottom are nearly circular. An aluminum frame holds the components of the turbulence box together. The turbulence has been generated by eight impellers driven by independently controlled servo motors (Lenz-model: MCS), which were mounted at the eight corners of the cube and point to the center of the tank. The rotation rate of each servo motor is adjustable over a range of 100-4500 rpm at a gearing rate of 0.075. For the used dataset in this study the speed is 1000 rpm. The motion-view filtering software that came with these motors was used to monitor and set up the suited speed of each impeller. The degassing system was used to remove bubbles and solid dust from the water before starting the experiment. The tank has been specifically designed for studying turbulence (Lagrangian and Eulerian motion at moderate Reynolds numbers). The flow facility produces a nearly stationary homogeneous isotropic turbulence near the center of the tank, where measurements have been performed. Circular plates, shown at the top and bottom (see Figure 1), generate the strain motions; a linear actuator drove each plate with a piston rod (Parker, model: ETV32M10PA47JMA400A). When they are moved towards the center with a pre-described rate, a nearly constant strain rate is ensured in the fluid. Spherical and hollow glass beads with a median diameter of 8-10 μm and a specific gravity of 1.1 g/cm 3 seed the flow. The recording area is located in the center of the tank with a size of 24.5 24.5 mm 2 (see Figure 1). The particle image velocimetry technique is applied and extracts the flow properties before strain deformation. Thus, via a second-order longitudinal velocity structure function, the turbulent flow dissipation rate is obtained; therefore, the Kolmogorov time scale is calculated based on Equation (6) and a Stokes number is obtained from Equation (4). The Taylor microscale Reynolds number is achieved in the range of 90 < < 110 in the performed experiment. For the dataset used, the strain rate produced by the Circular plates, shown at the top and bottom (see Figure 1), generate the strain motions; a linear actuator drove each plate with a piston rod (Parker, model: ETV32M10PA47JMA400A). When they are moved towards the center with a pre-described rate, a nearly constant strain rate is ensured in the fluid. Spherical and hollow glass beads with a median diameter of 8-10 µm and a specific gravity of 1.1 g/cm 3 seed the flow. The recording area is located in the center of the tank with a size of 24.5 × 24.5 mm 2 (see Figure 1). The particle image velocimetry technique is applied and extracts the flow properties before strain deformation. Thus, via a second-order longitudinal velocity structure function, the turbulent flow dissipation rate is obtained; therefore, the Kolmogorov time scale is calculated based on Equation (6) and a Stokes number is obtained from Equation (4). The Taylor microscale Reynolds number is achieved in the range of 90 < R λ < 110 in the performed experiment. For the dataset used, the strain rate produced by the two abovedescribed circular plates is 8 s −1 , and the Lagrangian particle-tracking techniques are applied to inscribe the data. A high-speed camera used as a detection system was set at 10 kHz (10,000 fps) for well-resolved particle velocity statistics. This very high temporal resolution (0.1-0.2 ms) is considerably smaller than the Kolmogorov time τ η (35-99 ms) of the smallest eddies present in the flow; therefore, the properties of the dissipation range in the flow are resolved. However, in contrast to the previous numerical works, this empirical study considers the velocity field explored by tracer particles in the presence of gravity. Each video has 2000 frames, and to collect sufficient statistical data, the strain motion was repeated 20 times to record 20 videos. All videos together have created 40,000 frames. Each of the videos was statistically independent, as the flow is given a generous time to recover to near isotropy between different strokes.

Sequential Velocity Dataset
The recorded videos from the experiment are processed via programming based on Lagrangian particle-tracking method and are not subject of this study, and the detail can be seen in [25,26]. Each particle has vector of velocity and displacement during the strain motion. According to denotation (1) and (2) in the Lagrangian view, these statistics could be used to investigate the turbulent flow. As one of the reviewers mentioned, some statistical features can be calculated from extracted data from the experiment [27]. In this study, however, our focus is on velocity because it is a feature that can be measured in many applications, and its prediction model can be helpful. For example, wind speed is a crucial issue in wind energy production; it is, therefore, essential to have a prediction model for speed itself so producers can forecast the power production in the following period, long or short term.
The dataset of this study composed of 2,862,119 tracking points for every vector is as follows: These tracking points consist of all particles' velocity vectors achieved via 20 recordings, and every video recording includes several particles. Moreover, it is expected to observe several tracking lines, as it is presented for the velocity in the results in Section 5; every tracking line specifies a particle.
Although the dataset is sequential, we split it into training data and test data for the first model, 80% and 20%, respectively, and for the second model, 60% and 40%, proportionally. Therefore, we assessed and validated the velocity prediction of the following period with the test data for LSTM and GRU models.

LSTM and GRU Model Set Up
We coded the models in Python and used the TensorFlow platform [28,29]. The LSTM model is set up with 100 layers and one dense layer, and Adam is specified as an optimizer. The GRU model has also been set up with the same layers and optimizer. The dataset was normalized by MinMaxScaler transformation [30]. The MinMaxScaler is a type of scaler that scales the minimum and maximum values to be 0 and 1, respectively [30]. Since the modeling was implemented on the DEEP-DAM module [31] parallel computing machine, we have applied a distributed strategy application programming interface from the TensorFlow platform abstraction to distribute the training across multiple custom training loops [32]. The strategy has been set up with one to four GPUs on one node. The result of the computing and the models' performance distinction are reported in Section 5.

Meseured Turbulent Flow Velocity
As is described in Section 3, from the experiments, based on the Lagrangian-particle tracking, the recorded videos included velocity vectors for particles moving in the flow velocity field. Figures 5 and 6 illustrate the velocity component in the y and x directions, respectively. Since the initial strain rate was generated in the flow in the y direction, as was expected, the velocity component in the y direction has an inclined average velocity relative to the velocity in the x direction. In this study, these extracted data have been used to train LSTM and GRU with a ratio of the training data and assess the prediction with the test data proportion. According to Equation (3), which defines the velocity field for laminar flow, it can be seen in Figures 5 and 6 that the turbulent flow behaves differently than the laminar equation because of velocity fluctuations. was expected, the velocity component in the y direction has an inclined average velocity relative to the velocity in the x direction. In this study, these extracted data have been used to train LSTM and GRU with a ratio of the training data and assess the prediction with the test data proportion. According to Equation (3), which defines the velocity field for laminar flow, it can be seen in Figures 5 and 6 that the turbulent flow behaves differently than the laminar equation because of velocity fluctuations.

Turbulent Flow Velocity Predicition via LSTM and GRU
Figures 7-10 display the velocity prediction via GRU and LSTM model for the y and the x directions. The models trained and assessed with two distinct data ratios. In the first model, 80% of the data have applied as training and 20% rest of the dataset validated the prediction. In second model, 60% of the training data have been applied and 40% used as test data. was expected, the velocity component in the y direction has an inclined average velocity relative to the velocity in the x direction. In this study, these extracted data have been used to train LSTM and GRU with a ratio of the training data and assess the prediction with the test data proportion. According to Equation (3), which defines the velocity field for laminar flow, it can be seen in Figures 5 and 6 that the turbulent flow behaves differently than the laminar equation because of velocity fluctuations.

Turbulent Flow Velocity Predicition via LSTM and GRU
Figures 7-10 display the velocity prediction via GRU and LSTM model for the y and the x directions. The models trained and assessed with two distinct data ratios. In the first model, 80% of the data have applied as training and 20% rest of the dataset validated the prediction. In second model, 60% of the training data have been applied and 40% used as test data.

Turbulent Flow Velocity Predicition via LSTM and GRU
Figures 7-10 display the velocity prediction via GRU and LSTM model for the y and the x directions. The models trained and assessed with two distinct data ratios. In the first model, 80% of the data have applied as training and 20% rest of the dataset validated the prediction. In second model, 60% of the training data have been applied and 40% used as test data.          LSTM and GRU models have provided accurate predictions of a strained turbulent flow velocity with no known pattern in theory. We must notice that the period of the experiment and data used in this study are short, but proportionally the prediction model could be used for similar velocity field application. Figures 7 and 8 represent the velocity in y direction, forecasting 80% training and 60% training. As can be seen, the prediction section is an impressive match for the test LSTM and GRU models have provided accurate predictions of a strained turbulent flow velocity with no known pattern in theory. We must notice that the period of the experiment and data used in this study are short, but proportionally the prediction model could be used for similar velocity field application. Figures 7 and 8 represent the velocity in y direction, forecasting 80% training and 60% training. As can be seen, the prediction section is an impressive match for the test data for LSTM and GRU model with two training ratios. The mean average error (MAE) = 0.001-0.002 and R 2 score is in range of 0.983-0.987 for both models. Figures 9 and 10 illustrate the prediction of the velocity in the x direction with two different training data ratios. The MAE and R 2 for x direction forecasting has the same range and with an outstanding match.
The result of the velocity prediction of the turbulent flow represents the capability of LSTM and GRU models, which can forecast unknown sequential data. The Lagrangian view provides temporal data, and it appears possible to apply this approach in similar turbulent flow with a longer period.

LSTM and GRU Models Performance
It has been reported in the literature that the GRU model has faster performance than the LSTM model [10,11]. For 80% training, GRU is 8-12% faster than LSTM and for 60% training it is 5-10% faster. However, when the number of GPU increases, LSTM performs modeling faster than GRU, which can be explained by the application of GPUs that provides much more memory for the LSTM. We investigated the performance of LSTM and GRU on DEEP-DAM module on one node with four GPUs. For all models in this study, LSTM executed 7-8% faster than the GRU model. Table 1 shows the result of this evaluation.

Conclusions
The subject of this study was using LSTM and GRU models to provide a prediction for distortion turbulent flow performed in a laboratory with specific turbulent intensity and mean strain rate in the primary direction. For two training efforts, the dataset was split into 80% first and secondly 60%. Every ratio of training in the rest of the data was applied for test and prediction validation. LSTM and GRU models were applied and executed on the DEEP-DAM module of parallel computing machine at Juelich supercomputing center. Two different GPU set ups were applied to assess the model's performance.
The result of this study shows that LSTM and GRU models can predict the straining turbulence flow appropriately and match in quality and quantity. The mean average error (MAE) = 0.001-0.002 and R 2 score is in the range of 0.983-0.987 for both models. Without GPU, the GRU model has faster performance than the LSTM and, with less training ratio (60%), can provide prediction with the same performance of training with 80%. Nevertheless, we must notice the period of the dataset used was short, so the forecast was also brief. However, with GPUs set up, LSTM gets faster performance than GRU, which is related to GPUs memory, which strengthens the LSTM memory.
In many applications of fluid dynamics, there is a possibility to collect the velocity field data in the Lagrangian framework in which data are sequential. It seems this advantage of the Lagrangian view could be applied to predict the velocity field via such LSTM and GRU models.