Extracting fundamental parameters of 2D natural thermal convection using convolutional neural networks

Ali Boroumand, Mohammad; Morra, Gabriele; Mora, Peter

doi:10.1063/5.0198004

The Lattice Boltzmann Method (LBM) is an approach for modeling mesoscopic fluid flow and heat transfer, based on modeling distributions of particles moving and colliding on a lattice. Using a perturbative formulation of the Boltzmann equation, it scales to the macroscopic Navier–Stokes equation. We simulate natural thermal convection via LBM in a 2D rectangular box being heated from below and cooled from above, and use the results as training, testing, and generalization datasets to build a deep learning model. GoogLeNet, a convolutional neural network, is used to classify the simulation results based on two parameters: Rayleigh (⁠ $R a$ ⁠) and Prandtl (⁠ $P r$ ⁠) numbers, from a single snapshot of either the entire modeling field of resolution $1024 \times 1024$ ⁠, or a $224 \times 224$ crop. For each fixed $P r$ in a range from 1 to 128, increasing by a factor of 2, we estimate $R a$ with an accuracy varying from 40% to 90%, depending on the chosen augmentation strategy. For each fixed $R a$ in the range from $10^{5}$ to $10^{9}$ ⁠, increasing of a factor $\sqrt{10}$ ⁠, the method predicts $P r$ with a systematically lower accuracy ranging from 30% to 80%. This approach has great potential for industrial applications like being able to control the industrial flow or scientific research on geophysical ones including the transport of heat in the earth’s interiors, ocean, and atmosphere.

I. INTRODUCTION

Convection, a mechanism of heat transfer in a fluid due to the diffusion–advection of heat, is a sophisticated phenomenon that has attracted much attention because of both its fundamental interest and its practical applications. Categorized as natural or forced, natural convection is caused by external or internal heat sources, which cause buoyancy imbalance, the rise of warmer fluid, and the downwelling of the cooler fluid.¹

Natural convection has a major role in many industrial and geophysical processes. Industrial examples include the transport of heat in chemical reactors and electronic cooling systems. Geophysical flow comprises heat transport in the atmosphere and the ocean, which determines weather, climate, and ocean circulation. In all these cases, the ability to control and anticipate the flow behavior requires real-time access to the critical parameters of the flow based on the boundary conditions and geometries.

The Rayleigh number (⁠

R a

⁠) is a dimensionless number associated with the intensity of natural convection: a low

R a

is associated with laminar flow while a high

R a

to turbulent behavior. Below a critical value

R a_{c}

⁠, heat transfer dominates over advection, therefore convection does not happen. The Rayleigh number is defined as the product of the Grashof number (⁠

G r

⁠), which describes the ratio between buoyancy and viscosity, and the Prandtl number (⁠

P r

⁠), the ratio between momentum and thermal diffusivity,

R a = G r \times P r .

(1)

Any couple of these three parameters uniquely determine the convection characteristics, most commonly $R a$ and $P r$ ⁠. With the present development of machine learning, it is tempting to examine the possibility of extracting $R a$ and $P r$ from the convection patterns. Our research is in line with previous work where fundamental parameters for convection such as Yield stress and viscosity were extrapolated from numerical results² and attempts at generating depth profiles of steady-state convecting systems from machine learning.^3–5 Our work more generally explores the prediction of fundamental fluid-dynamic parameters, which has applications in engineering as well as for the geo/planetary community.

Progress in machine learning leaves open several challenges and decisions on how to implement this task. First, there is no general recipe to create an effective dataset designed to capture the fundamental characteristics of the physical model. Second, it is unknown whether the complexity emerging in fluid-dynamic systems when chaos, or turbulence, forms, is an obstacle due to its unpredictability or an advantage due to its richness of details when combined with machine learning algorithms such as neural networks. Finally, the generation of datasets of computational fluid-dynamic simulations for machine learning applications requires the amount of computational resources that can be inaccessible to many scientists.⁶

In this work, we generate training and testing sets for a popular CNN algorithm, called GoogLeNet, by simulating 2D natural thermal convection between two plates, using the Lattice Boltzmann method (LBM). LBM allows modeling incompressible convection with a wide range of $P r$ and $R a$ numbers (Sec. II). By using traditional data augmentation,⁷ we either fix $R a$ or $P r$ and extract the other fundamental convection parameter (Sec. III). Finally, our approach is tested for different boundary conditions, sampling rates, and augmentation methods (Sec. IV).

II. SIMULATION

The Lattice Boltzmann method (LBM) is an alternative approach to classical numerical solutions of PDEs for modeling fluid dynamics based on simulating the Boltzmann equation on a discrete lattice. Since the first inception as the Lattice gas method in which it was proven that the Navier–Stokes equations are recovered at the macroscopic scale by modeling discrete particles moving and colliding on a discrete lattice,⁸ the more efficient Lattice Boltzmann Method (LBM) was developed⁹ in which particle number densities are moved on a discrete lattice with collisions being achieved by relaxing the number densities toward the Boltzmann equilibrium distribution.¹⁰ The LBM, has since been used to model a wide range of fluid dynamics problems¹¹ including mantle convection,^12–15 magneto-hydrodynamics,¹⁶ multiphase flow in porous media,^17,18 and aerospace engineering.¹⁹ In the following, we overview the thermal Lattice Boltzmann method which can be used to simulate natural convection and present 2D simulations with a linear rheology.

A. The thermal Lattice Boltzmann method

The thermal LBM models two distributions $f_{α}$ and $g_{α}$ representing the mass density and energy density moving one lattice spacing along the $α$ directions (i.e., orthogonally and diagonally) of a Cartesian lattice. After these number densities move by one lattice spacing, they are “relaxed” toward the Boltzmann equilibrium distribution. These two steps applied to $f_{α}$ which relates to the fluid density $ρ$ are found to yield the Navier–Stokes equations for incompressible fluids.⁹ The two steps applied to $g_{α}$ the advection–diffusion equation for energy density $E$ which relates to the temperature $T$ ⁠.²⁰

The combined movement and collision steps can be written as

f_{α} (x + c_{α} Δ t, t + Δ t) = f_{α} (x, t) + Δ f_{α}^{C} (x, t),

(2)

and

g_{α} (x + c_{α} Δ t, t + Δ t) = g_{α} (x, t) + Δ g_{α}^{C} (x, t),

(3)

where

c_{α}

is the velocity vector pointing between adjacent lattice sites (i.e.,

c_{α} Δ t

equals one lattice spacing orthogonally or diagonally in a square or cubic lattice), and

Δ_{α}^{C} f

and

Δ_{α}^{C} g

are the collision terms which redistribute number densities between directions due to collisions. These collision terms can be calculated by an efficient method of relaxing the distributions toward the equilibrium distribution due to Bhatnagar et al.¹⁰ using

Δ f_{α}^{C} = (\frac{1}{τ_{f}}) (f_{α}^{e q} - f_{α}),

(4)

Δ g_{α}^{C} = (\frac{1}{τ_{g}}) (g_{α}^{e q} - g_{α}),

(5)

where

τ_{f}

and

τ_{g}

are the relaxation times for the two number densities

f_{α}

and

g_{α}

⁠, and

f_{α}^{e q}

and

g_{α}^{e q}

are, respectively, the corresponding equilibrium distributions which can be calculated as a Taylor expansion to the second order of the Boltzmann distribution, namely,

f_{α}^{e q} = ρ w_{α} (1 + \frac{c_{α} \cdot u}{c_{s}^{2}} + \frac{(c_{α} \cdot u)^{2}}{2 c_{s}^{4}} - \frac{u^{2}}{2 c_{s}^{2}}),

(6)

and

g_{α}^{e q} = ρ ε w_{α} (1 + \frac{c_{α} \cdot u}{c_{s}^{2}} + \frac{(c_{α} \cdot u)^{2}}{2 c_{s}^{4}} - \frac{u^{2}}{2 c_{s}^{2}}),

(7)

where

ρ

is the macroscopic density,

u

is the macroscopic fluid velocity,

ε

is the energy density,

w_{α}

are weights which depend on the lattice (to be specified shortly) and

c_{s} = s / \sqrt{3}

is the speed of sound in the lattice where

s = Δ x / Δ t

is the lattice speed (i.e.,

Δ x

is the lattice spacing of the cartesian lattice and

Δ t

is the time step). In the following, we use non-dimensionalized simulations and hence, we set

Δ x = Δ t = 1

which leads to equilibrium distributions given by

f_{α}^{e q} = ρ w_{α} (1 + 3 (c_{α} \cdot u) + \frac{9}{2} {(c_{α} \cdot u)}^{2} - \frac{3}{2} u^{2}),

(8)

and

g_{α}^{e q} = ρ ϵ w_{α} (1 + 3 (c_{α} \cdot u) + \frac{9}{2} {(c_{α} \cdot u)}^{2} - \frac{3}{2} u^{2}) .

(9)

The relaxation times in the above equilibrium distributions relate to the kinematic viscosity and the thermal diffusivity as

τ_{f} = \frac{ν_{f}}{c_{s}^{2} Δ t} + 0.5,

(10)

where

ν_{f}

is the kinematic viscosity and

τ_{g} = \frac{ν_{g}}{c_{s}^{2} Δ t} + 0.5,

(11)

where

ν_{g} = κ

is the thermal diffusivity.

So far, the above equations are valid for 2D or 3D. However, in the following numerical study, we will perform simulations in 2D. We use the so-called D2Q9 lattice (2 dimensions, 9 lattice velocities) in which number densities move on a square lattice along the axes or in the diagonal directions. We define the velocity vectors in the D2Q9 lattice

c_{α}

as

c_{α} = \begin{matrix} [(0, 0), (1, 0), (- 1, 0), (0, 1), (0, - 1), \\ (1, 1), (- 1, - 1), (1, - 1), (- 1, 1)] \frac{Δ x}{Δ t} \end{matrix},

(12)

and the weights vector

w_{α}

for this lattice is given by

w_{α} = (4 / 9, 1 / 9, 1 / 9, 1 / 9, 1 / 9, 1 / 36, 1 / 36, 1 / 36, 1 / 36),

(13)

i.e., a weight of 4/9 for the zero velocity vector (i.e., the velocity vector for no movement), 1/9 for directions along the cartesian axes, and 1/36 along diagonals (note that

\sum_{α} w_{α} = 1

⁠) (Fig. 1). It is well known that the above choice leads to the Navier–Stokes equations for incompressible fluid flow⁹ where the macroscopic properties can be obtained from the number densities

f_{α}

and

g_{α}

using

ρ = \sum_{α} f_{α},

(14)

ρ u = \sum_{α} f_{α} c_{α},

(15)

and

ρ ε = \sum_{α} g_{α},

(16)

where the energy density

ε

relates to temperature through

ε = D R T / 2

where

D = 2

is the number of dimensions,

T

is the temperature, and

R

is the gas constant, which we set to unity for our non-dimensionalized simulation runs in this paper.

FIG. 1.

View large Download slide

(a) Space discretization of the 2D convection field and (b) weighted discretized velocity space in D2Q9 lattice.

1. Boussinesq buoyancy term

To model convection, we need to add a buoyancy force and we adopt the usual Boussinesq approximation where density variations have a fixed part plus a perturbation that linearly depends on the temperature given by

(ρ + δ ρ) = ρ (1 - β Δ T) .

(17)

We assume gravity and hence, under the Boussinesq approximation, we need to add a force term of

G = - ρ β Δ T g,

(18)

where

g

is the acceleration due to gravity,

β

is the coefficient of thermal expansion, and

G

is the gravitational force due to gravity. As a practice in LBM, the physical quantities used in the simulations are dimensionless. Here,

ρ

is a perturbative value around

1

⁠.

The above gravitational force can be added to the LBM by adding a forcing term to the Lattice Boltzmann equations for

f_{α}

⁠. Equation including the gravitational force becomes

f_{α} (x + c_{α} Δ t, t + Δ t) = f_{α} (x, t) + Δ f_{α}^{C} (x, t) + F_{α}^{b} (x, t),

(19)

where the Boussinesq buoyancy forcing term

F^{b}

is given by

F_{α}^{b} = w_{α} 3 c_{α} \cdot G = w_{α} 3 c_{α_{z}} G Δ t,

(20)

where

G = | G | = G_{z} = - ρ β Δ T g

⁠. Equation can be verified by calculating the change in velocity during one time step due to the Boussinesq term using Eq. .

B. Simulation setup and results

For setting up the simulation, we select a suitable range for $R a$ and $P r$ numbers. Looking for a range for $P r$ ⁠, we choose $P r = 1$ – $128$ to encompass a broad range of fluids, from air to heavy oils.²¹ Convection patterns converge toward an asymptotic solution for higher Prandtl numbers, normally obtained for infinite $P r$ ⁠.²² We vary $R a$ in the range from $10^{5}$ ⁠, which is sufficiently above the critical $R a$ number (for fixed boundary conditions $R a_{c} = 1708$ ⁠) to guarantee a rapid initiation of convection, to $10^{9}$ ⁠. The maximum $R a$ is determined by resolution since high $R a$ convection is characterized by progressively thin boundary layers, which need to be well resolved for the validity of the solution.²¹

Once

P r

and

R a

are chosen, and assumed the other physical parameters to be fixed, the viscosity and thermal diffusivity are uniquely determined. Using Eq. and the definition of the Grashof number (Gr) and of the Prandtl number (Pr) we have¹³

R a = G r \cdot P r = \frac{g β (T_{l} - T_{u}) L^{3}}{κ ν},

(21)

with

G r = \frac{g β (T_{l} - T_{u}) L^{3}}{ν^{2}}, P r = \frac{ν}{κ},

(22)

where

ν

is the kinematic viscosity and

κ

is the thermal diffusivity. Using Eqs. and , one obtains for all our simulations,

ν = \sqrt{(\frac{P r}{R a}) g β (T_{l} - T_{u}) L^{3}},

(23)

κ = \sqrt{\frac{g β (T_{l} - T_{u}) L^{3}}{P r R a}} .

(24)

The range of values obtained through this formulation varying of several orders of magnitude is summarized in Table I.

TABLE I.

The values and ranges of the parameters used for the simulations.

Symbol	Definition	Value
ρ	Density	≃1
g	Acceleration due to gravity	0.42
β	Coefficient of thermal expansion	0.1
T_l	Temperature of the lower plate	0.0005
T_u	Temperature of the upper plate	−0.0005
L	The characteristic length	1024
Pr	Prandtl number	[1 − 128]
Ra	Rayleigh number	[10⁵ − 10⁹]
ν	Kinematic viscosity	[6.7 × 10⁻³ − 7.6]
κ	Thermal diffusivity	[5.9 × 10⁻⁴ − 0.67]

Symbol	Definition	Value
ρ	Density	≃1
g	Acceleration due to gravity	0.42
β	Coefficient of thermal expansion	0.1
T_l	Temperature of the lower plate	0.0005
T_u	Temperature of the upper plate	−0.0005
L	The characteristic length	1024
Pr	Prandtl number	[1 − 128]
Ra	Rayleigh number	[10⁵ − 10⁹]
ν	Kinematic viscosity	[6.7 × 10⁻³ − 7.6]
κ	Thermal diffusivity	[5.9 × 10⁻⁴ − 0.67]

The boundary conditions that we use for simulation purposes are either (i) fixed or (ii) periodic. In fixed boundary conditions, the values for $T$ ⁠, $v_{x}$ ⁠, and $v_{z}$ are set on the right and left side of the modeling domain equal to 0. For periodic boundary conditions, these values are the same on the left and right sides of the simulation domain.

All simulations in this work are executed on a lattice of resolution $1024 \times 1024$ ⁠. $R a$ varies from $10^{5}$ to $10^{9}$ ⁠, increasing by factor of $\sqrt{10} \approx 3.16$ for each fixed $P r$ number. $P r$ number varies from 1 to $P r = 128$ ⁠, increasing it of a factor 2 at each iteration. Using $3.16$ as a factor, $10^{9}$ is therefore approximated by $9.89 \times 10^{8}$ ⁠. For each couple of $R a$ and $P r$ ⁠, we run two simulations, one from $t = 0$ to $t = 2.0 \times 10^{5}$ ⁠, saving the results each 1000 steps, and a second one from $t = 0$ to $t = 2.0 \times 10^{6}$ ⁠, saving the solution each 10 000 steps. Three scalars are saved for each computing node: (1) the temperature field (⁠ $T$ ⁠), (2) the x-component of the velocity (⁠ $v_{x}$ ⁠), and (3) the z-component of the velocity (⁠ $v_{z}$ ⁠). In this way, for every $R a$ and $P r$ number, two sets of 603 images are generated, each used to train a different neural network and perform another ML experiment.

Examples of results ranging from laminar flow to turbulent one are shown in Fig. 2.

FIG. 2.

View large Download slide

Simulation results for laminar and turbulent convection with different boundary conditions and time steps. (a) Convection field for $R a = 10^{5}$ and $P r = 128$ and fixed boundary conditions at $t = 2 000 000$ ⁠, (b) Convection field for $R a = 10^{9}$ and $P r = 1$ and fixed boundary conditions at $t = 200 000$ ⁠, (c) Convection field for $R a = 10^{5}$ and $P r = 128$ and periodic boundary conditions at $t = 2 000 000$ ⁠, (d) Convection field for $R a = 10^{9}$ and $P r = 1$ and periodic boundary conditions at $t = 200 000$ ⁠.

III. CONVOLUTIONAL NEURAL NETWORKS

A. Algorithm

In the last several years, mainly due to the advances of deep learning, and convolutional networks principally, the quality of image recognition and object detection has been progressing at an exceptional pace.²⁴ Most of this progress is not just the result of more powerful hardware, expanded datasets, and larger models, but primarily a consequence of new ideas, improved algorithms, and innovative network architectures. The most successful methods of object detection instead of coming from more powerful deep networks alone, emerged from the combination of deep architectures and classical computer vision algorithms, such as the R-CNN algorithm by Girshick et al.²⁵ In this paper, we use a widely used deep neural network architecture for computer vision, codenamed GoogLeNet,²³ which is rooted in pioneering work by Lin et al.,²⁶ who experimented with stacking micro neural networks to improve feature map generation.

The simplest way to improve the performance of a deep neural network is to increase its size (depth, i.e., the number of levels of the network, and width, i.e., the number of units per level). With a sufficiently large training set, this approach is successful, but with some caveats. First of all, if the training set is not sufficiently large, the large number of parameters will make the greater network more prone to overfitting. In works like ours, the generation of the training dataset is related to the use of limited computational resources. By the similarity of the results in some of the numerical models, once a steady state is reached, the number of required training images is increased.

Another aspect in which computational resources limit the size of a suitable network for practical application is that the training of a network requires resources that scale quadratically with the network size.²⁷ On top of that, for specific and limited applications such as the one presented in this paper, where the categories are of the order of 10, many weights of the network are irrelevant. They would be set to close to zero values, implying a waste of computational resources.²⁸ Overall, an optimal algorithm uses all the available computational resources, minimizing waste.

GoogLeNet addressed these challenges in Szegedy et al.²³ as part of the ImageNet Large-Scale Visual Recognition Challenge 2014 (ILSVRC14). The solution proposed requires the replacement of fully connected architectures with primarily sparsely connected ones, which allows to flexibly adapt to features at many scales.²⁹ Considering this idea, inspired by the biological interpretation of neural networks (e.g., Arora et al.³⁰), the optimal network topology is layer by layer constructed by correlating clustered neurons with highly correlated outputs. In Ref. 23, this is described with the well known Hebbian principle that “Neurons that fire together, wire together,” which is a more algorithmic way to describe the same principle.

From the computational point of view, the connectivity between neural layers can be either sparse or dense. Sparse connectivity is more challenging than dense. Constantly improving numerical libraries for fast dense matrix multiplication, based on state-of-the-art standard hierarchical CPU/GPU hardware^31,32 allows hardware-based rapid progress of traditional computational approaches. For this reason, the Inception architecture (GoogLeNet) that we employ approximates sparse structures for vision networks with dense, computationally efficient components. This approach allows more uniformity of structures, and filters, by employing efficient dense computations.

A key aspect responsible for the quality of the classification is the choice of the dataset used for pre-training the network. Training from scratch a neural network such as GoogLeNet, characterized by a complex structure and vast number of parameters, is computationally overly expensive and not necessary, given the common and well-tested use of pretraining.²⁷ Even if pretraining is done with radically different datasets from the ones for which it is used, as in our case, its result is optimal because of the layered structures of networks. The network layers are responsible for detecting single features, simpler for the lowest layer and more abstract for the apical ones. For instance, for a lower layer, the curvature of the temperature field in a fluid-dynamic simulation is not different from the curvature of the body of a car. Hence, using a network pre-trained with a general purpose very large dataset can be trained much more rapidly than a blank network. Experience has shown that only the last layer needs to be reconstructed, based on the specific dataset of interest. This approach is commonly called transfer learning in ML literature.³³ We use a version of GoogLeNet pre-trained with the ImageNet dataset, composed of over 10 million images (Fig. 3). The use of ImageNet has two important advantages: (1) the large amount of data. (2) the diversity and fairness employed to design the composition of the dataset.^34,35

FIG. 3.

View large Download slide

An overview of the GoogLeNet internal structure. The green area shows the layers that their weights are not trained during transfer learning. The inception modules have different parameters based on Ref. 23. The linear layer is adopted upon the number of classes. Then, it is trained using a dataset designed by simulation results. The figure shows the situation for fixed $P r = 1$ ⁠. $S$ and $P$ in Convolution, max pooling, and average pooling layers are abbreviations of patch size and stride.

B. Dataset

1. Design

Since GoogLeNet, as well as most pre-trained networks, has been developed to process RGB images (three fields for each pixel), we designed a dataset that would optimally fit with our Neural Network. For each $R a$ ⁠, $P r$ ⁠, and time step, we created an RGB image as a combination of the three fields $T$ ⁠, $v_{x}$ ⁠, and $v_{z}$ ⁠. Four examples of the RGB combined images of the $200 k$ step of the simulation for the four end-members (minimum $R a$ and maximum $P r$ vs maximum $R a$ and minimum $P r$ ⁠, for both fixed and periodic boundary conditions) of the convection regime are shown in Fig. 4.

FIG. 4.

The RGB images from a combination of T (red), v x (green) and v z (blue) fields with (a), (b) fixed and (c) and (d) periodic boundary condition. (a) R a = 10 5 , P r = 128 t = 2 × 10 6, (b) R a = 10 9 , P r = 1 t = 2 × 10 5, (c) R a = 10 5 , P r = 128 t = 2 × 10 6, (d) R a = 10 9 , P r = 1 t = 2 × 10 5.

View large Download slide

The RGB images from a combination of $T$ (red), $v_{x}$ (green) and $v_{z}$ (blue) fields with (a), (b) fixed and (c) and (d) periodic boundary condition. (a) $R a = 10^{5}, P r = 128$ $t = 2 \times 10^{6}$ ⁠, (b) $R a = 10^{9}, P r = 1$ $t = 2 \times 10^{5}$ ⁠, (c) $R a = 10^{5}, P r = 128$ $t = 2 \times 10^{6}$ ⁠, (d) $R a = 10^{9}, P r = 1$ $t = 2 \times 10^{5}$ ⁠.

2. Image mapping

Since we use the pre-trained version of the GoogLeNet, only images with a resolution of $224 \times 224$ can be fed to the network. We adopted three strategies to map the $1024 \times 1024$ field to the required resolution.

The first approach uses the bilinear interpolation, which is a resampling method that uses the distance-weighted average of the four nearest pixel values to estimate a new pixel value, to map $1024 \times 1024$ field to $224 \times 224$ ⁠.³⁶ This implies a certain reduction of resolution, which reduces the sensibility of the neural network to the smallest details.
In a second approach, we cropped always the same $224 \times 224$ sized central region to the domain. Although a smaller region implies a loss of information on the overall convection field, our method tests whether the NN can extrapolate missing information from a subregion, which is expected particularly for the complex flow characteristic of a high Rayleigh number.
The third approach, random cropping, is based on the random selection of $224 \times 224$ regions at casual x and y positions of the convection field. This choice is expected to increase the information fed into the Network since data from the entire domain are used for training, but with the disadvantage of not giving information on the position to the network.

3. Image augmentation

Present machine learning algorithms are known to need huge amounts of data to perform satisfactorily, contrary to human-based intelligence, which can instead extrapolate conclusions from small inputs. To obtain the datasets necessary to train an ML, two approaches exist: one is to produce large low-quality data, like unfiltered Web pages, containing many errors, to train large language methods, or augment a small set of curated structured data.⁷ We use this second approach, as has been proven effective by several recent works.^37–39

Given that initial tests of random cropping gave poor results, as later shown in the results section, we implemented an augmentation technique for the third, random cropping, design. For each $1024 \times 1024$ model output, we cropped 20 $224 \times 224$ RGB images each at a different random position. This represented our fourth dataset.

Four datasets have been so created for each type of boundary condition, fixed and periodic (Fig. 5).

FIG. 5.

View large Download slide

Different methods of image preparation for making dataset. (a) Bilinear interpolation, (b) center cropping, (c) random cropping, (d) multiple cropping.

C. Implementation

Using GoogLeNet, we perform classification on $R a$ and $P r$ classes. For each test, the dataset was divided into two parts, randomly picked each time: $80 %$ for training and $20 %$ for testing. The training was executed for $40$ epochs with a batch size equal to 1. This minimum batch was chosen as a guarantee of accuracy over performance.

We performed training and testing separately on two setups. In one, we fixed the $P r$ number and trained the network to predict the $R a$ number from each image, and in the second case, we fixed $R a$ and tried to predict the $P r$ number.

Detailed information on the augmentation and mapping is given in the precious subsection and all the information are summarized in Table II.

TABLE II.

Detailed information about machine learning dataset.

Definition	Value
Boundary conditions	Fixed, periodic
Sampling rate (time steps)	10³,a 10⁴b
Simulation duration (time steps)	2 × 10⁵, 2 × 10⁶
No. of images per simulation	200
No. of image: resize	1 image
No. of image: center cropped	1 tile
No. of image: random cropped	1 tile
No. of image: augmented random cropped	20 tiles
Dataset size (fixed Pr)	$[1800, 1800, 1800, 36 000]$ c
Dataset size (Fixed Ra)	$[1600, 1600, 1600, 32 000]$

Definition	Value
Boundary conditions	Fixed, periodic
Sampling rate (time steps)	10³,a 10⁴b
Simulation duration (time steps)	2 × 10⁵, 2 × 10⁶
No. of images per simulation	200
No. of image: resize	1 image
No. of image: center cropped	1 tile
No. of image: random cropped	1 tile
No. of image: augmented random cropped	20 tiles
Dataset size (fixed Pr)	$[1800, 1800, 1800, 36 000]$ c
Dataset size (Fixed Ra)	$[1600, 1600, 1600, 32 000]$

^a

Refers to the 2 × 10⁵ long models.

^b

Refers to the 2 × 10⁶ long models.

^c

Each number refers to each of the four approaches.

IV. RESULTS AND DISCUSSION

Given the large number of numerical experiments performed, we synthesize the results in two plots in which each line represents the average of four models: (1) for 1000 and 10 000 time step intervals for each (2) fixed and periodic lateral boundary conditions. The fine-grained effect of each of these choices is then shown in the figures in Appendix A and discussed later. The main pictures emerging from the four types of datasets: (a) resized image from the entire domain, (b) center cropped image, (c) random cropped image and (d) augmented random cropped (x 20 times). Figure 6 presents the results for (a) fixed $R a$ and variable $P r$ ⁠, and (b) fixed $P r$ and variable $R a$ ⁠.

FIG. 6.

View large Download slide

Average classification accuracy for fixed $P r$ and $R a$ numbers. (a) Average classification accuracy of obtaining $R a$ number at fixed $P r$ number, (b) Average classification accuracy of obtaining $P r$ number at fixed $R a$ number.

An immediate result is that regardless of the augmentation mechanism, the classification accuracy is always superior when estimating Ra (fixed $P r$ ⁠) vs estimating $P r$ (given a fixed $R a$ ⁠). This can be associated with the greater factor increase for Ra (⁠ $\sqrt{10}$ ⁠) than for $P r$ (⁠ $2$ ⁠). We performed tests with a factor $4$ for $P r$ in the “Resized” case and verified that performance drastically improved. However, it is not clear whether this is due to the greater factor or to the reduced number of options.

Another aspect related to the superior performance of the Ra classifier is that high $P r$ solutions asymptotically converge toward infinite $P r$ solution. This implies that high $P r$ solutions are more similar to each other than different $R a$ solutions. For example, the complexities of the patterns determined by the fluid flow vary with $R a$ (it is known the general relationship between Reynolds and Rayleigh number, $R e \approx R a^{1 / 2}$ ⁠), therefore an increase in turbulence can be likely picked by the CNN. On the contrary an increase in $P r$ produces smoother solutions (more laminar flow) more and more similar to each other, with increasing $P r$ ⁠.

For both classifications of $R a$ and $P r$ ⁠, a common result is that for both the physical parameters, GoogleNet displays consistently much better results for the bilinear dataset compared to the others. The most obvious explanation is that the mapping of the convection field features (⁠ $T$ ⁠, $v_{x}$ ⁠, $v_{z}$ ⁠) maintains the macroscopic patterns even if $\sim 20$ pixels are mapped into 1. When looking more in detail into the $P r$ and $R a$ dependent results, one observes that for fixed $P r$ [Fig. 6(b)], the performance of the network systematically improves for increasing $R a$ ⁠, until it saturates for $R a \geq 10^{8}$ ⁠. The increase is likely due to the greater amount of details of high $R a$ convection flow vs the low $R a$ convection, however, the very small scale features that characterized the highly turbulent convection above $R a = 10^{8}$ are most likely smoothed by the bilinear algorithm. On the contrary, for fixed $R a$ ⁠, Fig. 6(a), performance is very uniform for varying $P r$ ⁠, with a mild improvement for high $P r$ ⁠.

Our work does not investigate which features are extracted by the trained neural network to infer $R a$ and $P r$ ⁠. Fluid-dynamic theory predicts that the boundary layer thickness decreases with increasing $R a$ ⁠, for a fixed $P r$ ⁠. We estimate that for the extreme case of $P r = 1$ and $R a = 10^{9}$ the boundary layer at the $1024 \times 1024$ resolution is about $10$ pixels thick so in its mapping to a $224 \times 224$ image the boundary layer will be only about 2 pixels thick. However, we not have information whether at very high $R a$ the network infers $R a$ and $P r$ from the thickness of the boundary layers. Since we do not observe a worsening of the performance for high $R a$ ⁠, we expect that these values emerge from the patterns of the turbulence in the rest of the modeling domain. Our work does not put a final word on this regard, and future work will have reverse the network to find the active features.

Of the other three cases in which we cropped part of the domain, GoogleNet systematically performs better for the center cropped field dataset. albeit always much worse than the bilinear mapping. We attribute this relative success of the Network learning to interpret this small portion, 20 times smaller than the entire domain, to the fact that features in the same location, at equal distance from the top and bottom boundary, and from the sides for the non-periodic case, are much more similar compared to a random location in the domain.

We challenged the network by testing its ability to detect $P r$ and $R a$ from a random cropped subset of the convection domain. Unsurprisingly for the same dataset size of the bilinear and center cropped case, performance dropped drastically to below 50% for fixed $R a$ and below 30% for fixed $P r$ ⁠. However, we observed a substantial improvement by using image augmentation. By selecting 20 crops for each image, in turn by increasing twenty-folds the training dataset, performance systematically improved by 20% for both cases. We concluded from this last test, that even for the extremely challenging task of extracting fundamental physical properties from a small patch of a large-scale fluid-dynamic experiment, satisfactory performances can be achieved if a sufficiently augmented dataset is used.

The network performance is superior for $d t = 10^{3}$ compared to $d t = 10^{4}$ ⁠. A more detailed analysis of the role of the timesteps and the boundary conditions shows that the first has a substantial effect on the final result (Fig. 7), while boundary conditions have not (Fig. 8). Systematically, choosing a smaller time-step interval improves the performance of $\sim 10 %$ for most scenarios. A detailed analysis of the images has shown that this difference is likely due to greater variety among the $d t = 10^{3}$ time-step because most images are collected before reaching a steady state, for low Ra numbers. This is confirmed also by the matching between $d t = 10^{3}$ and $d t = 10^{4}$ results for cropped $P r = 1$ (Fig. 7) and low Ra numbers (Fig. 8). Overall, this result is encouraging for future application to natural systems where a steady state is not expected to be reached. In turn, training the network with a denser dataset will be preferred for generalizing our approach to datasets of different origins.

FIG. 7.

View large Download slide

Average classification accuracy for fixed $P r$ and $R a$ numbers (solid lines) represent the average of the values for two tested cases with different time sampling rate: $d t = 1000$ and $d t = 10 000$ (dashed lines). (a) Average classification accuracy of obtaining $R a$ number at fixed $P r$ number, (b) Average classification accuracy of obtaining $P r$ number at fixed $R a$ number.

FIG. 8.

View large Download slide

Average classification accuracy for fixed $P r$ and $R a$ numbers (solid lines) represent the average of the values for either fixed or periodic boundary conditions (dashed lines). (a) Average classification accuracy of obtaining $R a$ number at fixed $P r$ number, (b) Average classification accuracy of obtaining $P r$ number at fixed $R a$ number.

On the contrary, we observed little effect by changing the choice of BC, however even if in small measure, training sets with fixed boundary conditions exhibit superior performance. Overall, we believe that this is due to the limited freedom that the flow has with fixed BCs, whereas with periodic boundary conditions, the fluid dynamics can be more complex. Overall this effect will likely vanish for a wider domain, where the effect of lateral boundary conditions will play a minor role.

These models are just a small sample of the opportunities offered by machine learning to solve the inverse problem of extracting fluid-dynamic properties from numerical (and potentially experimental) results. Future tests will be necessary to test consistency for the non-linear flow and temperature-dependent parameters such as fluid viscosity, which have wide applications in Geophysics, Planetary Science, and Engineering. Further, our results suggest that generative machine learning will likely succeed at offering even better performance than the one that we obtained since our approach is limited by a predefined number of categories.

ACKNOWLEDGMENTS

This work was supported by NASA through NASA Emerging World Program Grant (No. 20-EW20_2-0026) as well as Louisiana Optical Network Infrastructure (LONI).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Mohammad Ali Boroumand: Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Gabriele Morra: Conceptualization (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Peter Mora: Software (equal); Supervision (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

APPENDIX A: REFINED ANALYSIS OF THE NETWORK PERFORMANCE

Figure 7 provides more information about the effect of choosing different time steps on the classification results. Every point on the dashed line graphs is the average classification accuracy value for fixed and periodic boundary conditions at the chosen time step. To present the same analysis for different boundary conditions, the dashed line in Fig. 8 exhibits average classification results for different time steps at selected boundary conditions. Both analyses have been done for the situations where the dataset is comprised of images with fixed $P r$ number and changing $R a$ and vice versa.

APPENDIX B: STEADY-STATE REGIMES OF THE CONVECTION

Figure 9 has illustrated the effect of changing fundamental parameters on the starting point for the stability of convection. For a lower

P r

number and a higher

R a

number, the convection reaches the steady state at an earlier time. This can be seen from the time that

V_{r m s}

and Nusselt number,

N u

⁠, are constant or exhibit less fluctuation. In this study,

N u

was calculated based on the following formula:⁴⁰

N u = - L_{z} \frac{\int_{0}^{L_{x}} \partial_{z} T (x, z = L_{z}) d x}{\int_{0}^{L_{x}} T (x, z = 0) d x}

(B1)

where

T

is temperature and

L_{z}

and

L_{x}

are height and width of the convection field. We conducted experiments in the rectangular domain with a cold upper lid with temperature

T_{u}

and a hot lower base with temperature

T_{l}

⁠. In our simulations, we investigated convection with Rayleigh numbers far from critical Rayleigh number

R a_{c} = 1708

(a high Rayleigh number corresponding to vigorous convection). When the Rayleigh number is similar to but greater than

R a_{c}

⁠, broad Rayleigh-Bernard convection cells will tend to form. For high Rayleigh numbers with

R a ≫ R a_{c}

⁠, convection will become increasingly vigorous.²¹

FIG. 9.

View large Download slide

Root mean square of the velocity and Nusselt number through time for two end-member fluid-dynamic solutions with fixed boundary condition. (a) $V_{r m s}$ and $N u$ vs time for $1024 \times 1024$ convection field with $R a = 10^{5}$ and $P r = 128$ ⁠, (b) $V_{r m s}$ and $N u$ vs time for $1024 \times 1024$ convection field with $R a = 10^{9}$ and $P r = 1$ ⁠.

Results in 9 show that while for high $R a$ the majority of the images used for this work are in a dynamic steady state, half or more of the images for the low $R a$ are taken from the transient time while it reaches steady state. Our predicting approach shows robustness by combining transient and steady-state images.

REFERENCES

1.

Y.

Çengel

and

A.

Ghajar

,

Heat and Mass Transfer: Fundamentals & Applications

(

McGraw-Hill Education

,

2019

).

Google Scholar

2.

S.

Atkins

,

A. P.

Valentine

,

P. J.

Tackley

, and

J.

Trampert

, “

Using pattern recognition to infer parameters governing mantle convection

,”

Phys. Earth Planetary Interiors

257

,

171

–

186

(

2016

).

https://doi.org/10.1016/j.pepi.2016.05.016

Google Scholar

3.

S.

Agarwal

,

N.

Tosi

,

D.

Breuer

,

S.

Padovan

,

P.

Kessel

, and

G.

Montavon

, “

A machine-learning-based surrogate model of mars’ thermal evolution

,”

Geophys. J. Int.

222

,

1656

–

1670

(

2020

).

https://doi.org/10.1093/gji/ggaa234

Google Scholar

4.

S.

Agarwal

,

N.

Tosi

,

P.

Kessel

,

S.

Padovan

,

D.

Breuer

, and

G.

Montavon

, “

Toward constraining mars’ thermal evolution using machine learning

,”

Earth Space Sci.

8

,

e2020EA001484

(

2021

).

https://doi.org/10.1029/2020EA001484

Google Scholar

5.

M.

Shahnas

,

D.

Yuen

, and

R.

Pysklywec

, “

Inverse problems in geodynamics using machine learning algorithms

,”

J. Geophys. Res.: Solid Earth

123

,

296

–

310

, https://doi.org/10.1002/2017JB014846 (

2018

).

https://doi.org/10.1002/2017JB014846

Google Scholar

6.

S. L.

Brunton

,

B. R.

Noack

, and

P.

Koumoutsakos

, “

Machine learning for fluid mechanics

,”

Annu. Rev. Fluid Mech.

52

,

477

–

508

(

2020

).

https://doi.org/10.1146/annurev-fluid-010719-060214

Google Scholar

7.

L.

Perez

and

J.

Wang

. “

The effectiveness of data augmentation in image classification using deep learning

,” arXiv preprint arXiv:1712.04621 (2017)

8.

U.

Frisch

,

B.

Hasslacher

, and

Y.

Pomeau

, “

Lattice-gas automata for the Navier–Stokes equation

,”

Phys. Rev. Lett.

56

,

1505

(

1986

).

https://doi.org/10.1103/PhysRevLett.56.1505

Google Scholar

9.

S.

Chen

and

G. D.

Doolen

, “

Lattice Boltzmann method for fluid flows

,”

Annu. Rev. Fluid Mech.

30

,

329

–

364

(

1998

).

https://doi.org/10.1146/annurev.fluid.30.1.329

Google Scholar

10.

P. L.

Bhatnagar

,

E. P.

Gross

, and

M.

Krook

, “

A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems

,”

Phys. Rev.

94

,

511

(

1954

).

https://doi.org/10.1103/PhysRev.94.511

Google Scholar

11.

S.

Succi

and

S.

Succi

,

The Lattice Boltzmann Equation: For Complex States of Flowing Matter

(

Oxford University Press

,

2018

).

Google Scholar

12.

P.

Mora

and

D. A.

Yuen

, “

Simulation of plume dynamics by the Lattice Boltzmann method

,”

Geophys. J. Int.

210

,

1932

–

1937

(

2017

).

https://doi.org/10.1093/gji/ggx279

Google Scholar

13.

P.

Mora

and

D. A.

Yuen

, “

Simulation of regimes of convection and plume dynamics by the thermal Lattice Boltzmann method

,”

Phys. Earth Planetary Interiors

275

,

69

–

79

(

2018

).

https://doi.org/10.1016/j.pepi.2018.01.003

Google Scholar

14.

P.

Mora

and

D.

Yuen

, “

Comparison of convection for Reynolds and Arrhenius temperature dependent viscosities

,”

Fluid Mech. Res. Int.

2

,

99

–

107

(

2018

).

https://doi.org/10.15406/fmrij.2018.02.00025

Google Scholar

15.

P.

Mora

,

G.

Morra

, and

D. A.

Yuen

, “

Models of plate tectonics with the Lattice Boltzmann method

,”

Artif. Intellig. Geosci.

4

,

47

–

58

(

2023

).

https://doi.org/10.1016/j.aiig.2023.03.002

Google Scholar

16.

P. J.

Dellar

, “

Lattice kinetic schemes for magnetohydrodynamics

,”

J. Comput. Phys.

179

,

95

–

126

(

2002

).

https://doi.org/10.1006/jcph.2002.7044

Google Scholar

17.

H.

Huang

,

M.

Sukop

, and

X.

Lu

,

Multiphase Lattice Boltzmann Methods: Theory and Application

(

John Wiley & Sons

,

2015

).

Google Scholar

18.

P.

Mora

,

G.

Morra

,

D. A.

Yuen

, and

R.

Juanes

, “

Influence of wetting on viscous fingering via 2D Lattice Boltzmann simulations

,”

Transport Porous Media

138

,

511

–

538

(

2021

).

https://doi.org/10.1007/s11242-021-01629-8

Google Scholar

19.

K. V.

Sharma

,

R.

Straka

, and

F. W.

Tavares

, “

Current status of Lattice Boltzmann methods applied to aerodynamic, aeroacoustic, and thermal flows

,”

Prog. Aeronaut. Sci.

115

,

100616

(

2020

).

https://doi.org/10.1016/j.paerosci.2020.100616

Google Scholar

20.

T.

Krüger

,

H.

Kusumaatmaja

,

A.

Kuzmin

,

O.

Shardt

,

G.

Silva

, and

E. M.

Viggen

, “

The Lattice Boltzmann method

,”

Springer International Publishing

10

,

4

–

15

(

2017

).

https://doi.org/10.1007/978-3-319-44649-3

Google Scholar

21.

D.

Turcotte

and

G.

Schubert

,

Geodynamics

(

Cambridge University Press

,

2014

).

Google Scholar

22.

P.

Constantin

and

C. R.

Doering

, “

Infinite Prandtl number convection

,”

J. Stat. Phys.

94

,

159

–

172

(

1999

).

https://doi.org/10.1023/A:1004511312885

Google Scholar

23.

C.

Szegedy

,

W.

Liu

,

Y.

Jia

,

P.

Sermanet

,

S.

Reed

,

D.

Anguelov

,

D.

Erhan

,

V.

Vanhoucke

, and

A.

Rabinovich

, “Going deeper with convolutions,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2015), pp. 1–9.

24.

Y.

LeCun

,

B.

Boser

,

J. S.

Denker

,

D.

Henderson

,

R. E.

Howard

,

W.

Hubbard

, and

L. D.

Jackel

, “

Backpropagation applied to handwritten zip code recognition

,”

Neural Comput.

1

,

541

–

551

(

1989

).

https://doi.org/10.1162/neco.1989.1.4.541

Google Scholar

25.

R.

Girshick

,

J.

Donahue

,

T.

Darrell

, and

J.

Malik

, “Rich feature hierarchies for accurate object detection and semantic segmentation,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2014), pp. 580–587.

26.

M.

Lin

,

Q.

Chen

, and

S.

Yan

, “Network in network,” arXiv:1312.4400 (2013).

27.

N. C.

Thompson

,

K.

Greenewald

,

K.

Lee

, and

G. F.

Manso

, “The computational limits of deep learning,” arXiv:2007.05558 (2020).

28.

G.

MacDonald

,

A.

Godbout

,

B.

Gillcash

, and

S.

Cairns

, “Volume-preserving neural networks,” in 2021 International Joint Conference on Neural Networks (IJCNN) (IEEE, 2021), pp. 1–9.

29.

Y.

LeCun

,

L.

Bottou

,

Y.

Bengio

, and

P.

Haffner

, “

Gradient-based learning applied to document recognition

,”

Proc. IEEE

86

,

2278

–

2324

(

1998

).

https://doi.org/10.1109/5.726791

Google Scholar

30.

S.

Arora

,

A.

Bhaskara

,

R.

Ge

, and

T.

Ma

, “Provable bounds for learning some deep representations,” in International Conference on Machine Learning (PMLR, 2014), pp. 584–592.

31.

F.

Song

and

J.

Dongarra

, “Scaling up matrix computations on shared-memory many core systems with 1000 CPU cores,” in

Proceedings of the 28th ACM International Conference on Supercomputing

(Association for Computing Machinery, New York, 2014), pp. 333–342.

https://doi.org/10.1145/2597652.2597670

32.

A.

Krizhevsky

,

I.

Sutskever

, and

G. E.

Hinton

, “

ImageNet classification with deep convolutional neural networks

,”

Commun. ACM

60(6), 84–90 (2017).

https://doi.org/10.1145/30

33.

F.

Zhuang

,

Z.

Qi

,

K.

Duan

,

D.

Xi

,

Y.

Zhu

,

H.

Zhu

,

H.

Xiong

, and

Q.

He

, “

A comprehensive survey on transfer learning

,”

Proc. IEEE

109

,

43

–

76

(

2020

).

https://doi.org/10.1109/JPROC.2020.3004555

Google Scholar

34.

O.

Russakovsky

,

J.

Deng

,

H.

Su

,

J.

Krause

,

S.

Satheesh

,

S.

Ma

,

Z.

Huang

,

A.

Karpathy

,

A.

Khosla

,

M.

Bernstein

,

A. C.

Berg

, and

L.

Fei-Fei

, “

ImageNet large scale visual recognition challenge

,”

International Journal of Computer Vision (IJCV)

115

,

211

–

252

(

2015

).

https://doi.org/10.1007/s11263-015-0816-y

Google Scholar

35.

K.

Yang

,

K.

Qinami

,

L.

Fei-Fei

,

J.

Deng

, and

O.

Russakovsky

, “Towards fairer datasets: Filtering and balancing the distribution of the people subtree in the imagenet hierarchy,” in

Conference on Fairness, Accountability, and Transparency

(Association for Computing Machinery, New York, 2020).

https://doi.org/10.1145/3351095.3375709

36.

K. T.

Gribbon

and

D. G.

Bailey

, “A novel approach to real-time bilinear interpolation,” in Proceedings of DELTA 2004. Second IEEE International Workshop on Electronic Design, Test and Applications (IEEE, 2004), pp. 126–131.

37.

C. N.

Vasconcelos

and

B. N.

Vasconcelos

, “Convolutional neural network committees for melanoma classification with classical and expert knowledge based image transforms data augmentation,” arXiv:1702.07025 (2017).

38.

Y.

Xu

,

R.

Jia

,

L.

Mou

,

G.

Li

,

Y.

Chen

,

Y.

Lu

, and

Z.

Jin

, “Improved relation classification by deep recurrent neural networks with data augmentation,” arXiv:1601.03651 (2016).

39.

S. C.

Wong

,

A.

Gatt

,

V.

Stamatescu

, and

M. D.

McDonnell

, “Understanding data augmentation for classification: When to warp?,” in 2016 International Conference on Digital Image Computing: Techniques and Applications (DICTA) (IEEE, 2016), pp. 1–6.

40.

B.

Blankenbach

,

F.

Busse

,

U.

Christensen

,

L.

Cserepes

,

D.

Gunkel

,

U.

Hansen

,

H.

Harder

,

G.

Jarvis

,

M.

Koch

,

G.

Marquart

et al., “

A benchmark comparison for mantle convection codes

,”

Geophys. J. Int.

98

,

23

–

38

(

1989

).

https://doi.org/10.1111/j.1365-246X.1989.tb05511.x

Google Scholar

Extracting fundamental parameters of 2D natural thermal convection using convolutional neural networks

I. INTRODUCTION

II. SIMULATION

A. The thermal Lattice Boltzmann method

1. Boussinesq buoyancy term

B. Simulation setup and results

III. CONVOLUTIONAL NEURAL NETWORKS

A. Algorithm

B. Dataset

1. Design

2. Image mapping

3. Image augmentation

C. Implementation

IV. RESULTS AND DISCUSSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: REFINED ANALYSIS OF THE NETWORK PERFORMANCE

APPENDIX B: STEADY-STATE REGIMES OF THE CONVECTION

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

Extracting fundamental parameters of 2D natural thermal convection using convolutional neural networks

I. INTRODUCTION

II. SIMULATION

A. The thermal Lattice Boltzmann method

1. Boussinesq buoyancy term

B. Simulation setup and results

III. CONVOLUTIONAL NEURAL NETWORKS

A. Algorithm

B. Dataset

1. Design

2. Image mapping

3. Image augmentation

C. Implementation

IV. RESULTS AND DISCUSSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: REFINED ANALYSIS OF THE NETWORK PERFORMANCE

APPENDIX B: STEADY-STATE REGIMES OF THE CONVECTION

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only