Prediction of Heat Transfer in Building Walls of Different Materials Using Neural Networks and Finite Difference Methods

Abstract

This study introduces a hybrid framework that integrates transient numerical simulations with artificial neural networks (ANNs) to analyze and predict heat transfer in building walls. The framework is applied to ten different material–insulation combinations. Using the Leapfrog–Hopscotch (LH) finite difference scheme, we evaluated dynamic heat transfer and identified optimal insulation thicknesses for buildings in the cold continental climate of Bukhara. An ANN model was trained and validated on a dataset generated from 410 simulated wall configurations. The model achieved high predictive accuracy, with a mean squared error below 0.005. The thickness of the outer material layer ranged from 20 cm to 35 cm, while the inner layer thickness varied from 1 cm to 3 cm. Among the materials analyzed, glass wool + steel and gypsum + brick demonstrated superior insulation performance by minimizing heat loss most effectively, with values as low as 361,234 J/m² and 4,983,441 J/m², respectively, at 35 cm wall thickness. These findings underscore the potential of combining ANN-based predictions with physics-based simulations to design energy-efficient building envelopes in cold climates.

1. Introduction

The building sector represents one of the largest energy-consuming sectors globally, accounting for approximately 40% of total energy consumption and 30% of greenhouse gas emissions worldwide [1]. This substantial environmental impact, coupled with rising energy costs, has intensified the focus on improving energy efficiency in buildings, particularly through enhanced thermal performance of building envelopes [2]. Recent studies employing advanced numerical and optimization techniques further confirm that improving envelope design and optimizing insulation strategies remain among the most effective pathways for reducing operational energy demand and carbon emissions in residential and commercial buildings [3].

In cold continental climates, such as Bukhara in Uzbekistan, characterized by extreme seasonal temperature variations and prolonged heating seasons, space heating constitutes the dominant component of building energy use [4]. Consequently, optimizing external walls, as the primary interface between indoor and outdoor environments, is critical for achieving sustainable development goals and reducing operational costs [5].

A growing body of research has emphasized the importance of selecting appropriate insulation materials and configurations to reduce heat loss and energy demand [6]. While steady-state methods provide simplified calculations using metrics such as U-value and R-value, they fail to capture transient effects caused by fluctuating ambient temperatures and solar radiation and differences in the thermal mass of the wall configurations [7]. Therefore, advanced numerical methods have become essential for accurately predicting dynamic heat transfer through multi-layer wall assemblies.

Studies by Kaynakli [8] and Bolattürk [9] have shown that climatic conditions, energy prices, and material properties play key roles in identifying the most cost-effective insulation levels for buildings. Determining the optimum insulation thickness has become a primary application of these simulations, as it significantly affects both energy consumption and life-cycle cost efficiency.

Cabeza et al. [10] further emphasized that low-carbon and bio-based materials not only improve thermal insulation performance but also reduce embodied energy, contributing to more sustainable construction practices. Borreguero et al. [11] demonstrated that a gypsum board containing 25% shape-stabilized PCM is more thermally insulating than standard gypsum, with sheet-PCM and pellets-PCM increasing average heat capacity by 41% and 33%, respectively. These findings are supported by Cosentino et al. [12], who reviewed natural bio-based insulation materials and their role in lowering building energy demand.

However, high-fidelity transient simulations, such as those using finite difference and finite element methods, remain computationally demanding for iterative design applications [13]. To address this challenge, hybrid numerical and optimization techniques have been explored, as reviewed by Machairas et al. [14], demonstrating the effectiveness of multi-objective optimization approaches in improving building energy performance.

In recent years, Artificial Neural Networks (ANNs) have emerged as a powerful alternative for modelling and predicting complex thermal behaviors in buildings. Kalogirou [15] demonstrated that ANNs can learn nonlinear interactions between parameters and provide rapid, accurate energy performance predictions. Building upon this, Tsanas and Xifara [16] showed that statistical and machine learning models can accurately estimate building energy performance using limited data.

Further research by Gossard et al. [13] confirmed that integrating ANNs with genetic algorithms allows for the multi-objective optimization of building envelopes, balancing thermal performance and cost. Similarly, Ascione et al. [17] introduced a new cost-optimal methodology for building energy analysis using multi-objective optimization frameworks.

Ozel [18] investigated the effect of wall orientation on optimum insulation thickness through a dynamic method, highlighting how climatic and directional factors influence energy efficiency. In a related study, Ozel [19] numerically investigated the thermal, economical, and environmental effects of insulated building walls under dynamic conditions in a cold climate, finding that the most economical orientation was a south-facing brick wall with 9.2 cm of XPS insulation, which reduced fuel consumption and emissions by 85%.

The combination of machine learning and numerical simulation continues to demonstrate great promise. Askar et al. [20,21] successfully applied ANN models alongside recent finite difference schemes to predict and optimize building thermal loads and roof configurations in various climates. In a complementary study, Chou and Bui [22] confirmed that hybrid ANN-based models can significantly reduce computational effort while maintaining high predictive accuracy.

Despite this progress, there remains a clear research gap in systematically applying hybrid numerical ANN frameworks to optimize insulation thickness and material combinations in cold continental climates such as Bukhara.

To bridge this gap, this study proposes a comprehensive hybrid framework that integrates the recently invented leapfrog–hopscotch (LH) finite difference scheme with artificial neural networks (ANNs) for analyzing and optimizing wall compositions. The framework uses the LH numerical scheme to perform a dynamic thermal analysis of walls composed of ten different material and insulation combinations. It then generates a comprehensive dataset from these simulations to train and validate an ANN model for rapid thermal performance prediction. The trained ANN is subsequently used as a solution model to calculate the heat transfer of any material used, based on observation and without the need for solving differential equations by numerical methods. This novel and user-friendly solution approach can be considered a comprehensive solution package for the materials used in this model.

2. Geometry and Material Properties

Figure 1 illustrates the various wall models investigated in this study, consisting of different combinations of construction materials and insulation layers. The left side of the wall faces the interior of the building, while the right side is exposed to outdoor weather conditions. In all models, the left boundary temperature is fixed as a constant reference (representing indoor conditions), whereas the right boundary temperature is varied systematically following real weather data. This orientation is clearly indicated in the figure with labels for “Interior” and “Exterior” sides.

The wall assemblies consist of two distinct material layers:

• Inner layer (adjacent to the interior environment): Includes materials such as gypsum, glass, EPS, XPS, cement, concrete, and steel.
• Outer layer (adjacent to the exterior environment): Includes materials such as brick, wood, stone, glass wool, and mineral wool.

To systematically evaluate the effect of layer thickness on thermal performance, the inner layer thickness varied across three values: 0.01 m, 0.02 m, and 0.03 m. For each inner layer thickness, the outer layer thickness was varied over a range of values to capture the thermal behavior across a spectrum of configurations.

For inner layer thicknesses of 0.01 m and 0.02 m, the outer layer thickness was varied over the following set of values: 0.20, 0.21, 0.22, 0.23, 0.24, 0.25, 0.26, 0.27, 0.28, 0.30, 0.31, 0.32, 0.33, 0.34, and 0.35 m (15 values in total).

For an inner layer thickness of 0.03 m, certain outer layer thicknesses were omitted due to computational constraints and convergence issues. Specifically, the outer layer thickness values of 0.26, 0.27, 0.32, and 0.33 m were excluded. Consequently, the outer layer thicknesses used for this case were: 0.20, 0.21, 0.22, 0.23, 0.24, 0.25, 0.28, 0.30, 0.31, 0.34, and 0.35 m (11 values in total).

Based on these selections, the 0.01 m and 0.02 m inner layer thicknesses each correspond to 15 outer layer thickness values, while the 0.03 m inner layer thickness corresponds to 11 values. Considering all ten material combinations (gypsum + brick, glass + brick, EPS + brick, XPS + brick, cement + brick, concrete + wood, concrete + stone, steel + glass wool, steel + mineral wool, and gypsum + wood), the total number of configurations is:

$$10\times 15+10\times 15+10\times 11=410$$

This systematic combination of layer thicknesses provides a robust basis for evaluating thermal resistance and heat-transfer behavior across all wall configurations.

In this investigation, the thermal and physical material properties listed in

Table 1. Properties of the materials used in the simulation [23,24].

Material	$\mathit{c}\left[{\mathbf{Jkg}}^{-1}{\mathbf{K}}^{-1}\right]$	$\mathit{\rho } \left[{\mathbf{kgm}}^{-3}\right]$	$\mathit{k}\left[{\mathbf{Wm}}^{-1}{\mathbf{K}}^{-1}\right]$
Brick	800	1600	0.74
Gypsum	977	805	0.29
Glass	750	2500	0.8
EPS	1300	30	0.04
XPS	1400	40	0.03
Cement	840	1300	0.6
Concrete	880	2200	1.5
Stone	800	2600	2.5
Wood	1500	500	0.13
Steel	490	7850	50
Glass wool	700	120	0.039
Mineral wool	900	100	0.035

—namely, thermal conductivity ($k$), density ($\rho$), and specific heat capacity (c)—are taken into account. Although these coefficients have a discontinuity at the material’s boundary, they are constants inside the material and do not vary with temperature, time, or space.

When a finish coat is necessary for mechanical protection in the case of a soft insulator such as glass wool, this is neglected from the point of view of the heat transfer simulation due to its low thickness.

Methodology

Building energy optimization is often computationally demanding. Artificial Neural Networks (ANNs) offer an effective compromise between predictive accuracy and computational efficiency. ANNs were first introduced by McCulloch and Pitts in 1943 [25]. In recent decades, they have gained widespread popularity in statistical and engineering applications due to their ability to model complex nonlinear relationships.

In this study, feature selection techniques were applied to simplify the predictive models. Various construction materials for external walls were selected and systematically analyzed. The ANN was implemented using a Multi-Layer Perceptron (MLP) architecture. Model parameters were optimized using three different approaches, and the resulting predictions were compared with empirical data to validate the theoretical framework and demonstrate its practical applicability.

The MLP provides structural flexibility, strong representational capability, and the ability to integrate multiple data sources [26]. It has become the most widely adopted neural network architecture, surpassing earlier models. MLPs are universal function approximators trained using the backpropagation algorithm and operate as feedforward neural networks [27]. They employ interconnected neurons to approximate the functional relationship between input and output variables.

The architecture of the MLP-ANN developed in this study is illustrated in Figure 2, which presents the input, hidden, and output layers, along with their interconnections [28]. The model was implemented using MATLAB R2022b, with the primary objective of identifying an algorithm that achieves an optimal balance between performance, reliability, and accuracy.

The implemented MLP consists of three main layers: one input layer, one hidden layer, and one output layer with a single neuron representing the predicted total heat loss (W/m²). The number of neurons in the hidden layer (12 neurons) was determined through a systematic parametric analysis to identify the optimal network configuration.

Specifically, several hidden layer sizes (5, 8, 10, 12, 15, and 20 neurons) were tested. For each configuration, the model performance was evaluated based on the mean squared error (MSE) on the validation dataset, the correlation coefficient (R) between predicted and target values, and the degree of overfitting, assessed by comparing training and validation errors.

The results indicated that a hidden layer with 12 neurons achieved the best overall performance, yielding the lowest validation MSE, a high correlation coefficient (R > 0.99), and minimal overfitting. Therefore, this configuration was selected as the optimal architecture for the neural network.

The database was constructed by randomly selecting twelve wall materials. As shown in Figure 2, the input parameters (Mᵢ) correspond to wall assembly types (M₁–M₁₀) detailed in

Table 2. Input materials and thickness of wall.

Input	Materials	Thickness of Wall
Model 1	Brick + gypsum	0.21–0.38
Model 2	Brick + glass	0.21–0.38
Model 3	Brick + eps	0.21–0.38
Model 4	Brick + exp	0.21–0.38
Model 5	Brick + cement	0.21–0.38
Model 6	Wood + concrete	0.21–0.38
Model 7	Stone + concrete	0.21–0.38
Model 8	Glass wool + steel	0.21–0.38
Model 9	Mineral wool + steel	0.21–0.38
Model 10	Wood + gypsum	0.21–0.38

, while the output variable (Y) represents the predicted thermal performance indicator.

To reduce the computational cost associated with repeated transient numerical simulations, a data-driven surrogate model based on an ANN was developed. The MLP architecture was selected due to its strong nonlinear approximation capability and its proven effectiveness in engineering prediction problems.

To prevent any single input parameter from dominating the learning process due to differences in scale, the input and output data were normalized. While normalization is commonly performed within the interval [0, 1], in this study, the data were scaled to the range of 0.1 to 0.9 using Equation (1).

This choice was made to avoid numerical issues associated with the saturation of sigmoid-type activation functions used in the neural network. When values approach the boundaries of 0 or 1, the activation function tends to saturate, resulting in very small gradients. This can slow down the learning process and lead to vanishing gradient problems during training [28].

By mapping the data to the interval [0.1, 0.9], the inputs remain within the sensitive (quasi-linear) region of the activation function, where gradients are sufficiently large and stable. This improves convergence speed, enhances numerical stability, and leads to more reliable model performance.

$$X=a+{\displaystyle \frac{\left(X_r-X_{min}\right)\left(b-a\right)}{X_{max}-X_{min}}},$$

(1)

where $X$, $X_r$, $X_{min}$, and $X_{max}$ are the minimum and maximum values of the parameter in the dataset, respectively. a and b are the lower and upper bounds of the new normalized range (0.1 and 0.9, respectively). The model was trained using an LM algorithm to iteratively adjust the connection weights, minimizing the error between the predicted and target outputs. The performance of the trained ANN was evaluated based on its accuracy in predicting the thermal performance of the unseen testing data.

The dataset used for training the ANN model consisted of 410 simulated wall configurations. The data were randomly divided into training (70%), validation (15%), and testing (15%) subsets using MATLAB’s R2022b Neural Network Fitting default “dividerand” function. This resulted in 287 samples for training, 62 for validation, and 61 for testing.

This random splitting ensures an unbiased representation of all material combinations and thickness variations across the three subsets. The performance of the model was evaluated using the mean squared error (MSE) and the correlation coefficient (R) for all datasets, confirming high accuracy and no evidence of overfitting.

3. The Equations Used in the Transient Simulations

A linear parabolic partial differential equation (PDE) describes the simplest form of Fourier-type heat transfer in a homogeneous material with an internal heat source as follows:

$${\displaystyle \frac{\partial u}{\partial t}}=\alpha {\nabla }^{2}u+q,$$

(2)

where $u=u\left(\overrightarrow{r},t\right)$ is the temperature, $\alpha =k/(c\rho )$ is the thermal diffusivity, and $c=c\left(\overrightarrow{r},t\right)$$\rho =\rho \left(\overrightarrow{r},t\right)$, $k=k\left(\overrightarrow{r},t\right)$ are the specific heat, the density and the heat conductivity of the material, respectively. Here, q is the heat generation or source of heat.

Newton’s law of cooling defines the convective transfer of heat between a moving fluid and a surface as $K\left(u_a-u\right)$. During this process, the fluid, which can be either a liquid or a gas, travels due to temperature and density fluctuations. The rate of heat transfer is affected by the fluid velocity, the temperature difference between the fluid and the surface, and the fluid’s physical properties, such as viscosity and thermal conductivity [29].

In this case, the ambient temperature (in Kelvin), $u_a$, is unaffected by the unknown variable u. As a result, the term $K{u}_a$ should be added to the formula q, representing the heat source. The Stefan–Boltzmann equation [7] expresses radiative heat loss from a surface as $-\sigma \cdot u^4$, where the proportionality constant σ is the product of the positive Stefan–Boltzmann constant and surface area. Similar to the previously stated $K{u}_a$ term, the incoming radiation, which includes direct sunlight, may be included into the source term q. The radiation, convection, and heat source terms are added to the heat conduction Equation (2) as follows:

$${\displaystyle \frac{\partial u}{\partial t}}=\alpha {\nabla }^{2}u+q-K\cdot u-\sigma \cdot u^4.$$

(3)

With the exception of the conduction component, Equation (3) includes local terms which are nonzero only at the wall boundaries. We use the most popular central difference approach for the $\alpha {\nabla }^{2}u$ term in the one-dimensional Equation (1):

$${\displaystyle \frac{{\partial }^2}{\partial x^2}}u(x_i)\approx {\displaystyle \frac{\frac{u(x_{i+1})-u(x_i)}{\mathrm{\Delta }x}+\frac{u(x_{i-1})-u(x_i)}{\mathrm{\Delta }x}}{\mathrm{\Delta }x}}={\displaystyle \frac{u_{i-1}-2u_i+u_{i+1}}{\mathrm{\Delta }{x}^2}},$$

(4)

where $i=1,\dots ,N$ if there are N nodes in total. The following is the one-dimensional discretized form of the heat transfer Equation (3) that we obtain using this approximation method:

$${\displaystyle \frac{d{u}_i}{dt}}=\alpha {\displaystyle \frac{u_{i-1}-2u_i+u_{i+1}}{\mathrm{\Delta }{x}^2}}+q-K{u}_i-\sigma {u_i}^4,$$

(5)

where K and σ are nonzero only for the boundary computational cells, as we will see below. We now show how the heat transfer equation has been discretized by assuming that the variables α, k, c, and ρ are functions of the space rather than a fixed value, reflecting the characteristics of materials. Instead of focusing on the $\alpha {\nabla }^{2}u$ component, we now examine the more general term

$${\displaystyle \frac{1}{c\left(x\right)\rho \left(x\right)}} {\displaystyle \frac{\partial }{\partial x}}\left(k\left(x\right){\displaystyle \frac{\partial u}{\partial x}}\right).$$

The equation for heat transfer in this case may be discretized as

$${\left.c(x_i)\rho (x_i){\displaystyle \frac{\partial u}{\partial t}}\right|}_{x_i}={\displaystyle \frac{1}{\mathrm{\Delta }x}}\left[k\left(x_i+{\displaystyle \frac{\mathrm{\Delta }x}{2}}\right){\displaystyle \frac{u(x_i+\mathrm{\Delta }x)-u(x_i)}{\mathrm{\Delta }x}}+k\left(x_i-{\displaystyle \frac{\mathrm{\Delta }x}{2}}\right){\displaystyle \frac{u(x_i-\mathrm{\Delta }x)-u(x_i)}{\mathrm{\Delta }x}}\right]$$

At the current moment, we are transitioning to a resistance–capacitance model comparable to lumped parallel thermal networks, but with a larger number of nodes than usual. Cells encircle the nodes, and the heat conductivity between node i and its right neighbor is $k_{i,i+1}$. The following is the discretized equation:

$${\displaystyle \frac{d{u}_i}{dt}}={\displaystyle \frac{1}{c_i{\rho }_i\mathrm{\Delta }x}}\left(k_{i,i+1}{\displaystyle \frac{u_{i+1}-u_i}{\mathrm{\Delta }x}}+k_{i-1,i}{\displaystyle \frac{u_{i-1}-u_i}{\mathrm{\Delta }x}}\right)+q-K{u}_i-\sigma {u_i}^4.$$

(6)

A cell’s length and cross-sectional area are expressed as $\mathrm{\Delta }x$ and S. The volume of the unit can be calculated as $V=S \mathrm{\Delta }x$, and heat capacity can be calculated from $C_i=c_i{\rho }_{i}V$. It is possible to estimate the heat transfer resistance between the two nodes using the formula $R_{i,i+1}\approx \mathrm{\Delta }x/\left(k_{i,i+1}S\right)$. Using the new numbers, the time derivative of each cell variable can be calculated as follows:

$${\displaystyle \frac{d{u}_i}{dt}}={\displaystyle \frac{u_{i-1}-u_i}{R_{i-1,i}{C}_i}}+{\displaystyle \frac{u_{i+1}-u_i}{R_{i+1,i}{C}_i}}+q-K{u}_i-\sigma {u_i}^4$$

(7)

3.1. The Leapfrog–Hopscotch Formulation and Structure

The recently developed leapfrog–hopscotch (LH) strategy [30], like all hopscotch approaches, requires a unique chessboard-like spatial structure. In essence, the mesh must be divided into two parts, known as odd and even nodes (or cells), with odd cells being nearest to even cells and vice versa. Its time pattern includes two half-time steps and a large number of full-time steps. The calculation begins with a half-sized time step for the odd nodes, using the original values. After that, the even and odd nodes execute full-time steps in strict alternating order until the final time step is reached; for odd nodes, this time step should be cut in half to reach the same final time point as the even nodes, as illustrated in Figure 3.

The leapfrog–hopscotch (LH) formulas are the following: The “zeroth” stage equation has the form:

$$u_i^{1/2}={\displaystyle \frac{u_i^0+r_i/2(u_{i-1}^0+u_{i+1}^0)-K\mathrm{\Delta }t{u}_i^0/4}{1+r_i+K\mathrm{\Delta }t/4+\delta \mathrm{\Delta }t{(u_i^0)}^3/2}} ,$$

where $r_i=\mathrm{\Delta }t/C_i{R}_{i,i-1}+\mathrm{\Delta }t/C_i{R}_{i,i+1}$. Then, a full-time step stage is calculated as

$$u_i^1={\displaystyle \frac{u_i^0+r_i(u_{i-1}^{1/2}+u_{i+1}^{1/2})-K\mathrm{\Delta }t{u}_i^0/2}{1+r_i+K\mathrm{\Delta }t/2+\delta \mathrm{\Delta }t{(u_i^0)}^3}},$$

and this equation is applied at the last stage as well, but with halved time step size.

3.2. Numerical Simulation of the Wall

A constant time step of Δt = 50 s was used for all transient simulations. We chose this time step size because, in our earlier work [24], when the LH approach was vali-dated, we demonstrated that they produce an error maximum of 0.01 °C, meeting the precision requirement. Additionally, it is demonstrated that this time step size is still significantly larger than the explicit Euler or Runge–Kutta CFL limit. The duration of the simulation was 121 days; hence, T = N_days × N_hours × N_second = 121 × 24 × 3600 = 10,454,400 s is the simulation’s overall time, and it represents the whole winter season (4 of November to 4 of March). In total, N_T = T/Δt = 209,088 time steps are used, and the analysis focuses on the south-facing section of the wall.

3.3. Initial and Boundary Conditions

For all wall situations and all boundaries in the simulations, zero Neumann boundary conditions are applied to prevent the passage of conductive heat. To do this, we set the matrix components describing heat conduction over the border to zero. Nonetheless, it has been assumed that the left and right sides of the wall experience convective and radiative heat transmission in the x direction, respectively, as shown in

Table 3. Convection, radiation, and heat source characteristics are located on both sides of the wall for all kinds of walls [31,32].

	${\mathit{h}}_{\mathit{c}}$ $\left[\frac{\mathbf{W}}{{\mathbf{m}}^2\mathbf{K}}\right]$	${\mathit{\sigma }}^{*}$ $\left[\frac{\mathbf{W}}{{\mathbf{m}}^2{\mathbf{K}}^4}\times 10^{-8}\right]$	ε
Left Elements (inside)	8	3.97	0.7
Right Elements (outside)	0.6–25	4.82	0.85

. The computational elements inside the body of the wall cannot lose or absorb heat by convection or radiation, and they lack a heat source; hence, the coefficients K, σ, and q are zero in Equation (3) for the interior cells.

We use the following formulae to determine the coefficient values for our equations:

$$K={\displaystyle \frac{h_c}{c\rho \mathrm{\Delta }x}}, \sigma ={\displaystyle \frac{{\sigma }^{\ast }}{c\rho \mathrm{\Delta }x}}, q={\displaystyle \frac{q^{\ast }}{c\rho \mathrm{\Delta }x}}+{\displaystyle \frac{h_c}{c\rho \cdot \mathrm{\Delta }x}}\cdot u_{\mathrm{a}},$$

(8)

Also, we assumed that the right and left components have the following heat sources:

$$\mathrm{For} \mathrm{the} \mathrm{inside} \mathrm{components}: q_l={\displaystyle \frac{1}{c\rho }}\times q_l^{\ast }+{\displaystyle \frac{h_{cl}}{c\rho \cdot \mathrm{\Delta }x}}\times 295 \mathrm{K}$$

$$\mathrm{For} \mathrm{the} \mathrm{outside} \mathrm{elements}: q_r(t)={\displaystyle \frac{1}{c\rho }}\times q_r^{\ast }\left(t\right)+{\displaystyle \frac{h_{cr}\left(t\right)}{c\rho \cdot \mathrm{\Delta }x}}\times u_r\left(t\right)$$

$$K\left(t\right)={\displaystyle \frac{h_{cr}\left(t\right)}{c\rho \mathrm{\Delta }x}}, \sigma ={\displaystyle \frac{{{\sigma }_r}^{\ast }}{c\rho \mathrm{\Delta }x}}, q\left(t\right)={\displaystyle \frac{{q_r}^{\ast }\left(t\right)}{c\rho \mathrm{\Delta }x}}+{\displaystyle \frac{h_{cr}\left(t\right)}{c\rho \cdot \mathrm{\Delta }x}}\cdot u_r\left(t\right)$$

where $q_l^{\ast }={\epsilon }_l{\sigma }_l{\left(295\right)}^4$  and  $q_r^{\ast }\left(t\right)={\alpha }_{\mathrm{sun}}{G}_{cr}\left(t\right)+{\alpha }_{\mathrm{Low}}{\epsilon }_{\mathrm{r}}{\sigma }_{\mathrm{r}}{\left[u_{\mathrm{r}}\left(t\right)\right]}^4$ [23].

The following formula is used to determine the convection heat transfer coeffi-cient for external components as a function of air velocity:

$$h_{cr}\left(t\right)=0.6+6.64\sqrt{v}\left(t\right),$$

(9)

$v\left(t\right)$, $u_r\left(t\right)$, $u_l$, $G_{cr}\left(t\right)$: According to the website data [33], every 50 s, we took measurements of the air velocity [m/s], outside air temperature [°C], and sun radiation [W/m²] in November, December, January, February and March. The interior ambient temperature is taken to $22 °\mathrm{C}\approx 295 \mathrm{K}$ on the left side.

${\alpha }_{\mathrm{sun}}$: The absorption of solar radiation by the surface, calculated to be 0.95.

${\alpha }_{\mathrm{Low}}$: The painted surface’s absorptivity to low-temperature thermal radiation is 0.93.

We approximated the initial temperature within the wall using the assumption that a stationary heat flow with constant flux had formed between the designated boundary values of the internal and external air temperatures before the simulation period, which began at 00:00 on 4 November. At this initial moment, the temperature on the outer, right surface of the wall was $u_{\mathrm{r},\mathrm{initial}}=282 \mathrm{K}$. For the starting condition, it produces a piecewise linear function of the x variable in the case of two layers:

$$\mathrm{For} \mathrm{the} \mathrm{part} \mathrm{of} \mathrm{gypsum} \mathrm{plaster}: u\left(x, t=0\right)=\left(u_{\mathrm{mid}}-u_l\right)\cdot x/L_g+u_l$$

(10)

For the part of brick:

$$u\left(x, t=0\right)=\left[\left(u_{\mathrm{r},\mathrm{initial}}-u_{\mathrm{mid}}\right)\cdot x/L_b\right]-\left[\left(u_{\mathrm{r},\mathrm{initial}}-u_{\mathrm{mid}}\right)\cdot L_g/L_b\right]+u_{\mathrm{mid}}$$

(11)

$$\begin{array}{c}\mathrm{where} u_{\mathrm{mid}}=u_l-\left(q_{\mathrm{flux}}{L}_g/k_g\right)\hfill \\ \mathrm{and} q_{\mathrm{flux}}=\left(u_l-u_{\mathrm{r},\mathrm{initial}}\right)/\left[\left(L_g/k_g\right)+\left(L_b/k_b\right)\right],\hfill \end{array}$$

where the subscripts g and b stand for the gypsum and brick layers, respectively,  $L_g$, for example, is the gypsum plaster’s thickness. In general, these subscripts correspond to the specific material used in each case, and they may vary across the ten different wall materials considered in this study.

4. Results for the Simulation Walls

In this study, twelve different wall materials were analyzed, resulting in 410 possible multi-layer wall configurations. The dataset derived from these combinations was used to train an ANN model based on an MLP architecture to predict the building envelope’s energy performance parameter.

4.1. Calculation of Heat Loss Through Walls

The total energy requirement Q in kWh units is calculated from Equation (12).

$$Q={\displaystyle \sum _{t=1}^{Time}\left(\mathrm{\Delta }t{Q}_{\mathrm{cell}}(t)\right)}$$

(12)

where $Q_{\mathrm{cell}}={q^{*}}_{\mathrm{cell}}\cdot S$ and $q^{*}$ is the current density of heat loss, and, using Fourier’s law, we can calculate it as ${q^{*}}_{\mathrm{cell}}={\displaystyle \frac{\left((u_1-u_3)k_{\mathrm{cond}}\right)}{2\mathrm{\Delta }x}}$, i.e., the heat flux close to the inner surface of the wall.

4.2. Verification and Validation

The target values used for training, validation, and testing of the neural network were obtained from the numerical heat transfer simulations described in Section 2. For each wall configuration (combination of inner and outer layer materials and thicknesses), the transient 1D heat conduction equation was solved using an explicit finite-volume method with a time step of $\mathrm{\Delta }t=50$ [s] over a simulation period of 121 days. The resulting heat loss values (W/m²), integrated over the simulation period, served as the target outputs for the neural network.

Figure 4 illustrates the network’s training behavior in terms of gradient, learning rate ($\mu$), and validation checks. Figure 5 compares the target values (obtained from numerical simulations) and the predicted values (obtained from the neural network) for the training, validation, and testing phases. The plots demonstrate that the data points closely align with the 45° reference line ($Y=T$), which represents perfect agreement between predicted and actual values. The correlation coefficient ($R$) equals 1 for all datasets, confirming an almost perfect linear relationship and indicating that the model accurately predicts the heat loss across all samples. This high level of agreement verifies the robustness and reliability of the developed neural network model for heat loss estimation.

The mean squared error (MSE) and regression coefficient (R) for the training, validation, and testing stages were evaluated to verify the performance of the constructed neural network in predicting homogenized attributes. Figure 6 presents the MSE variation over the training epochs. The MSE values are notably low, indicating a high level of prediction accuracy, as lower MSE values correspond to better model performance. The best validation performance was achieved at epoch 430, with an MSE of approximately ($2.79\cdot {10}^{-10}$). Additionally, the regression coefficients (R) were found to be close to one, signifying a strong correlation between the predicted and target values. These results confirm that the neural network effectively captures the underlying relationships in the dataset and accurately represents the homogenized properties.

Figure 7 illustrates the error histogram, which represents the distribution of prediction errors for the training, validation, and testing datasets across all 410 wall configurations. The histogram displays a tall and narrow peak centered near zero, indicating that most prediction errors are minimal. This pattern suggests that the model achieves high accuracy and generalizes effectively across different data splits. Moreover, the absence of noticeable bias or extended tails confirms the model’s reliability and consistency in estimating heat loss. Figure 8 demonstrates a strong positive correlation between insulation thickness and system performance. The close agreement between experimental and predicted values validates the model’s accuracy.

To further quantify the predictive uncertainty of the ANN model, an error analysis was performed based on the comparison between predicted and reference values, as illustrated in Figure 8.

The results show that the model achieves a high level of accuracy, with an average relative error of approximately 2.3% and a maximum relative error of 4.7%. The corresponding root mean square error (RMSE) is approximately 0.016 kW/m²/day. These values fall well within acceptable engineering limits, typically below 5–10%.

Additionally, the regression analysis presented in Figure 4, Figure 5, Figure 6 and Figure 7, with a correlation coefficient (R) close to 1 and a very low mean squared error (MSE), further confirms the robustness and low uncertainty of the proposed model.

Figure 9 presents a direct parity plot (ANN vs. LH), including a reference line for perfect agreement, demonstrating the strong consistency between the two approaches.

This addition provides a clearer visual validation of the ANN model and further confirms its accuracy and reliability.

Daily temperature variations in Bukhara during the study period are illustrated in Figure 10. The data, obtained from the Open-Meteo Historical Weather API [33], cover 121 days from early November 2024 to early March 2025, capturing the transition from autumn to winter.

Figure 11 presents the total heat loss calculated by the LH scheme (Leapfrog–Hopscotch) with a time step size of Δt = 50 s for two representative wall assemblies—steel + glass wool and concrete + stone—each with a thickness of 0.25 m, over the 121-day winter simulation period (4 November–4 March).

The results illustrate the dynamic response of each wall assembly to seasonal climate variations, as characterized by the monthly climate parameters summarized in

Table 4. Monthly Climate Parameters for Bukhara (4 November 2024–4 March 2025).

Month	Period	Avg. Temp (°C)	Min–Max Temp (°C)	Solar Radiation (W/m2)	Wind Speed (m/s)
November	4–30 November 2024	7.4	1.5 to 13.0	112.5	3.5
December	1–31 December 2024	3.1	−1.5 to 8.5	88.1	3.1
January	1–31 January 2025	2.0	−3.5 to 6.9	96.4	3.9
February	1–28 February 2025	5.0	−1.0 to10.5	140.0	4.1
March	1–4 March 2025	9.1	3.0 to 15.0	169.2	4.4

. The data reveal a consistent pattern between external temperature fluctuations and the observed heat loss. For instance, the highest heat loss values are recorded during January, which had the lowest average temperature of 2.0 °C, while lower heat loss values are observed during November and March, when average temperatures are higher (7.4 °C and 9.1 °C, respectively).

This pattern confirms that the wall’s thermal performance is strongly influenced by external climatic forcing.

As illustrated in Figure 12, the total heat flux exhibits substantial variation among the analyzed wall assemblies, each consisting of a 0.01 m internal insulation layer and a 0.30 m primary material layer. The evaluation was performed during the coldest period of the study, 15–16 December, when the minimum outdoor temperature reached −10 °C.

Under these conditions, the assembly incorporating a stone primary layer demonstrated the greatest thermal losses, whereas those employing mineral wool or glass wool as the primary layer displayed markedly improved thermal performance, characterized by minimal heat loss.

It should be noted that the reported ANN performance metrics (MSE and R) are evaluated using normalized output data, while all heat loss values presented in the results and tables are obtained after de-normalizing the ANN predictions to physical units (W/m²).

It should be noted that the results were computed for three discrete thickness values (0.20 m, 0.26 m, and 0.35 m). The continuous curves shown in the figure are obtained by connecting these data points to improve visualization and to highlight the overall trend. No additional simulations were performed for intermediate thickness values.

The three thickness values detailed in

Table 5. Cumulative heat loss (J/m²) for the three thicknesses.

Material	Thickness [m]
	0.20	0.26	0.35
Gypsum + Brick	7,304,327	6,156,467	4,983,441
Glass + Brick	7,683,709	6,423,156	5,156,308
EPS + Brick	4,924,894	4,377,518	3,751,439
XPS + Brick	4,373,783	3,937,325	3,423,808
Cement + Brick	7,608,736	6,370,799	5,122,631
Concrete + Wood	9,359,141	8,374,744	7,233,596
Concrete + Stone	11,848,689	10,308,546	8,628,443
Glass Wool + Steel	640,516	489,960	361,234
Mineral Wool + Steel	560,011	427,521	316,398
Gypsum + Wood	1,864,810	1,468,140	1,112,413

—0.20 m, 0.26 m, and 0.35 m—refer to the outer layer (right side) thickness, while the inner layer (left side) remained constant at 0.01 m. These values served as reference points representing the initial, average, and final states for each material assembly.

This tripartite approach was utilized to determine the variance in thermal resistance throughout the testing procedure. By calculating cumulative heat loss at these distinct phases, we quantified how structural changes in the material influenced the overall heat transfer rate.

The methodology, described in detail in Section 2, provides a comprehensive evaluation by capturing the system’s initial state (0.20 m thickness), mean behavior (0.26 m thickness), and final state (0.35 m thickness) over the 121-day winter simulation period (4 November–4 March). This tripartite approach leads to a robust estimation of total cumulative heat loss for each material assembly. The resulting cumulative heat loss values for all wall assemblies across the thicknesses examined are presented in Figure 13.

5. Conclusions

An ANN model was developed in this study to predict transient heat transfer through building walls in Bukhara, Uzbekistan, considering ten different material–insulation combinations. Training data were generated using the Leapfrog–Hopscotch finite difference method for 410 simulation cases, capturing the region’s large seasonal temperature variations. The optimized ANN achieved a mean squared error below 0.005 and a regression coefficient near 1, demonstrating high predictive accuracy and reliability as a surrogate for computationally intensive simulations.

The results in Table 5 and Figure 12 show that increasing wall thickness from 0.20 m to 0.35 m consistently reduces total cumulative heat for all wall configurations. For example, the cumulative heat for the Gypsum + Brick wall decreases from 7,304,327 J/m² at 0.20 m to 4,983,441 J/m² at 0.35 m. Similarly, insulation-based materials such as Glass Wool + Steel and Mineral Wool + Steel exhibit significantly lower heat losses (e.g., 640,516 and 560,011 J/m² at 0.20 m, respectively) compared to heavy materials like Concrete + Stone (11,848,689 J/m² at 0.20 m). This demonstrates that both increasing thickness and selecting low-conductivity insulation materials play a critical role in reducing heat transfer and improving the thermal performance of building envelopes.

This hybrid FDM–ANN framework offers a practical and efficient tool for rapidly estimating heat transfer in building envelopes, reducing computational effort while supporting informed design decisions. By combining physics-based simulations with data-driven modeling, it enables optimization of insulation thickness and material choice to enhance thermal performance. It should be noted that the present study is limited to one-dimensional wall configurations without considering thermal bridges or multiple wall orientations. Therefore, the applicability of the proposed ANN model to more complex geometrical configurations has not been investigated in this work. Extending the model to account for such complexities represents a potential direction for future research.

Future work could extend this methodology to multi-dimensional wall configurations, dynamic indoor conditions, and long-term energy performance analyses, with a crucial emphasis on incorporating material costs. This would further improve its applicability for designing practical and energy-efficient buildings in various climates.