Airborne sound insulation prediction of masonry walls using artificial neural networks

Abstract

In this paper, an alternative method for the calculation of masonry walls sound insulation is investigated. Thirty-four simple monolithic brick walls were examined. For the considered walls, the measurements made by accredited laboratories in compliance with ISO 10140 standard were collected. For each building element, the experimental measurements and all the information concerning the geometric and physical characteristics of all the components (material, dimensions, mass, density, Young’s modulus, Poisson’s module, hole content, etc.) were classified. Then, from the collection of measurements obtained in the standard laboratories and the division of them into homogeneous groups, a statistical sensitivity analysis to determine the most statistically significant parameter for the sound insulation of building walls was carried out. This analysis was a useful tool for selecting parameters to be used in forecast models and for calculating statistical effects. Finally, an alternative method based on the use of artificial neural networks (ANN) was proposed. This paper discusses the results obtained by applying the models to a specific kind of construction: the masonry walls. The results will show a good correlation of the values obtained on this first type of building construction. This encourages the extension of the method explained in this work to other types of walls.

Keywords

Sound Insulation artificial neural network building acoustic

Introduction

The airborne sound insulation of a wall depends on the properties of materials used (i.e. thickness, mass, Young’s modulus, etc.) and by the construction techniques. Several options are available for calculation of the sound insulation of single homogeneous wall.

In the international scientific literature, various models of sound insulation of building elements are presented and they are generally referred to ISO standard methods.^1–4 Mathematical models, theoretical or empirical, are generally function of one or more significant variables and are diversified according to the type of building element (masonry, brick, dry, etc.) under examination. These correlations are usually derived through multiple regression analysis starting form experimental measurements.

Applying these models to data measured according to ISO 10140 standard¹ and coming from different laboratories, notable gaps in the results are evident.

Starting from the description of sound field in coupled rooms and the structural wave field in the wall, applying a modal analysis approach it is possible to find an analytical solution. The following equations can be used^2–4:

R = - 10 \log (τ)

(1)

τ = {(\frac{2 ρ_{0} c_{0}}{2 π f m^{'}})}^{2} \cdot \frac{π f_{c} σ^{2}}{2 f η_{t o t}} f > f_{c}

τ = {(\frac{2 ρ_{0} c_{0}}{2 π f m^{'}})}^{2} \cdot \frac{π σ^{2}}{2 η_{t o t}} f \approx f_{c}

τ = {(\frac{2 ρ_{0} c_{0}}{2 π f m^{'}})}^{2} \cdot

\cdot [2 σ_{f} + \frac{{(l_{1} + l_{2})}^{2}}{l_{1}^{2} + l_{2}^{2}} \sqrt{\frac{f_{c}}{f}} \frac{σ^{2}}{η_{t o t}}] f < f_{c}

where $τ$ is the transmission factor, $m^{'}$ is the mass per unit area (kg/m²), $f$ is the frequency (Hz), $f_{c}$ is the critical frequency (Hz), $η_{t o t}$ is the total loss factor, $σ$ is the radiation factor for free bending waves, $σ_{f}$ is the radiation factor for forced transmission, $l_{1}$ , $l_{2}$ are the lengths (m) of the borders of the (rectangular) element.

Recently, a modified prediction model was published in the new version of the standard ISO 12354-Part 1.⁴ In the new ISO standard version, for frequencies below $f_{c}$ the transmission factor is calculated as:

\begin{array}{l} τ = {(\frac{2 ρ_{0} c_{0}}{2 π f m^{'}})}^{2} \cdot \\ \cdot [2 σ_{f} + (\frac{1 - f^{2}}{f_{c}^{2}}) + 2 \frac{π f_{c}}{4 f} \frac{σ^{2}}{η_{t o t}}] f < f_{c} \end{array}

(2)

Differently from the previous version of the standard, the coefficient $τ$ does not depend on the dimensions of the borders of the element ( $l_{1}$ , $l_{2}$ ).

The relations (1) and (2) introduce some physical quantities related to materials adopted in a wall and to the way in which it deforms when subjected to vibro-acoustic stress. The accuracy of the results is strictly related to the quality and quantity of the input data. It is essential to have, with good accuracy, all the necessary construction information such as density, Young’s modulus, axial velocity in the material, internal damping factor.

In the last 40 years of literature,^2,3,5–8 several corrective terms and proposals for modifying these relationships were proposed aiming to better represent different monolithic construction types. A comparison with measurement results gathered in different laboratories over the past years showed that the measured results lie in a range around the given lines from –4 till +8.⁴

Background

In the last years, Artificial Neural Network (ANN) models were used in different field application of building energy prediction (Chari and Christodoulou⁹ and Li et al.¹⁰). In literature, many authors applied the ANN to predict the energy behaviour of buildings^11–15 and the energy consumption^16–20 obtaining satisfactory results. Ayata et al.,²¹ Soleimani et al.²² and Qi et al.²³ in 2008 have used ANN in prediction of the temperature distribution in buildings and have achieved strong linear association between the predicted and calculated results.

Neural Network is one of the methods employed in various fields of chemical engineering to predict thermophysical properties such as thermal conductivity^24,25 and surface tension.^26,27 Among all the methods used often, Neural Network is one of the more accurate and it is employed as a benchmark for all the other methods.

In acoustic applications, extensive use of ANN was made in the field of audio engineering by Mohamed et al.,²⁸ Di Loreto et al.^29,30 and Sak et al.³¹. Using ANN, they performed discriminative fine-tuning using back propagation in phone recognition. In environmental acoustic Iannace et al. in 2019^32,33 applied ANN to sound emissions of wind farm, for automatic recognition of the position of wind turbines. The same authors in 2020 applied ANN to experiments of sound evaluation of noise by public car parks.³⁴ Ciaburro et al.^35,36 applied ANN in the study of sound absorption properties of different materials starting by their physical properties.

In building acoustic field, ANN were used for predicting sound insulation through multi-layered sandwich gypsum partition panels by Garg et al.;³⁷ the objective of the work was to develop an Artificial Neural Network (ANN) model to estimate the $R_{w}$ and STC value of sandwich gypsum constructions. The developed ANN model showed a prediction error of $\pm 3$ with a confidence level higher than $95 %$ but it was limited to a single index value calculation and it did not allow the frequency values of sound insulation prediction.

In a similar work, Buratti et al.³⁸ developed an ANN model to estimate the single index $R_{w}$ value of wooden windows starting by a limited number of windows parameters. Based on the results obtained by means of a preliminary training and test campaign of several ANN architectures, five main parameters were selected as network inputs: window typology, frame and shutters thickness, number of gaskets, $R_{w}$ of glazing; $R_{w}$ value of the window is the network output. Different ANN configurations were trained obtaining very good results, comparable to measurement uncertainty. Nevertheless, again, this work doesn’t allow the frequency values prediction of sound insulation.

Even if the structure of the human brain encourages this accurate computational method, some lacks dealing with ANN are also appearing: one of the most important features is that sometimes produces an unexplained behaviour that reduces the reliability of the network. Moreover, ANN cannot deal with uncertainties. For all these reasons, an accurate tuning of the architecture of the network is necessary. Neurons are the smaller structure of the Neural Network. They are mutually interconnected, and we call input neurons those neurons receiving the proper input of the network and output neurons those producing the final outputs.³⁹

A neural network⁴⁰ could be represented as a function that given some input data $x$ , return an output which is a calculating $y$ of a continuous output data:

y = f (x; θ)

(3)

where $θ$ is the set of all the parameters of the network, especially all the values of the weights matrices and bias vectors.

Given a set of examples made of input data and corresponding output data, the so-called training set, training an ANN means to find the best set of values for q so that some loss function, representing the prediction error of the network, is minimized. In particular we split the entire dataset in three subsets called: training, validation and test set. While the training set is used to establish the ANN features (weights, bias, etc.), the validation set is employed to withdraw overfitting and analyse the performances of different ANNs and choose the suitable model. Consequently, both training and validation data sets are ways of determining the architecture of the model. Lastly, a different data set, separate from the training and validation ones, called test set, and nevermore employed in the model representation, is utilized to estimate the predictive capabilities of the model after it has been chosen.

Dealing with continuous values like in our case, the Root Mean-Squared Error can be used as the loss function:

RMSE = \sqrt{\sum_{i = 1}^{N} {(\hat{y} - f (x_{i}, θ))}^{2}}

(4)

The value of the parameters is renewed through the so-called backpropagation algorithm. For a complete description see Rumelhart et al.⁴¹

Material and method

Collection of laboratory measurements

In order to apply the calculation methods ISO 12354 and the ANNs, a homogeneous family of walls, of acoustic performances known from laboratory measurements, was chosen. A 34 simple monolithic brick walls were taken into account.⁴² The physical properties and the construction techniques of each wall were known. A simple brick wall generally consists of brick or concrete blocks, with or without plaster on the sides and on the joints between the blocks. Furthermore, for the considered walls, the measurements made by accredited laboratories in compliance with ISO 10140 standard were available. For each wall, the experimental measurements and all the information concerning the geometric and physical characteristics of all the components (material, dimensions, mass, density, Young’s modulus, Poisson’s module, hole content, etc.) were collected and classified. In Table 1, a sample of data report is shown, in which are specified: Wall Identification, test laboratory, certificate number, wall description, wall image and sound insulation values measured in the accredited laboratory.⁴² The example shows a 12-cm thick wall, made with Poroton blocks 12 cm × 50 cm × 25 cm, tested by the laboratory of the University of Padua, Italy.

Table 1.

Laboratory report of wall ID no. 4, certification no. 182, University of Padua, Italy.

ID	004	Picture
Laboratory	University of Padua
Certification ID	No. 182—Date: 15/02/05
Description
Sound reduction index: $R_{w} = 36 d B$
Layers: single wall 12 cm thick, with Poroton 800 brick blocks with vertical holes
Dimensions: thickness 12 cm; length 50 cm; height 25 cm
Percentage of drilling: 50%
Young’s modulus: 5000 N/mm²
Mass/area: 106.6 kg/m²
Joints type: horizontal joint of continuous bedding mortar (average thickness 1 cm), continuous vertical mortar joint (average thickness 1 cm)
Plaster: not plastered

In Table 2 the frequency sound reduction of 34 masonry single walls classified with the method described in Table 1 is shown. In Table 3, the properties of the 34 walls used for calculations are reported.

Table 2.

Sound reduction of 34 masonry walls by laboratory certifications.

Wall ID	Frequency (Hz)
Wall ID	100	125	160	200	250	315	400	500	630	800	1000	1250	1600	2000	2500	3150	4000	5000
1	23.4	32.6	32.9	32.2	32.2	32.6	30.7	35.5	37.0	39.2	42.4	44.0	45.4	47.4	47.9	49.2	49.4	49.4
2	27.5	22.7	29.2	32.1	29.3	32.8	33.7	35.0	38.2	40.3	43.1	45.1	46.5	48.6	49.2	51.6	52.9	55.2
3	24.3	21.2	24.8	24.1	26.1	27.3	24.8	26.5	27.5	29.6	32.5	34.7	36.7	38.0	38.4	40.0	42.2	44.2
4	24.8	24.3	27.5	31.1	27.0	26.7	28.1	31.9	33.0	35.1	37.1	39.9	41.1	42.4	42.2	42.1	42.8	45.2
5	31.1	38.5	40.1	40.4	36.8	35.3	41.4	42.3	45.3	45.8	49.5	51.6	53.2	55.6	57.1	59.0	57.4	56.6
6	42.5	48.0	42.5	46.5	48.0	48.5	50.0	49.5	51.5	54.5	55.5	57.0	59.5	60.0	57.5	60.0	62.5	65.0
7	30.3	32.1	30.7	31.0	30.2	30.6	31.3	30.5	28.9	30.4	33.6	36.7	39.1	41.3	42.8	44.7	46.1	46.2
8	42.9	40.6	43.6	46.6	42.1	45.3	46.5	49.0	50.2	52.0	54.6	56.5	58.0	60.0	61.2	62.4	60.3	62.5
9	45.3	40.6	46.2	46.8	44.8	42.7	47.2	48.7	51.0	53.2	55.5	58.0	59.9	61.2	62.7	64.5	62.4	59.8
10	45.4	43.3	36.6	48.3	45.0	45.0	47.2	47.7	49.8	52.2	55.6	56.1	58.3	60.5	61.6	60.5	57.4	59.1
11	43.1	43.8	39.1	45.9	45.3	45.9	48.2	49.7	50.3	52.8	53.2	56.6	57.7	60.2	59.5	59.5	61.0	63.9
12	42.6	41.1	40.8	46.3	47.7	48.0	48.8	50.3	51.2	52.9	55.8	56.7	59.0	61.6	59.5	57.4	59.9	62.1
13	51.0	50.1	41.7	51.6	49.9	49.0	49.8	50.2	54.2	56.0	58.4	58.9	60.9	63.1	63.4	58.5	59.4	61.2
14	49.7	47.5	39.0	49.8	48.1	48.2	52.4	52.1	53.9	56.1	58.4	58.6	59.4	61.2	59.7	61.4	63.0	63.6
15	47.9	45.8	32.2	47.5	49.5	48.7	50.4	49.9	51.5	53.3	55.8	56.6	57.8	54.1	53.7	54.6	60.6	62.1
16	47.7	48.0	40.7	49.8	49.9	48.1	49.8	50.7	51.1	52.8	55.5	56.2	56.1	52.9	53.4	56.3	59.9	63.3
17	47.5	48.9	37.2	50.1	48.4	48.1	52.2	42.5	53.8	54.9	57.0	57.8	58.2	53.1	52.5	53.8	56.0	54.4
18	45.0	42.7	41.4	34.3	38.8	39.7	40.0	44.5	46.2	49.0	49.8	52.3	53.1	53.7	52.3	52.8	55.2	56.4
19	39.1	32.1	43.5	39.0	41.9	42.6	44.4	42.4	45.5	48.5	48.3	44.5	42.8	43.9	45.0	48.0	49.9	52.2
20	48.0	47.2	40.5	49.9	46.0	46.4	45.8	50.6	50.9	53.7	53.9	56.5	52.6	50.1	52.8	55.3	58.2	60.1
21	37.1	41.3	38.4	41.5	41.2	43.7	45.1	46.6	47.5	49.2	50.5	51.3	53.6	53.0	52.1	54.2	55.7	58.6
22	34.4	28.8	28.5	30.3	34.3	33.1	33.5	34.9	39.4	42.0	44.4	46.3	48.2	50.3	53.1	54.5	51.9	54.2
23	31.5	29.2	29.2	26.8	33.8	32.1	32.7	34.3	35.7	40.0	42.6	44.6	46.7	47.6	48.5	50.4	51.0	53.2
24	35.4	29.9	29.9	28.4	30.8	32.5	33.8	35.7	36.8	37.8	39.7	42.0	45.3	46.9	49.8	49.1	51.4	55.5
25	41.5	42.9	42.0	40.7	47.0	47.7	49.2	50.9	49.9	52.0	49.9	52.2	53.6	54.7	54.3	55.7	58.2	60.3
26	51.5	39.9	45.5	41.0	44.0	43.8	48.3	50.9	54.2	55.3	56.1	58.2	59.2	59.4	58.4	61.4	63.0	63.6
27	52.6	43.9	46.7	46.8	44.9	48.2	47.7	53.4	55.2	57.2	57.8	57.4	58.7	60.5	61.3	61.8	58.4	59.6
28	43.7	38.6	43.4	40.8	45.3	45.2	47.4	48.0	50.5	52.0	53.2	55.2	55.3	55.0	54.9	57.6	59.6	62.6
29	46.9	38.0	43.1	43.6	44.9	46.2	48.2	49.2	51.6	53.4	53.8	54.9	56.4	57.3	57.7	56.8	59.7	60.1
30	42.0	43.9	38.4	44.1	46.9	48.1	50.2	53.1	54.9	56.2	57.9	59.5	59.8	59.3	58.0	60.0	62.2	65.2
31	42.4	44.0	41.2	46.2	47.8	48.9	52.3	53.7	56.1	56.3	58.4	59.3	57.4	57.2	60.0	61.2	62.8	64.7
32	34.2	32.1	35.7	31.4	32.2	30.9	34.7	35.2	38.6	41.4	43.9	46.1	47.9	50.0	51.6	54.3	56.6	58.4
33	29.5	36.4	32.3	34.4	34.9	36.2	36.3	41.8	43.7	46.5	49.4	50.0	53.6	54.4	56.5	56.9	57.8	60.1
34	42.5	39.5	39.2	38.4	40.1	45.3	43.7	46.2	47.3	49.7	51.2	53.1	54.1	52.8	52.5	54.9	56.6	58.8

Table 3.

Properties of 34 walls used for calculation.

ID	Wall		Plaster				Block						Joints	Mortar
ID	Thickness (cm)	Mass per unit area (kg/m²)	Bides (none/1/2)	Density (kg/m³)	Type	Drilling % (%)	Young’s modulus (N/mm²)	Density (kg/m³)	t (cm)	l (cm)	h (cm)	Mass (kg)	Vertical (yes/no)	Horizontal (yes/no)
1	13.5	144.3	1	1800	Hollow bricks	50	5000	600	12	25	25	7.0	Y	Y
2	13.5	137.2	1	1800	Hollow bricks	50	5000	760	12	50	25	12.7	Y	Y
3	8	97.5	no	0	Hollow bricks	50	5000	760	8	50	25	8.9	Y	Y
4	12	106.6	no	0	Hollow bricks	50	5000	860	12	50	25	12.7	Y	Y
5	17	256.2	no	0	Hollow bricks	50	5000	1500	17	30	19	9.5	Y	Y
6	34	452.9	2	1800	Hollow bricks	⩽45	5000	860	30	24	19	12.8	Y	Y
7	8	57.9	no	0	Concrete (hollow)	No	1500	480	8	62.5	25	6.0	Y	Y
8	20	321	2	1800	Hollow bricks	⩽45	5000	800–860	17	30	19	9.5	Y	Y
9	21	363.3	2	1800	Hollow bricks	⩽50	3000	800–860	13	50	20	13.0	N	Y
10	28	370.7	2	1800	Hollow bricks	⩽45	5000	800–860	25	30	19	12.4	Y	Y
11	33	396.5	2	1800	Hollow bricks	⩽45	5000	860	30	25	19	12.8	Y	Y
12	33	465.5	2	1800	Hollow bricks	⩽45	5000	800–860	30	30	18	14.6	Y	Y
13	33	443.3	2	1800	Hollow bricks	⩽45	5000	800–860	30	25	17	11.1	Y	Y
14	33	509.9	2	1800	Hollow bricks	⩽40	5000	860	30	20	18.5	12.5	Y	Y
15	36	443.S	2	1800	Hollow bricks	⩽45	5000	800–860	33	25	18.5	13.3	Y	Y
16	41	512.5	2	1800	Hollow bricks	⩽45	5000	800–860	33	25	19	15.5	Y	Y
17	41	549.S	2	1800	Hollow bricks	⩽45	5000	800–860	38	25	19	15.0	Y	Y
18	21.5	256	1	1800	Hollow bricks	⩽45	5000	800–860	25	35	19	15.5	Y	Y
19	31.5	375.6	1	1800	Hollow bricks	50	3000	700–760	30	40	25	30.4	N	Y
20	33	393	2	1800	Hollow bricks	50	3000	870	30	25	25	20.1	Y	Y
21	22	246.3	2	1800	Hollow bricks	42.9	5000	900	20	25	19	8.1	Y	Y
22	11	33	2	1800	Hollow bricks	40	5000	900	8	50	25	10.5	N	Y
23	11	61.3	2	1800	Hollow bricks	65	5000	600	8	25	25	3.0	Y	Y
24	15	92	2	1800	Hollow bricks	65	5000	600	12	25	25	4.9	Y	Y
25	28	265	2	1800	Hollow bricks	⩽45	5000	1368	25	25	19	12.6	N	Y
26	33	320	2	1800	Hollow bricks	⩽45	5000	1221	30	25	19	14.0	N	Y
27	33	291	2	1800	Hollow bricks	⩽45	5000	1238	30	30	19	16.6	N	Y
28	28	280	2	1800	Expanded clay and concrete	22	3300	1000	25	25	20	10.0	Y	Y
29	23	280	2	1800	Expanded clay and concrete	25	3300	1400	20	25	20	10.5	Y	Y
30	28	350	2	1800	Expanded clay and concrete	22	3300	1200	25	25	20	12.5	Y	Y
31	33	396	2	1800	Expanded clay and concrete	20	3300	1200	30	25	20	15.5	Y	Y
32	11	115	2	1800	Expanded clay and concrete	0	3300	750	8	55	28	11.0	Y	Y
33	13	148	2	1800	Expanded clay and concrete	0	3300	800	10	55	28	14.5	N	Y
34	33	303	2	1800	Hollow bricks	44.8	5000	900	30	25	19	13.2	Y	Y

Table 4.

Sound reduction of 34 masonry walls using ANNs.

Wall ID	Frequency (Hz)
Wall ID	100	125	160	200	250	315	400	500	630	800	1000	1250	1600	2000	2500	3150	4000	5000
1	28.2	28.6	29.2	29.9	30.8	31.8	33.2	34.8	36.7	39.2	41.7	44.4	47.1	48.7	49.4	49.2	49.2	52.1
2	28.6	29.1	29.7	30.4	31.3	32.4	33.7	35.3	37.2	39.6	42.0	44.5	47.0	48.5	49.1	49.0	49.2	52.3
3	24.0	24.2	24.5	24.7	25.1	25.6	26.3	27.1	28.4	30.1	32.2	34.6	36.9	38.0	38.4	38.8	39.1	41.8
4	23.7	24.2	24.8	25.5	26.4	27.5	28.9	30.5	32.5	34.7	37.1	39.4	41.6	42.9	43.5	43.9	44.9	48.1
5	36.4	37.0	37.7	38.5	39.4	40.6	41.9	43.3	44.9	46.7	48.5	50.6	53.0	55.3	57.2	58.1	57.9	58.4
6	44.6	45.1	45.7	46.4	47.3	48.5	50.0	51.7	53.6	55.6	57.3	58.6	59.5	59.8	59.9	59.9	60.3	62.4
7	31.7	31.4	31.1	30.7	30.4	30.1	29.9	29.9	30.4	31.5	33.3	35.8	39.1	41.6	42.6	41.4	40.7	42.3
8	41.8	42.4	43.2	44.0	45.0	46.2	47.7	49.2	50.9	52.8	54.7	56.6	58.6	60.1	61.2	61.6	61.4	62.1
9	42.8	43.2	43.7	44.2	44.9	45.8	47.0	48.3	50.2	52.7	55.6	58.5	60.7	61.7	61.9	61.8	61.7	63.2
10	42.6	43.2	43.9	44.8	45.8	47.0	48.5	50.0	51.7	53.4	54.8	56.0	57.1	58.0	58.8	59.6	60.5	62.1
11	41.8	42.3	43.1	43.9	44.9	46.2	47.7	49.5	51.4	53.4	55.1	56.6	57.7	58.5	59.0	59.5	60.7	63.6
12	43.5	43.9	44.6	45.3	46.2	47.4	48.9	50.6	52.4	54.4	56.1	57.3	58.2	58.5	58.7	58.9	59.4	61.0
13	44.9	45.4	46.1	46.8	47.8	49.0	50.5	52.1	54.0	56.0	57.7	58.9	59.9	60.3	60.6	60.8	61.3	63.1
14	45.4	45.8	46.5	47.2	48.1	49.3	50.8	52.5	54.4	56.5	58.2	59.5	60.3	60.5	60.7	61.0	61.9	63.9
15	40.6	41.0	41.6	42.3	43.2	44.4	45.9	47.6	49.6	51.8	53.6	55.0	55.9	56.2	56.2	56.4	57.9	62.1
16	45.8	46.0	46.4	46.8	47.3	48.1	49.2	50.5	52.1	54.1	55.9	57.1	57.2	56.4	55.2	54.7	57.2	63.4
17	45.4	45.6	45.9	46.2	46.7	47.3	48.2	49.3	50.8	52.7	54.4	55.6	55.5	54.2	52.0	50.2	50.9	55.8
18	35.6	36.3	37.2	38.3	39.5	40.9	42.6	44.3	46.2	48.0	49.5	50.8	51.8	52.3	52.4	52.1	52.0	53.6
19	37.5	37.9	38.5	39.2	40.0	41.1	42.4	43.9	45.4	46.7	47.5	47.5	47.0	46.5	46.4	47.6	49.6	52.6
20	45.0	45.4	46.0	46.6	47.4	48.4	49.7	51.0	52.4	53.7	54.2	54.0	53.0	51.9	51.3	51.9	53.4	55.8
21	37.5	38.2	39.2	40.2	41.4	42.8	44.5	46.2	48.0	49.8	51.2	52.4	53.3	53.8	53.8	53.6	53.5	54.7
22	30.1	30.6	31.2	31.9	32.8	33.9	35.3	36.9	38.9	41.3	43.9	46.6	49.1	50.5	51.1	51.7	52.7	55.3
23	29.0	29.4	30.0	30.6	31.3	32.3	33.4	34.8	36.5	38.8	41.3	44.2	47.2	49.0	49.6	49.6	50.4	52.8
24	29.6	30.1	30.6	31.3	32.0	33.0	34.2	35.5	37.0	39.0	41.0	43.1	45.5	47.3	48.7	49.5	50.6	53.6
25	41.5	42.1	42.9	43.7	44.8	46.0	47.4	48.9	50.5	52.1	53.5	54.6	55.5	56.1	56.5	56.7	56.9	58.2
26	43.5	44.1	44.8	45.6	46.7	47.9	49.5	51.2	53.0	55.0	56.6	58.0	59.0	59.6	60.0	60.5	61.6	64.9
27	44.1	44.6	45.4	46.2	47.2	48.5	50.1	51.7	53.5	55.4	56.9	58.1	59.1	59.6	59.9	60.3	61.7	65.7
28	40.6	41.3	42.1	43.1	44.2	45.5	46.9	48.3	49.7	51.0	52.1	53.0	53.9	54.9	56.4	58.5	60.1	62.1
29	41.7	42.4	43.3	44.3	45.5	46.9	48.5	50.1	51.7	53.2	54.4	55.2	55.8	56.1	56.6	57.6	59.5	62.2
30	42.9	43.6	44.5	45.5	46.7	48.2	50.0	51.8	53.8	55.6	56.9	57.9	58.6	59.1	59.8	61.1	63.2	65.1
31	43.7	44.3	45.2	46.1	47.3	48.8	50.5	52.3	54.3	56.0	57.2	57.9	58.3	58.5	59.3	60.9	62.9	64.2
32	31.6	31.9	32.3	32.8	33.4	34.2	35.2	36.3	37.9	40.0	42.5	45.4	48.6	51.0	52.6	54.0	55.9	59.9
33	31.9	32.5	33.4	34.4	35.6	37.1	38.9	41.0	43.5	46.3	49.1	51.7	54.0	55.2	55.8	56.2	57.4	60.8
34	38.6	39.2	40.0	40.9	42.0	43.3	44.8	46.4	48.2	49.9	51.3	52.4	53.2	53.7	54.0	54.8	56.8	62.1

Prediction using standard calculation model

The standardized calculation model proposed by the ISO 12354-1 standard was applied to the 34 walls for which all the construction data were collected. The results obtained in the frequency range of 100Hz – 5000Hz were compared with the experimental measurements obtained from original laboratory certificates.

For each wall, the single $R_{w}$ index was also calculated from the values of the airborne sound insulation $R$ at different frequencies according to ISO 717-1,⁴³ and then compared with the sound insulation index shown in the corresponding laboratory test certificates. In Figure 1 it is shown, as example of the performed comparison, the wall ID 4.

Figure 1.

Airborne sound insulation of wall ID 4. Laboratory certificate (black-points) compared to ISO 12354-1 calculation (grey-squares).

The differences between the measured and the calculated value at fixed frequency bands for some walls are high, as for the wall in the example in Figure 1. The single index value $R_{w}$ differs for the 34 walls from 1 dB to 4 dB, in accordance with ISO 12354-1 which specifies a range from −4 dB to +8 dB.^4,7 All these differences are due to how the wall was built, from the type of brick, mortar, plaster, assembly, etc. Detailed information and values for each wall are reported in Serpilli et al.⁴².

To assess the validity of the calculation model in comparison with the experimental data, the Average Absolute Deviation (AAD%) was adopted as indicator, as also was done in a previous paper of the same authors.²⁴

The AAD% is given by the average waste of each wall comparing the calculated acoustic insulation ( $R_{c a l c}$ , dB) with the sound insulation measured in the accredited laboratory ( $R_{e x p}$ , dB):

AAD % = \frac{1}{N} \sum_{j = 0}^{n} (\frac{R_{e x p_{j}} - R_{c a l c_{j}}}{R_{e x p_{j}}}) \times 100 [%]

(5)

The AAD% was calculated first for each of the 34 walls under examination and then as the total average value of the 34 walls at every frequency. The results are shown in Table 5 for the investigated walls.

Table 5.

AAD% between ISO 12354-1 and laboratory measurements for the investigated walls.

F (Hz)	100	125	160	200	250	315	400	500	630	800	1000	1250	1600	2000	2500	3150	4000	5000
AAD%	21.40	17.55	14.66	14.64	9.50	7.82	7.77	7.41	6.52	6.61	6.74	7.54	9.38	11.07	12.09	10.93	12.39	10.87

The calculation model ISO 12354-1 is found to be more reliable in the mid-frequencies (AAD 6%–8%), while at low frequencies the deviations reach up to 15 dB (AAD% from 7% to 21.4%). At medium-high frequencies the deviations are from 6 dB to 8 dB (AAD% from 9% to 12.4%). In several cases, the most significant deviations occur at the critical frequency. For masonry walls the critical frequency is generally at medium-low frequencies. It depends on the construction details, materials and plaster joints, which clearly influence the laboratory measurement. This is not directly considered by the ISO standard formula for $f_{c}$ , that considers only the Young’s modulus E.

Prediction using artificial neural networks

Given the above results and the criticisms appearing adopting the traditional calculation methodology as shown in the previous paragraphs, a new calculation method based on the neural networks approach was used. We have therefore adopted the method of neural networks trying to have better results than the methods traditionally used in acoustics.

Starting with evaluated data from known or unknown origin, a neural network may be trained to perform simulation, estimation, classification, confirmation of hidden relationship in data and regression of data. This work aims to calculate the performance of the walls of know acoustic characteristic. For this reason, a Multilayer Perceptron network able to solve this particular problem was built. This class of networks is generally adopted for the majority of practical application.^41,44

In general, several activation functions could be employed: hyperbolic tangent function, the sigmoid function, the inverse tangent function, and the saturated linear function. The activation functions can be divided into two groups: linear activation function and non-linear activation functions. In this specific non- linear case, the sigmoid function was utilized as the activation function as follows:

σ (x) = \frac{1}{1 + e^{- x}}

(6)

This particular non-linear function allows²⁴ the model to create complex mappings between the network’s inputs and outputs, which are essential for learning and modelling complex data. Moreover, this function is differentiable, and its derivatives are quite simple to calculate, its smooth gradient, prevent abrupt jumps in data.

The network tuning consists of several steps: the choice of the amount of the layers to use, the number of the neurons to pick, the number of input parameters and how to divide the database to obtain better results.

One of the important issues in the ANN modelling is the overfitting. To overcome this problem, it is possible to split the database into the training, validation and test set. Therefore, before the trained network is accepted, it should be validated. The test set is a set of data that is independent of the training data. If a model fitted to the training set also fits the test set well, minimal overfitting has taken place. In this particular case, after some trials, it was decided to split the database into three parts. The training set contains 65% of the whole dataset which were used for training. The validation set contains the 25% of the points used to perform the validation test of the previous calculations. The test set the residual 10% which were used for the test, carried out to investigate the prediction capability of the network configuration. In all samples, data were randomly extracted from the database.

In this paper, considering that one hidden layer is sufficient for approximating any continuous function,⁴⁵ the architecture presented only one hidden layer and we tried to reduce the number of neurons to reduce the complexity of the network.

The first step applied was the statistical analysis of sensitivity starting from the physical characteristics of the building elements and the laboratory measurements, aimed at identifying the parameters, both single and combined, which most influence the value of the soundproofing power.

For this purpose, empirical relations were used through the Mode Frontier commercial software,⁴⁶ used for multi-objective and multi-disciplinary design.

To detect the most relevant parameters a factor analysis was performed,⁴⁷ that is an exploratory data analysis, which investigates the effects of multiple factors simultaneously. Then the result obtained from this preliminary analysis must be confirmed by the calculation with neural network.

Factor analysis is used for understanding the underlying conceptual structure of an instrument and relationships among variables, as well as for refining instruments. It involves analysing relationships among items of an instrument or large numbers of variables in order to obtain a smaller number of variables or factors. In this way, having a smaller number of variables facilitates theory development and testing. Thus, the factor analysis offers not only the possibility of gaining a clear view of the data, but also the possibility of using the output in subsequent analyses.

Let $x$ be a vector of independent variables with $x_{i} \in [x_{i}^{l b}, x_{i}^{u b}], i = 1, \dots, N$ and $y$ be the dependent variable. It is defined

\begin{array}{l} D_{j}^{+} = {x \in D | x_{j} > \frac{x_{j}^{u b} - x_{j}^{l b}}{2}} \\ D_{j}^{-} = {x \in D | x_{j} < \frac{x_{j}^{u b} - x_{j}^{l b}}{2}} \end{array}

(7)

then the effect of $x_{j}$ is:

Eff (x_{j}) = \frac{\sum_{x \in D_{j}^{+}} y (x)}{card (D_{j}^{+})} - \frac{\sum_{x \in D_{j}^{-}} y (x)}{card (D_{j}^{-})}

(8)

where $card (D_{j}^{+})$ denotes the cardinality of the set $A_{j}$ .

For all the experimental data, the factor analysis was performed, so as to identify which of the parameters were more likely to influence the output. The results are reported in Table 6.

Table 6.

Factor analysis results.

Factors	Abs. effect	Abs. coefficient
Mass per unit area	12.478	0.063
Sound insulation index	12.478	0.851
Frequency	12.061	0.001
Block thickness	11.933	0.405
Wall thickness	11.933	0.356
Young’s modulus	11.270	0.011
Bloch height	10.720	10.897
Bloch width	10.631	0.625
Frequency × Young’s modulus	10.135	0.000
Block thickness × frequency	9.381	0.000
Frequency × wall thickness	9.381	0.000
Block density	9.220	0.003
Frequency × block density	9.102	0.000

Afterwards, it has been performed the Student Analysis,^6,47 to acquire the parameters that most affect the output parameter in order to reduce the distance between the calculated and the experimental data (Figure 2).

Figure 2.

Results of the student analysis and of the factor relevance data.

In this kind of graph, a user can evaluate relationships between output and input values. The height of the bars is called Effect Size and gives the strength of the connection between the output and the input values. An Effect Size greater than zero states a direct relationship with the input variable, whereas an amount less than zero indicates an inverse relationship.

Table 7 shows the results obtained from the Student Analysis, that in addition to providing a numerical classification of the effects of the various parameters, identifies which mostly match to the physical characteristics of the building elements. The result shows the relationship between the most relevant and least relevant factors and the output, therefore between the physical characteristics of the mono-layer walls and the sound insulation.

Table 7.

Student analysis results.

Factor	Effect size
Mass per unit area	12.478
Sound insulation index	12.477
Frequency	12.061
Bloch thickness	11.933
Wall thickness	11.933
Young’s modulus	−11.269
Bloch height	−10.720
Bloch width	−10.630
Bloch density	9.2196
Plaster density	8.773
Bloch drilling %	−8.330
Wall density	4.919

The choice of the parameters in the neural network tuning is one of the most crucial aspect. In the past literature, authors generally tried different network configurations. Here, it was decided to combine the statistical analysis with the literature review. The factors that most influence the estimation of sound insulation are: the area-related mass of the building element, the frequency, the thickness of the element, the Young’s modulus, the length and height of the brick block. This analysis was also confirmed by the fact that the same parameters have been adopted by some other authors as Josse and Lamure⁴⁴ and Ljunggren.⁵

Moreover, from the results obtained it can be noticed that the combination of some parameters, such as the product between the frequency and the Young’s modulus or between the thickness of the element and the frequency are of minor effect. Mass per area unit is the factor with higher weight on the results. This well expected information witnessed the validity of the procedure for the choice of the parameters employed as input in the neural networks.

Results

Using all the information gathered with the statistical analysis, for all experiments, it was assumed the sound insulation index (in dB) as the property to be estimated and feed the network with samples characterized with the following input characteristics: mass per unit area (kg/m²), frequency (Hz), density of block (kg/m³), and Young’s modulus (N/m²).

Figure 3 shows a characteristic chart for one-hidden-layer network considering the following inputs: Mass per unit area, Frequency, density of block, and Young’s modulus. The output is the airborne sound insulation, and each arrow in the figure denotes a parameter in the network.

Figure 3.

Schematic diagram of the ANN model. Inputs: frequency (Hz); $ρ$ : density of the block (kg/m³); Young’s modulus (N/m²); mass per unit area (kg/m²). Outputs: $R$ : airborne sound insulation (dB).

Table 8 shows the average absolute deviation for the entire dataset and the three-dataset training, validation and test set obtained for the chosen network configuration after 1000 iterations. As it is evident, the minimum (3.51%) of the Absolute Average Deviation is reached for the neuron number 17 for the entire dataset. Going into details, the 17th neuron performed very well also for the Training (AAD = 1.96%), for the Validation (AAD = 0.92%) and the Test set (AAD = 5.18%). It means that this is the best configuration to choose. Table 4 at page 7 shows the values calculated at the frequencies range of interest 100 Hz–5000 Hz for all the 34 samples.

Table 8.

Neuron output on ANN.

Neurons	AAD%	AAD%	AAD%	AAD%
	Entire database	Training	Validation	Test
1	7.24	4.15	{0.72}	{7.50}
3	5.41	3.12	{0.89}	{6.61}
5	4.76	2.70	{0.91}	{6.21}
5	4.33	2.47	{0.91}	{5.63}
9	4.13	2.24	{0.90}	{6.08}
11	3.67	2.12	{0.93}	{5.30}
13	4.11	2.26	{0.89}	{5.72}
15	3.87	2.18	{0.91}	{5.50}
17	3.51	1.96	{0.92}	{5.18}
19	4.28	2.44	{0.91}	{6.07}
21	3.52	2.05	{0.91}	{4.71}
23	4.43	2.50	{0.88}	{5.94}
25	3.68	2.10	{0.91}	{5.04}
27	3.75	2.15	{0.89}	{5.17}
29	3.57	1.95	{0.87}	{4.97}
31	4.36	2.43	{0.88}	{5.62}
33	5.01	2.90	{0.90}	{6.27}
35	5.17	2.99	{0.88}	{6.32}
37	4.04	2.26	{0.91}	{5.99}
39	4.70	2.72	{0.91}	{6.43}
41	5.00	2.82	{0.90}	{6.90}
43	5.00	2.84	{0.88}	{6.19}
45	4.44	2.57	{0.91}	{6.21}
47	4.60	2.60	{0.90}	{6.09}
49	3.91	2.18	{0.88}	{5.41}

Table 9 shows the comparison with the results obtained by laboratory experimental measurements. The comparison is done as previously described, using the AAD% indicator.

Table 9.

AAD% between ANN calculations and laboratory measurements.

F (Hz)	100	125	160	200	250	315	400	500	630	800	1000	1250	1600	2000	2500	3150	4000	5000
AAD%	7.94	6.49	8.31	5.75	3.59	3.21	3.22	2.49	2.01	1.70	1.76	1.93	1.72	2.05	2.23	2.35	3.13	3.37

In Figure 4 the results obtained by calculating the ID 34 wall with the ANNs are shown. Results are also compared with the ISO 10140-1 certificate of the Italian laboratory Istituto Giordano Spa n.194471/04.

Figure 4.

Results on wall ID 34 at different frequencies (ANNs—blue line; laboratory test using ISO 10140—red line).

Discussion

The best performance of the ANN model, in terms of RMS error value, was obtained for the configuration characterized by neuron 17.

The calculation method based on neural networks produces significant improvements to all the range of frequencies. Table 10 shows the results of average AAD% using the two methods.

Table 10.

AAD% of neural networks and ISO 12354-1.

ANN—neuron 17
F (Hz)	100	125	160	200	250	315	400	500	630	800	1000	1250	1600	2000	2500	3150	4000	5000
AAD%	7.94	6.49	8.31	5.75	3.59	3.21	3.22	2.49	2.01	1.70	1.76	1.93	1.72	2.05	2.23	2.35	3.13	3.37
ISO 12354-1
F (Hz)	100	125	160	200	250	315	400	500	630	800	1000	1250	1600	2000	2500	3150	4000	5000
AAD%	21.40	17.55	14.66	14.64	9.50	7.82	7.77	7.41	6.52	6.61	6.74	7.54	9.38	11.07	12.09	10.93	12.39	10.87

The calculation with ANN, for this special case of masonry wall, showed better results than the traditional calculation methods founded in the technical literature and in the ISO international standards. The improvement in the calculation is glaring at all the frequency range considered from 100 Hz to 5000 Hz. The difference between the calculated value and the measured value is almost negligible above 250 Hz. A value of AAD% lower than 2% means a difference lower than 1 dB at a given frequency band.

The results confirm also the starting hypothesis that the main calculation difficulties can be found near the critical frequency and the issue doesn’t seem to depend by the adopted calculation method.

The causes of the higher inaccuracy of the ISO 12354-1 calculation method are the initial hypotheses that consider the walls made of homogeneous and isotropic materials.^2,4 The typical walls of the southern European constructions are made using bricks, plaster and mortars.^48,49 In these walls it is very difficult to correctly predict the variables that depend on how the materials are produced and how they are assembled in the building.⁵⁰ For these reasons, the ISO 12354-1 calculation methods, based on Cremer² and Sewell⁸ cannot be used without corrective terms of uncertainty, which are not provided by the methods based on the standard ISO.⁴³ For all these reasons, in the absence of laboratory measurements, the use of the calculation by ANN could be a valid and helpful tool to the designer.

Conclusions

The paper shows the potential of neural networks in optimizing the numerical methods used for the current standards and used by commercial software, for the case of masonry walls. The use of ANN can be a successful tool for the determination of sound insulation of building masonry walls. The AAD% index shows that ANN method gives better results when compared to the standardized methods based on ISO 12354-1 standard. The greatest criticality was found in determining the physical characteristics of different components and in particular the kind and the quantity of mortars used in brick structures. The comparison between the application of the traditional ISO 12354 calculation method and the application of the ANNs to the specific case shows an effectiveness of the ANNs up to 3 times greater. The numerical values of these results are shown in Tables 9 and 10 in the typical frequency range of the building acoustic, from 100 to 5000 Hz. These results encourage this kind of simulations, although the experimental data set is limited. When a larger data set is available, the model could be improved, and a better accuracy may be achieved. In this case, it could be possible to represent non-linear systems also with a limited number of parameters. The critical frequency should be treated in detail by applying the ANN near that frequency range, through a different model than the one used for the other octave bands. This observation could lead to a future work in which it could be analysed only this range of frequencies.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Serpilli Fabio

References

ISO. 10140-1. Acoustics - laboratory measurement of sound insulation of building elements – part 1: application rules for specific products, 2010.

Cremer

Heckl

Structure-borne sound: Structural vibrations and sound radiation at audio frequencies. Berlin: Springer Berlin Heidelberg, 1988.

Gerretsen

Calculation of airborne and impact sound insulation between dwellings. Appl Acoust 1986; 19(4): 245–264.

ISO. 12354-1. Building acoustics - estimation of acoustic performance of buildings from the performance of elements – part 1: airborne sound insulation between rooms, 2017.

Ljunggren

Airborne sound insulation of thick walls. J Acoust Soc Am 1991; 89(5): 2338–2345.

Rietveld

Van Hout

Statistical techniques for the study of language behaviour. Berlin, Germany: Mouton de Gruyter, 1993.

Warnock

Birta

Detailed report for consortium on fire resistance and sound insulation of floors: Sound transmission and impact insulation data in 1/3 octave bands. Ottawa, Canada: National Research Council Canada, Institute for Research in Construction, 2000.

Sewell

Transmission of reverberant sound through a single-leaf partition surrounded by an infinite rigid baffle. J Sound Vib 1970; 12(1): 21–32.

Chari

Christodoulou

Building energy performance prediction using neural networks. Energy Effic 2017; 10: 1315–1327.

10.

Chu

Forecasting building energy consumption using neural networks and hybrid neuro-fuzzy system: a comparative study. Energy Build 2011; 43(10): 2893–2899.

11.

Kelesoğlu

Fırat

. Determination of loss of heat in brick wall and installation with artificial neural network. Sci Eng J Fırat Univ 2006; 18: 133–141.

12.

Kumar

Aggarwal

Sharma

Energy analysis of a building using artificial neural network: a review. Energy Build 2013; 65: 352–358.

13.

Yang

Rivard

Zmeureanu

On-line building energy prediction using adaptive artificial neural networks. Energy Build 2005; 37(12): 1250–1259.

14.

Zhang

Liu

, et al. Application of neural network to building environmental prediction and control. Build Serv Eng Res Technol 2020; 41(1): 25–45.

15.

Sridharan

, et al. Application of generalized regression neural network in predicting the performance of solar photovoltaic thermal water collector. Ann Data Sci. Epub ahead of print May 2020. DOI: 10.1007/s40745-020-00273-1.

16.

González

Zamarreño

JM.

Prediction of hourly energy consumption in buildings based on a feedback artificial neural network. Energy Build 2005; 37(6): 595–601.

17.

Olofsson

Andersson

Long-term energy demand predictions based on short-term measured data. Energy Build 2001; 33(2): 85–91.

18.

Ben-Nakhi

Mahmoud

MA.

Cooling load prediction for buildings using general regression neural networks. Energy Convers Manage 2004; 45(13–14): 2127–2141.

19.

Ekici

Aksoy

UT.

Prediction of building energy consumption by using artificial neural networks. Adv Eng Software 2009; 40(5): 356–362.

20.

Mohandes

Zhang

Mahdiyar

A comprehensive review on the application of artificial neural networks in building energy analysis. Neurocomputing 2019; 340: 55–75.

21.

Ayata

Çavuşoglu

Arcakloglu

Predictions of temperature distributions on layered metal plates using artificial neural networks. Energy Convers Manage 2006; 47(15–16): 2361–2370.

22.

Soleimani-Mohseni

Thomas

Fahlen

Estimation of operative temperature in buildings using artificial neural networks. Energy Build 2006; 38(6): 635–640.

23.

Liu

Numerical simulation of shower cooling tower based on artificial neural network. Energy Convers Manage 2008; 49(4): 724–732.

24.

Pierantozzi

Mathematical modeling for thermodynamics: Thermophysical properties and equation of state. Ancona, Italy: Università Politecnica delle Marche, 2015.

25.

Di Nicola

Pierantozzi

Petrucci

, et al. Equation for the thermal conductivity of liquids and an artificial neural network. J Thermophys Heat Transfer 2016; 30(3): 651–660.

26.

Mulero

Pierantozzi

Cachadiña

, et al. An artificial neural network for the surface tension of alcohols. Fluid Phase Equilib 2017; 449: 28–40.

27.

Di Nicola

Pierantozzi

. Surface tension of alcohols: a scaled equation and an artificial neural network. Fluid Phase Equilib 2015; 389: 16–27.

28.

Mohamed

Dahl

Hinton

Acoustic modeling using deep belief networks. IEEE Trans Audio Speech Lang Process 2011; 20(1): 14–22.

29.

Di Loreto

Serpilli

Lori

, et al. Sound quality evaluation of kitchen hoods. Appl Acoust 2020; 168: 107415.

30.

Serpilli

Lori

Di Loreto

. Speech recognition assessment in italian pediatric schools using close-set speech tests: a case of study. In: INTER-NOISE and NOISE-CON congress and conference proceedings, Seoul, Korea, vol. 261. Reston, VA: Institute of Noise Control Engineering.

31.

Sak

Senior

Beaufays

Long short-term memory based recurrent neural network architectures for large vocabulary speech recognition. arXiv preprint arXiv:14021128 2014.

32.

Iannace

Trematerra

Ciaburro

Case study: automated recognition of wind farm sound using artificial neural networks. Noise Control Eng J 2020; 68(2): 157–167.

33.

Iannace

Ciaburro

Trematerra

Neural networks model to detect wind turbine dynamics. Int J Autom Smart Technol 2020; 10(1): 145–152.

34.

Ciaburro

Sound event detection in underground parking garage using convolutional neural network. Big Data Cognit Comput 2020; 4(3): 20.

35.

Ciaburro

Iannace

Passaro

, et al. Artificial neural network-based models for predicting the sound absorption coefficient of electrospun poly (vinyl pyrrolidone)/silica composite. Appl Acoust 2020; 169: 107472.

36.

Iannace

Ciaburro

Trematerra

Modelling sound absorption properties of broom fibers using artificial neural networks. Appl Acoust 2020; 163: 107239.

37.

Garg

Dhruw

Gandhi

Prediction of sound insulation of sandwich partition panels by means of artificial neural networks. Arch Acoust 2017; 42: 643–651.

38.

Buratti

Barelli

Moretti

Wooden windows: sound insulation evaluation by means of artificial neural networks. Appl Acoust 2013; 74(5): 740–745.

39.

White

Connectionist nonparametric regression: Multilayer feedforward networks can learn arbitrary mappings. Neural Networks 1990; 3(5): 535–549.

40.

Haykin

Neural networks: a comprehensive foundation/Simon Haykin. Singapore: Pearson Education, 1999.

41.

Rumelhart

Hinton

Williams

RJ.

Learning representations by back-propagating errors. Nature 1986; 323(6088): 533–536.

42.

Serpilli

Guerra

Cesini

Experimental optimization of computational models for acoustic performances of building elements. Riv Ital Acust 2018; 42(1–2): 17–28.

43.

ISO. 717-1. Acoustics - rating of sound insulation in buildings and of building elements – part 1: airborne sound insulation, 2013.

44.

Josse

Lamure

Transmission du son par une paroi simple. Acta Acust United Acust 1964; 14(5): 266–280.

45.

Hornik

Stinchcombe

White

, et al. Multilayer feedforward networks are universal approximators. Neural Networks 1989; 2(5): 359–366.

46.

Esteco

ModeFRONTIER 2014 user’s manual. Pune, India: ESTECO, 2014.

47.

Olkin

Sampson

AR.

Multivariate analysis: overview. London: Elsevier, 2001.

48.

TU0901

CA.

Integrating and harmonizing sound insulation aspects in sustainable urban housing constructions, 2009.

49.

Rasmussen

. Acoustic classification of buildings in Europe–main characteristics of national schemes for housing, schools, hospitals and office buildings. In: Association

(ed.) Proceedings of EuroNoise. Heraklion, Greece: Hellenic Institute of Acoustics.

50.

Ballagh

. Accuracy of prediction methods for sound transmission loss. In: INTER-NOISE and NOISE-CON congress and conference proceedings, New York City, Citeseer, vol. 2004, pp. 3095–3102. Reston, VA: Institute of Noise Control Engineering.