Prediction of sound absorption in TPMS-based acoustic metamaterials using convolutional neural networks and machine-learning-based feature importance analysis

Abstract

Noise pollution has become a major concern recently, and acoustic metamaterials have shown great potential to mitigate it. Among different acoustic metamaterials, sheet/shell lattices based on triply periodic minimal surfaces (TPMS) are a promising class known for their sound-absorbing properties. However, studying these materials through experiments, as well as theoretical approaches (such as numerical simulations and analytical solutions), can be very complex and expensive. To solve this, we developed a deep learning model using a 1D convolutional neural network (CNN) to predict the sound absorption coefficient (SAC) of TPMS structures. We generated training and validation data through theoretical predictions based on multi-physics finite element simulations using a thermoviscous acoustic constitutive model for three TPMS topologies: Gyroid, FRD, and FKS. The input features for each topology included porosity levels (70%, 80%, 90%), cell sizes (3 mm, 5 mm, 7 mm), and thicknesses (10 mm, 15 mm, 20 mm, 30 mm) across frequencies from 200 Hz to 6400 Hz (with 200 Hz intervals). After preparing the data, we built and trained the CNN model, which includes convolution, flatten, and feed-forward layers. The performance of the proposed model was compared with other machine learning models, like the random forest algorithm and XGBoost, where the CNN provided better prediction accuracy. The predicted SAC values matched well with cross-validation results and experimental values, with less than 5% error. We also performed machine learning-based feature importance analysis using permutation to determine how each input feature (frequency, cell size, porosity, and thickness) affects the SAC. The analysis revealed that frequency is the dominant factor influencing sound absorption across Gyroid, FRD, and FKS topologies, followed by sample thickness and cell size, while porosity showed minimal impact within the considered range of 70–90%. These findings were further validated by energy dissipation data from the simulations at resonance frequencies. The outcomes of this study provide researchers with an efficient framework to accurately predict and optimize sound absorption performance in cellular structures, offering a significant reduction in computational time and cost relative to conventional approaches.

Keywords

noise pollution acoustic metamaterial triply periodic minimal surfaces deep learning convolutional neural network permutation-based methods

Introduction

The advent of acoustic metamaterials has greatly expanded the scope of acoustics research, enabling innovative methods for manipulating acoustic wave propagation, including negative refraction, deep subwavelength focusing beyond the diffraction limit, and near-perfect absorption in compact structures.^1,2 Sound absorption is particularly critical among various applications, as excessive noise exposure poses significant public health concerns.^3–5 Consequently, acoustic-absorbing metamaterials have attracted considerable attention for effectively suppressing audible noise.^6–8 These acoustic-absorbing metamaterials are classified into perforated, slotted, and cellular structures.⁹ The widespread adoption of additive manufacturing has facilitated the development of intricate cellular structures, a key category within engineering metamaterials.¹⁰ Due to their multifunctional capabilities, cellular architectures, particularly triply periodic minimal surfaces (TPMS), have become key areas of interest in additive manufacturing research.¹¹ The industrial demand for structures offering superior noise reduction, mechanical strength, and lightweight characteristics has driven interest in such cellular structures.¹² Several cellular configurations have demonstrated excellent sound absorption properties, with TPMS-based architectures proving particularly effective, achieving good sound absorption at specific resonance frequencies.¹³

There are several methodologies for evaluating the acoustic absorption performance of these metamaterials. One approach involves the use of the thermoviscous acoustic module in the multi-physics finite element software COMSOL, which incorporates thermoviscous losses, especially in confined geometries, where sound interacts with viscous and thermal boundary layers. This module solves acoustic models by accounting for acoustic pressure, velocity fields, and temperature variations.¹⁴ The high fidelity of this approach necessitates a refined mesh to capture small-scale interactions, making it computationally expensive. Although it yields precise device response predictions, its application to complex structures is highly time-intensive.¹⁵ Alternatively, utilizing analytical models like Delany-Bazley,¹⁶ Johnson-Champoux-Allard,¹⁷ and the MAA model¹⁸ to predict SAC for porous and perforated acoustic absorbers. However, implementing these analytical models for intricate geometries will be very complex. Experimental measurement using an impedance tube or an anechoic chamber, reverberation room setup to determine acoustic absorption is another option.¹⁹ However, these experimental setups are often cost-prohibitive due to the specialized equipment and controlled environments required for accurate measurements. To mitigate the challenges in numerical, analytical, and experimental methods, a data-driven approach leveraging machine learning (ML) techniques is considered. ML has emerged as a transformative computational tool in recent years, efficiently addressing complex scientific problems across multiple domains.²⁰ By significantly reducing computational costs and time-consuming numerical and experimental methods, ML enables rapid performance predictions with limited resources when efficiently trained. Among various ML techniques, deep learning has demonstrated substantial promise,²¹ as it utilizes multi-layered neural networks trained via back-propagation to extract data representations autonomously. Given its foundation in universal approximation theorems,²² deep learning can model highly nonlinear relationships, making it an ideal tool for optimizing the acoustic performance of metamaterial structures.

Recent advancements in machine learning have revolutionized the predictive and inverse design methodologies for acoustic-absorbing metamaterials, enabling rapid and precise optimization of structural configurations for noise and vibration control applications. The following studies utilize machine learning architectures, including deep auto-encoders, deep neural networks, convolutional neural networks (CNN), artificial neural networks (ANN), and hybrid approaches integrating genetic algorithms (GA) and recurrent units. These studies have developed deep learning-based predictive models to estimate the acoustic properties of complex metamaterials, significantly reducing computational costs associated with conventional numerical simulations.^23–32 A deep auto-encoder was employed to learn the geometric and acoustic parameters of Helmholtz resonators, demonstrating high accuracy for the dataset samples.²³ Similarly, deep learning neural network models have been used to predict and design acoustic metamaterials. Gao et al.²⁴ have integrated theoretical modeling and deep neural networks to predict sound absorption in underwater coatings with extremely low prediction error, while Cao et al.²⁵ have identified key parameters for enhancing acoustic absorption and have accurately predicted the acoustic behavior of porous metamaterials across different frequency bands. Also, a deep learning CNN was implemented to predict the SAC of porous materials over a broad frequency range. By processing binary images representing internal structures, this model matched the accuracy of finite element method (FEM) simulations while drastically improving computational efficiency.²⁶ Additionally, CNN was utilized to predict the sound insulation performance of laminated plate-type acoustic metamaterials. By combining FEM and experimental validation, the model autonomously optimized the configurations for single-peak, dual-peak, and broadband insulation, addressing design complexity in multi-layered systems.²⁷ Inverse design remains a significant challenge due to the highly nonlinear and non-unique mapping between structural parameters and acoustic performance. To address this, a probabilistic generative network incorporating a gated recurrent unit was introduced. This method achieved a high accuracy for low-frequency sound absorption and demonstrated superior performance compared to alternative ML models.²⁸ A deep CNN-GA hybrid framework was developed to optimize porous acoustic materials by coupling CNN predictions with a GA-based iterative optimization. The proposed approach reduced computation time to 5–30 s, while maintaining high accuracy, validated through impedance tube experiments.²⁹ Another inverse design system employed a CNN-based encoder-decoder architecture to directly predict optimal geometric parameters for acoustic sinks, replacing traditional FEM and analytical methods. The model demonstrated a high correlation between predicted and target absorption curves, significantly reducing the design cycle.³⁰ Deep learning has also been used to design high-performance acoustic metasurfaces. A deep neural network-based inverse design optimized an inhomogeneous Helmholtz resonator has achieved broadband absorption. The optimized designs, validated through FEM simulations, achieved a high level of absorption with a compact thickness.³¹ A recent study introduced a multi-objective machine learning-driven inverse design approach to optimize both the acoustic performance and the mechanical energy absorption of shell-based lattice metamaterials. The ANN model accurately predicts the SAC of different lattice configurations, achieving a forward prediction accuracy of over 95% by optimizing hidden layer parameters and incorporating a dropout strategy. The proposed algorithm efficiently generates optimized designs, producing structures with high sound absorption near unity in specific frequency bands.³² While deep learning has been applied to optimize sound absorption in resonators and porous structures, CNN-based deep learning prediction of sound absorption for cellular structures, such as TPMS lattices, remains rarely explored. Hence, the current study introduces a deep learning CNN model, trained on numerical finite element simulations using the multi-physics thermoviscous acoustic constitutive model implemented in COMSOL software, to accurately predict SAC across various TPMS topologies with different porosities, cell sizes, and sample thicknesses, which is validated with experimental results. These TPMS topologies exhibit different sound absorption behaviors, predominantly governed by two mechanisms: resonance-based and porous-based behavior.³³ We selected three TPMS topologies, Gyroid as a representative porous-based topology, F-Rhombic Dodecahedron (FRD) as a resonance-based topology, and Fischer–Koch–Schoen (FKS), which exhibits intermediate behavior,³⁴ and performed feature importance assessment to gain deeper insights into the key input parameters governing their acoustic absorption. To the best of our knowledge, this is the first study to systematically evaluate the feature importance assessment (contribution of different features on sound absorption) in cellular metamaterials. This analysis is particularly crucial for such structures, as their complex geometries introduce multiple interacting factors that influence absorption performance.

The manuscript is organized into four sections. First section describes the data acquisition process, including data collection, distribution, pre-processing, model development, training, validation, and evaluation. Additionally, it elaborates on the methodology employed for feature importance analysis. The next section presents the results and discussion, offering a comprehensive assessment of the performance of the CNN model in predicting sound absorption compared to other machine learning models. This section also validates the predicted sound absorption by correlating it with numerical and experimental findings. The final section concludes the study by summarizing key findings and proposing directions for future research.

Methodology

Data collection

The dataset is crucial for the effective training and performance of deep learning models. The success of deep learning training relies heavily on the data volume, with larger datasets enhancing the ability of the model to generalize, resulting in improved accuracy.³⁵ Thus, data acquisition is a critical determinant of deep learning training efficiency. However, gathering large datasets through experimental or simulation-based methods can be challenging.³⁶ Hence, in this study, a limited dataset for training and validation of the deep learning model was generated using numerical simulation outcomes from COMSOL Multiphysics software, employing the thermoviscous acoustic model for three sheet-based TPMS lattice topologies: Gyroid, FRD, and FKS (see Figure 1(a)). A total of 3456 datasets, comprising 1152 datasets with the input variables such as porosities of 70%, 80%, and 90% (see Figure 1(b)), cell sizes of 3 mm, 5 mm, and 7 mm (see Figure 1(c)), and sample thicknesses of 10 mm, 15 mm, 20 mm, and 30 mm (see Figure 1(d)) in the frequency ranging from 200 Hz to 6400 Hz (in 200 Hz intervals) were used to train and validate the CNN model for each of the three TPMS topologies. The designs of these TPMS topologies were created using MSLattice software³⁷ based on the level-set equations; Equations (1)–(3) for Gyroid, FRD, and FKS, respectively,

f = \sin X \cos Y + \sin Y \cos Z + \sin Z \cos X = c

(1)

f = 4 (\cos X \cos Y \cos Z) - (\cos 2 X \cos 2 Y + \cos 2 Y \cos 2 Z + \cos 2 Z \cos 2 X) = c

(2)

f = \cos 2 X \sin Y \cos Z + \cos X \cos 2 Y s i n Z + \sin X \cos Y \cos 2 Z = c

(3)

where

f

is the iso-surface function,

X = 2 π x / L

Y = 2 π y / L

Z = 2 π z / L

with

x

y

, and

z

being the Cartesian coordinate and

L

is the cell size, and

c

is the level-set parameter that controls the porosity (Ø).³⁸ Among the many TPMS topologies, these three topologies are selected since they are very common and exhibit outstanding multifunctional properties.¹¹

Figure 1.

Data collection: (a) TPMS sheet-based lattice topologies selected for numerical study, (b) FRD topology as an example with different porosities, (c) FKS topology as an example with different cell sizes ( $L$ ), (d) Gyroid topology as an example with different sample thicknesses ( $T$ ), and (e) illustration of the numerical model for the Gyroid topology.

To study the effect of cell size on SAC for the TPMS topologies, the sample thickness (T) and porosity were fixed to 15 mm and 80%, respectively. To study the effect of sample thickness on SAC, the cell size and porosity were fixed at 5 mm and 80%, respectively. To study the effect of porosity on SAC, the cell size and sample thickness were fixed at 5 mm and 15 mm, respectively. We have used the data values from our previously published work,³⁴ along with a few additional combinations of input parameters that were newly simulated and are reported in section S1 of the Supplemental Material. When sound waves interact with these TPMS topologies, the incident energy is divided into reflected, transmitted, and absorbed components, with sound absorption influenced by impedance mismatch and thermoviscous effects.³⁹ The sound absorption of these topologies was numerically simulated using a model based on the finite element method (FEM), developed within the multi-physics software COMSOL, as detailed in 40. The methodology for the numerical simulations is briefly explained as follows. The thermoviscous acoustic model was employed, based on conservation laws of mass, momentum, and energy. The simulations consider air as the medium, and only the air cavity was modeled since the solid lattice is expected to reflect sound, due to an impedance mismatch. To prevent the sudden losses from the pressure acoustic domain to the lattice (thermoviscous), a small cuboid in the thermoviscous domain was added in front of the lattice (see Figure 1(e)). This setup ensures a smooth transition and accurate representation of viscous and thermal dissipation, which are the main contributors to sound absorption. Key outputs include the reflection coefficient ratio obtained from the port boundary and total energy dissipation (viscous and thermal) values, all extracted from COMSOL to calculate the SAC. The simulation uses standard air properties, and a tetrahedral mesh with normal refinement ensures accurate simulations, with results analyzed for the 200 to 6400 Hz frequency range. The detailed explanation regarding the descriptions of the boundary conditions, meshing strategy, convergence criteria, and SAC calculation can be found in section S1 of the Supplemental Information.

Data pre-processing

Data pre-processing starts with the statistical evaluation of input variables, such as frequency, cell size, porosity, thickness, and the output variable (SAC), to assess the distribution of the dataset. The kernel density estimation (KDE) technique, as shown in equation (5) provided a smoothed probability distribution function. In this study, we used Gaussian KDE, as shown in equation (6) along with the histogram plot, as shown in equation (4) providing a discretized approximation of the smoothed probability distribution function that shows enhanced trend representation, such that:

\hat{f_{h i s t}} (x) = \frac{n_{i} (x)}{n . h}

(4)

\hat{f_{h}} (x) = \frac{1}{n h} \sum_{i = 1}^{n} K (u)

(5)

K (u) = \frac{1}{\sqrt{2 π}} e^{- \frac{1}{2} u^{2}}

(6)

where

\hat{f_{h i s t}} (x)

is the histogram-based density estimate at the point

x

n_{i} (x)

is the number of observations within the bin containing

x

\hat{f_{h}} (x)

is the kernel density estimate at the point

x

n

is the total number of observations,

h

is the bin width (for the histogram) or bandwidth (for KDE), and

K (u)

is the Gaussian kernel function in which

u = (x - x_{i}) / h

, where

x_{i}

represents the individual data points. Bin-width (for histograms) or bandwidth (for KDE), which serves as the smoothing parameter governing the resolution of the density estimate by defining the interval size or the spread of the kernel around each data point, was selected adaptively using the interquartile range method for visualization clarity and data granularity, mitigating over-smoothing.⁴¹ The correlation heatmap was also presented to provide insights into the relationships between the input variables and their influence on the output variable, based on the Pearson correlation coefficient (

γ

), given by:

γ_{X, Y} = \frac{\frac{1}{n - 1} \sum_{i = 1}^{n} (X_{i} - \bar{X}) . (Y_{i} - \bar{Y})}{σ_{X} . σ_{Y}}

(7)

where

n

is the total number of observations,

X_{i}

and

Y_{i}

are the individual values,

\bar{X}

and

\bar{Y}

are the mean of

X

and

Y

, respectively, and

σ_{X}

and

σ_{Y}

are the standard deviation of

X

and

Y

, respectively. The correlation coefficient ranges from −1 to 1, where values closer to ±1 indicate a stronger linear relationship between the input and output variables. A positive value signifies that the variables increase or decrease together, while a negative value indicates an inverse relationship, meaning that as one variable increases, the other tends to decrease. The data pre-processing also involved standardizing and restructuring features to optimize model input. Initially, feature scaling, which is the standardization of data with zero mean and unit variance, was performed using the following equation:

X_{s c a l e d} = \frac{X - μ}{σ}

(8)

where

X

is the original feature value,

μ

is the mean of the training feature, and

σ

is the standard deviation of the training feature. This ensures uniformity across input values, facilitating faster model convergence during training. Additionally, the data was reshaped into a 3D format suitable for the Conv1D layer, enabling the model to capture sequential patterns within each feature set effectively.

Model development

Figure 2 shows the architecture of the proposed deep learning CNN model. The CNN architecture was chosen for its ability to capture linear and temporal relationships among the features.⁴² The proposed predictive deep learning model utilizes a 1D CNN to capture spatial dependencies across features within the input sequence. This model architecture includes a 1D convolutional layer with filters and a kernel size of 2, facilitating the extraction of essential feature patterns and interconnections within the data, such that:

y_{i} = \sum_{j = 0}^{k - 1} x_{i + j} . w_{j} + b_{i}

(9)

where

x

is the input feature sequence,

w

is the kernel weights learned during training,

k

is the kernel size,

b

is the bias, and

y_{i}

is the output of the convolution at position

i

. Following this, a flattening layer is applied to the output of the convolutional layer, preparing it for the subsequent dense layers. The flattened output is then fed into a fully connected feedforward network. In this network, dense layers refine the extracted features and learn high-level representations related to the SAC. During training, dropout is applied to deactivate neurons, mitigating the risk of overfitting randomly. In the final output layer of the feed-forward network, a single neuron is utilized to predict the SAC. This regression output continuously estimates the sound absorption characteristics of the TPMS topologies. Table 1 summarizes the parameters for the proposed deep learning CNN model for SAC prediction.

Figure 2.

Proposed deep learning CNN architecture for sound absorption coefficient (SAC) curve prediction.

Table 1.

Summary of the parameters for the proposed deep learning CNN model for SAC prediction.

Process	Parameter	Value/Description
Dataset: Input variables	Frequency	200 Hz to 6400 Hz
	Cell size	3 mm to 5 mm
	Sample thickness	10 mm to 30 mm
	Porosity	70% to 90%
Dataset: Output variable	Sound absorption coefficient (SAC)	0 to 1
Data collection	Size of dataset	3456 (1152 * 3)
Data pre-processing	Normalization	Standard scaler
Data pre-processing	Scaling method	Standardization with zero mean and unit variance
CNN model	Input shape	(4,1) reshaped for 1D
	Convolution 1D layer	Filters: 64, kernel size: 2, Activation: ReLU
	Flatten layer	-
	Dense layer 1	Neurons: 64, activation: ReLU
	Dropout layer	Rate: 0.3
	Dense layer 2	Neurons: 32, activation: ReLU
	Output layer	Neuron: 1
Model compilation	Optimizer	Adam
Model compilation	Loss function	Mean squared error (MSE)
Model training	Epochs	50
	Batch size	16
	Validation data	20%

Model training and validation

The dataset obtained for each topology is divided into training and validation sets, with 80% allocated for training and 20% for validation. We have considered this ratio to optimize the bias-variance tradeoff. A model with high bias tends to oversimplify the problem, leading to underfitting, while a model with high variance captures too much noise from the training data, resulting in overfitting.⁴³ The dataset underwent normalization to optimize model convergence and improve generalization. Layer configurations were carefully selected to balance prediction accuracy and processing efficiency. Weights and biases are fundamental parameters in a CNN that are refined during training.⁴⁴ These parameters significantly impact the network output and the ability of the model to approximate underlying patterns in the data. Table 2 provides a detailed breakdown of the weights and biases used within the CNN model shown in Figure 2 for predicting the SAC.

Table 2.

Learning parameters of the proposed CNN model.

Layer	Weights	Biases	Total parameters
Convolution layer	128	64	192
Flatten layer	8	0	0
Dense layer 1	3096	64	4160
Dropout layer	0	0	0
Dense layer 2	2048	32	2080
Output layer	32	1	33
Total			6265

Model evaluation

Loss analysis

The developed model is evaluated by tracking the reduction in the loss function ( $L_{M S E})$ , which is the mean square error (MSE) as shown in equation (10) throughout each epoch during the training and validation phases, such that:

L_{M S E} = \frac{1}{N} \sum_{i = 1}^{N} {(y_{d i} - y_{p i})}^{2}

(10)

where

y_{d i}

is the desired sound absorption value for the

i

^th sample in the dataset,

y_{p i}

is the predicted sound absorption value for the

i

^th sample by the CNN model, and

N

is the total number of data points. Monitoring the losses of the proposed model is essential for understanding and improving the performance of deep learning models,⁴⁵ such as the CNN, which is used to predict the SAC in Gyroid, FRD, and FKS TPMS topologies. The learning process is visually represented by loss curves plotting the training and validation losses over successive epochs. The training loss reflects the ability of the model to learn patterns from the dataset on which it was trained. In contrast, validation loss provides insights into the ability of the model to generalize to unseen data.⁴⁶ Analyzing these curves is crucial for identifying convergence patterns and detecting signs of overfitting or underfitting.

Performance analysis

The performance of any deep learning model can be validated by benchmarking it against other models commonly-used for similar applications. Accordingly, the random forest algorithm (RFA) and XGBoost were employed alongside CNN to predict SAC for these three TPMS topologies. The effectiveness of these three models (i.e., CNN, RFA, and XGBoost) in predicting SAC was evaluated and compared. Key metrics, including MSE, mean absolute error (MAE), root mean squared error (RMS), R-squared (R²), and explained variance score (EVS), as shown in equations (10)–(14) were utilized to measure prediction accuracy and explained variance.

M A E = \frac{1}{N} \sum_{i = 1}^{N} | y_{d i} - y_{p i} |

(11)

R M S E = \sqrt{M S E}

(12)

R^{2} = 1 - \frac{\sum_{i = 1}^{N} {(y_{d i} - y_{p i})}^{2}}{\sum_{i = 1}^{N} {(y_{d i} - y_{m})}^{2}}

(13)

E V S = 1 - \frac{\sum_{i = 1}^{N} {(y_{d i} - y_{p i} - \bar{e})}^{2}}{\sum_{i = 1}^{N} {(y_{d i} - y_{m})}^{2}}

(14)

where

y_{m}

is the mean of the desired absorption value and

\bar{e}

is the average of differences (

y_{d i} - y_{p i})

. These metrics form the foundation for the simulation and facilitate a comprehensive assessment of the performance of each model in predicting the SAC of the TPMS topologies.

Cross-validation

One of the random combinations of the three TPMS topologies, with a porosity of 80%, a cell size of 4 mm, and a sample thickness of 15 mm, was chosen for cross-validation. This combination of input variables was not part of the training dataset. The SAC across the 200 to 6400 Hz frequency range was predicted using the CNN model and subsequently validated using the numerical method. This cross-validation will confirm the reliability of the CNN model to predict the SAC for all possible combinations within the defined input variable limits.

Experimental validation

One of the combinations of the three TPMS topologies, defined by the porosity of 80%, the cell size of 5 mm, and the sample thickness of 20 mm, was selected (see Figure 3(a)–(c)). The SAC across the 200 to 6400 Hz frequency range was predicted using the CNN model and subsequently validated using an experimental impedance tube method. Samples of the selected combinations were fabricated using the CREALITY Halot Ray 3D printer (China) that employs the digital light processing (DLP) technique. The DLP technique is a vat photopolymerization-based additive manufacturing method that uses a digital projector to rapidly cure selected areas within the liquid resin vat, allowing for a much faster printing rate, as illustrated in Figure 3(e). A tough resin, with a density of 1.14 g/cm³ (as per the data provided by the supplier), was used to produce the test specimens. The printing process utilized a 50 µm layer thickness and a 50-s exposure time per layer. Following 3D printing, each specimen underwent an isopropyl alcohol cleansing and UV curing for 30 min.

Figure 3.

Experimental setup for measuring sound absorption coefficient. (a)–(c) Top view of the 3D printed samples for testing with 80% porosity, 5 mm cell size, and 20 mm sample thickness: (a) Gyroid, (b) FRD, (c) FKS, (d) side view of the 3D printed samples, (e) schematic of the DLP 3D printing process, and (f) impedance tube setup for sound absorption measurement.

Sound absorption measurements were conducted using the two-microphone impedance tube method according to ISO 10534-2 standards, using a Brüel & Kjær 4206 impedance tube (Denmark) with a 29 mm diameter at the Dassault Aviation testing facility, as shown in Figure 3(f). This impedance tube method creates a standing wave within the tube, with the sample positioned at one end, backed by a rigid termination. A speaker at the opposite end generates sound waves, while pressure values are recorded by microphones placed along the tube. These recorded values are processed to generate sound absorption curves in the 200–6400 Hz frequency range. As per the design methodology, we followed the unit cell pattern repeats until the entire cylindrical domain is filled. However, because the cylinder diameter is not an exact multiple of the unit cell size, partial unit cells are generated near the boundary. Thus, the number of encapsulated unit cells is around 6. In the numerical model, the acoustic performance was evaluated using a single unit cell with periodic boundary conditions, where a plane wave could be generated directly within the simulation domain as required. In contrast, for the experimental setup, maintaining a plane wave condition within the target frequency range required a sample diameter of 29 mm, consistent with impedance tube standards for accurate sound absorption measurements. To confirm the repeatability and reproducibility, three samples per topology were fabricated and tested three times, and the difference between them was insignificant.

Feature importance assessment

Features in this context refer to the input and output variables. The machine learning-based feature importance assessment was conducted using a permutation analysis,^47,48 which simplifies determining the significance of each input feature (cell size, thickness, and porosity) in predicting the target or the output feature (SAC) by providing a feature importance score ( ${F I}_{j}$ ), which is calculated using the following expression:

{F I}_{j} = {M S E}^{(j)} - M S E

(15)

where

{M S E}^{(j)} = \frac{1}{n} \sum_{i = 1}^{n} {(y_{d i} - {\hat{y_{p i}}}^{(j)})}^{2}

{\hat{y_{p i}}}^{(j)} = f (X^{(j)})

X^{(j)} = P e r m u t e (X, j)

in which

j

is the feature,

X^{(j)}

is the permuted version of the test data (

X

). The feature importance can be evaluated by observing the increase in

M S E

when features are randomly shuffled. The feature causing the larger increase in

M S E

tends to be more important. This step is vital for this study, since it enhances model interpretability by indicating which features exert the most influence and also aids in understanding the decision-making process of the model, validating the selection of relevant features that significantly affect the SAC of the TPMS topologies.

Results and discussion

Data distribution analysis

The histogram plots and KDE curves for the input variables remain consistent across all three topologies since we are providing the same input parameters for all three topologies (Refer to Section S2 in the Supplemental Information), and only the output variable (SAC) shows distinct distributions, as illustrated in Figure 4(a)–(c),. The SAC distribution demonstrates a broad spread, gradually tapering off at higher absorption values. The CNN model proves itself to be particularly effective under such non-uniform conditions, due to its ability to learn from asymmetric distributions.⁴⁹ Unlike traditional machine learning models that assume normality, CNN leverages its convolutional layers to capture local and global dependencies within the data. It adaptively learns the influence of cell size, porosity, and thickness across varying frequency levels, with its hierarchical feature extraction enabling better representation of complex interactions. CNN identifies patterns across multiple scales, enhancing its ability to generalize SAC predictions across different frequency bands. In contrast to models such as Random Forest, which depend heavily on engineered features, CNN autonomously learns spectral relationships, reducing overfitting risk and improving prediction fidelity. It effectively captures localized effects (e.g., cell size, porosity, and thickness) and global frequency-dependent trends, ensuring a good interpretation of acoustic absorption behavior. Its scalability and resilience to high-dimensional data make CNN particularly suited for complex acoustic datasets.

Figure 4.

Histogram plot and KDE curves of the SAC, and correlation heat maps for (a, d) Gyroid; (b, e) FRD; and (c, f) FKS, respectively.

The correlation heat maps presented in Figure 4(d)–(f) reveal a strong positive correlation between frequency and SAC. The correlation coefficient between frequency and SAC is 0.63 for both Gyroid and FRD, and 0.66 for FKS, suggesting that as frequency increases, so does SAC. The correlation coefficient between cell size and SAC is −0.45 for Gyroid, −0.35 for FRD, and −0.36 for FKS. The correlation result shows the inverse relationship, suggesting larger cell sizes tend to reduce SAC, suggesting topologies with smaller cell structures are more effective. The correlation coefficient between porosity and SAC is −0.07 for Gyroid, −0.20 for FRD, and −0.11 for FKS. The correlation results show that the porosity within the considered range has a minimal impact on SAC, possibly due to nonlinear interactions. The correlation coefficient between thickness and SAC is 0.06 for Gyroid, 0.13 for FRD, and 0.16 for FKS, showing a positive relation, suggesting that the thickness slightly improves SAC, following the acoustic trend, where thicker materials provide better SAC.⁵⁰ Interrelationships between features show no significant correlation (zero correlation coefficient) between cell size and frequency, and thickness does not directly depend on cell size. Porosity does not significantly change with frequency variations, and thickness is independent of frequency. The implications are that small cell sizes and higher frequencies favor sound absorption, additional factors, such as porosity, slightly contribute to absorption efficiency, and optimizing thickness can further enhance sound absorption.

Model performance

The characteristics of a model loss significantly influence generalization and predictive performance. Effective learning is indicated by a rapid decline in loss during the initial epochs, which then stabilizes around minimum values. The model predicts sound absorption using segmented datasets, with the corresponding losses shown in Figure 5(a), (d), and (g). The close alignment of training and validation losses towards the end of the training process suggests strong generalization and minimal overfitting, both of which are essential for accurate sound absorption predictions across all three topologies. This also indicates that the dataset provided for each topology was sufficient for the CNN model to learn meaningful patterns, contributing to reliable performance. A frequency step size of 200 Hz was selected, as using smaller intervals would primarily increase the dataset size without offering substantial additional insight. This resolution was sufficient to capture the overall trends and key features (e.g., resonant frequency) of the sound absorption behavior, as confirmed by the consistent performance of the model.

Figure 5.

Model performance: comparison of training and validation loss for (a) Gyroid, (d) FRD, (g) FKS, comparison of mean squared error (MSE), mean absolute error (MAE), root mean squared error (RMS) of CNN model with random forest algorithm (RFA), and extreme gradient boosting (XGBoost) models for (b) Gyroid, (e) FRD, (h) FKS, and comparison of R-squared (R²), and explained variance score (EVS) for (c) Gyroid, (f) FRD, and (i) FKS.

The performance data in Figure 5 demonstrates that the proposed CNN model outperforms the RFA and XGBoost models in predicting the sound absorption of TPMS topologies. This enhanced performance can be attributed to several inherent advantages of CNN. Primarily, CNN excels in complex feature extraction, due to its remarkable ability to detect spatial patterns within data autonomously. In contrast, traditional algorithms such as RFA and XGBoost may struggle to identify intricate patterns in input data without extensive feature engineering. Moreover, the convolutional layers in CNN process data in smaller, focused regions, allowing for the precise learning of localized patterns. This architectural design enables CNN to capture refined variations in input features associated with sound absorption, thereby reducing error rates and enhancing prediction accuracy. Additionally, the parameter-sharing characteristic of CNN layers decreases the overall number of parameters required, improving the efficiency of the model and its capacity to generalize to unseen data. The CNN-1D architecture effectively handles the challenge of multiple mappings in predicting the SAC by treating frequency as a sequential (time-like) variable, allowing the convolutional layers to capture both local patterns and global dependencies across the frequency range.⁵¹ This hierarchical feature extraction enables the network to automatically learn complex, nonlinear relationships between input and output parameters. As a result, the CNN performs better than RFA and XGBoost, as evidenced by higher R² and EVS, as seen in Figure 5(c),(f), and (i), alongside lower MSE, MAE, and RMS values, as seen in Figure 5(b), (e), and (h). These advantages highlight the effectiveness of CNN in performing complex predictive tasks and its suitability for this application.

Cross and experimental validation

The sound absorption coefficient is calculated as a point value for each combination of input parameters. As the frequency varies from 200 Hz to 6400 Hz, the CNN model predicts a series of point values across this range, which are then plotted to generate the sound absorption curve. For the comparison between numerical results and CNN predictions (cross-validation), we used unseen data that was not included during training and was not generated experimentally. For the comparison between experimental results and CNN predictions, a different set of parameters was selected that had been simulated numerically using COMSOL. Hence, we have plotted CNN, numerical, and experimental results in a single graph. This ensured a stronger validation of the CNN model. However, to ensure clarity, we have included the corresponding parameter values within Figure 6. The sound absorption curves that are derived from the CNN model, numerical simulations, and experimental studies for all three topologies exhibit a peak at a frequency within the mid-frequency range. This behavior is characteristic of cellular structures, particularly those with embedded cavities, as they inherently exhibit resonance phenomena, leading to enhanced sound absorption within this frequency spectrum. Figure 6(a), (c), and (e) compare the SAC obtained along the frequency for Gyroid, FRD, and FKS, respectively, from the CNN model with numerical results. The CNN model was tested on parameter combinations not included in the training, with an average observed error of 3.5% for Gyroid, 5% for FRD, and 4% for FKS compared to the numerical results. The SAC curves predicted by the CNN model show a strong agreement with the numerical data, once again confirming that the collected dataset and frequency discretization were sufficient to accurately capture and predict SAC behavior. The average error for all the topologies was less than or equal to 5%, which validates the ability of the CNN model to predict SAC within the defined input parameter limits, significantly reducing computational cost and time compared to the numerical approach. For instance, obtaining the sound absorption curve for a TPMS lattice with a given set of input parameters at an appropriate finite element mesh resolution requires a minimum of 2 hours using the numerical approach. In contrast, the experimental method achieves the same result in approximately 45 min, including design, fabrication, calibration, and testing. However, the proposed CNN model can generate the sound absorption curve for the given inputs in under a minute, demonstrating its computational efficiency and practicality for rapid assessments. Figure 6(b), (d), and (f) compare the SAC obtained along the frequency for Gyroid, FRD, and FKS, respectively, predicted by the CNN model, numerical, and experimental results. The predicted CNN SAC curve shows good agreement with the numerical SAC curve; however, a few deviations can be seen between the CNN and experimental SAC curves. The CNN model was trained using numerical simulation data, which explains the deviation observed between the predicted and experimental sound absorption coefficient curves. We anticipate that if the ML model is trained based on experimental data, then the CNN model predictions will be much better, but generating such experimental data is very costly, unlike the numerical data. Furthermore, the experimental results may show slight discrepancies due to minor fabrication-related factors, such as printing defects or residual resin blockage that may not have been completely removed after curing, despite efforts to minimize such issues. These small imperfections can slightly alter the acoustic behavior, especially around the absorption peaks. In fact, the experimental results between two nominally identical samples showed minor discrepancies, emphasizing that such variations are part of practical experimental limitations. Also, as seen in Figure 7, the predicted average SAC agrees more closely with the experimental results than the maximum SAC, showing that the CNN model effectively captures the overall absorption behavior, while the maximum SAC is more sensitive to minor fabrication and measurement uncertainties. Nevertheless, the predicted values closely align with the experimental data, exhibiting an average error of less than 5% for all three topologies, as seen in Figure 7(b), which indicates the reliability of the proposed CNN model and its potential for real-time applications.

Figure 6.

Comparison of SAC of the proposed CNN model with the numerical results for (a) Gyroid, (c) FRD, and (e) FKS, with input parameters being a porosity of 80%, cell size of 4 mm, and sample thickness of 15 mm in the frequency range of 200 Hz to 6400 Hz, and comparison of SAC of the proposed CNN model with the numerical, and experimental results for (b) Gyroid, (d) FRD, and (f) FKS, with input parameters being a porosity of 80%, cell size of 5 mm, and sample thickness of 20 mm in the frequency range of 200 Hz to 6400 Hz.

Figure 7.

Comparative analysis of average and maximum sound absorption coefficient for Gyroid, FRD, and FKS structures: (a) CNN-predicted versus numerical values with input parameters being a porosity of 80%, cell size of 4 mm, and sample thickness of 15 mm in the frequency range of 200 Hz to 6400 Hz, and (b) CNN-predicted versus experimental values with input parameters being a porosity of 80%, cell size of 5 mm, and sample thickness of 20 mm in the frequency range of 200 Hz to 6400 Hz.

Feature importance analysis

Analyzing the importance of the feature by permutation, described in Section 2.6, across the Gyroid, FRD, and FKS topologies provides critical insights into the factors driving sound absorption predictions. Notably, frequency emerges as the most significant determinant across all topologies, with a feature importance score of 0.0219 in the Gyroid topology, 0.0692 in the FRD topology, and 0.0387 in the FKS topology. This consistent frequency dominance underscores its essential role in shaping sound behavior, indicating a high sensitivity of sound absorption characteristics to frequency variations. Such findings highlight the need for a focused approach in material design to optimize frequency-related sound absorption properties. Sample thickness is also identified as a significant attribute, particularly in the FRD topology, which contributes a feature importance score of about 0.0157. In the Gyroid topology, thickness also plays a role in the predictive power, although it is less influential than cell size. In the FKS topology, thickness remains a significant factor, contributing to the feature importance score of 0.0113. Cell size also holds moderate importance, particularly in the FRD topology, indicated by the feature importance score of 0.0095. In the Gyroid topology, its importance value is 0.0080, and for the FKS topology, it reaches a significance value of 0.0096. Porosity consistently ranks as the least significant feature across all three topologies, with a feature importance score of 0.0034 in the Gyroid topology, approximately 0.0037 in the FRD topology, and 0.0045 in the FKS topology. These low values indicate that variations in porosity exert minimal influence on sound absorption predictions, suggesting that porosity might not be a critical parameter in optimizing these topologies for enhanced sound absorption performance.

The feature importance scores for features across all three topologies have been normalized as percentages and are presented in Figure 8(a)–(c). The results clearly indicate that these TPMS topologies are highly frequency-dependent, with frequency contributing to over 50% of the influence on SAC. Sample thickness and cell size impact SAC within the 10 to 20% range for all three topologies, while porosity remains the least influential parameter, accounting for less than 10% in each case.

Figure 8.

Feature importance analysis results based on permutation for (a) Gyroid, (b) FRD, (c) FKS, and (d) summary of the feature importance for all three topologies.

Figure 8(d) presents the feature importance rankings for the TPMS topologies based on the permutation method. It is evident that frequency is the most influential parameter affecting the SAC across all three topologies. In contrast, porosity consistently exhibits the lowest importance, indicating a minimal contribution to sound absorption behavior. Notably, in the Gyroid topology, cell size has a greater impact on SAC than sample thickness. However, this relationship is reversed in the FRD and FKS topologies, where sample thickness emerges as more influential than cell size in determining acoustic performance. This pattern is consistent with the correlation heat map analysis presented in Section 3.1, which evaluates the linear relationships between variables. While correlation analysis helps identify potential linear dependencies between features and the target variable, the feature importance assessment goes further by quantifying the actual contribution of each feature to the model’s predictions, considering complex interactions and nonlinear effects. Therefore, feature importance offers a more model-specific understanding of how each parameter influences SAC outcomes.

Dissipation data from numerical simulations were also analyzed further to confirm these trends at the frequency of maximum absorption (i.e., resonant frequency). Since sound absorption in TPMS topologies is mainly due to total (viscous and thermal) dissipation, changes in dissipation reflect changes in SAC. The total dissipation values for the topologies with different cell sizes at the resonant frequency are provided in Table 3. For the Gyroid topology, the total dissipation values for cell sizes 3 mm and 7 mm were 0.04332 W/m³ and 0.01199 W/m³ at 4200 Hz, respectively. For the FRD topology, the values were 0.06933 W/m³ and 0.02725 W/m³ at 4000 Hz, and for FKS, they were 0.0562 W/m³ and 0.01888 W/m³ at 4000 Hz, respectively. The percentage difference in dissipation across cell sizes was approximately 113% for Gyroid, 87% for FRD, and 99% for FKS, indicating that cell size has a more pronounced impact on total dissipation in Gyroid than the other two topologies. This trend is consistent with the feature importance ranking from the permutation analysis. Although the viscous dissipation in the Gyroid at the frequency of maximum absorption (see Figure 9(i)–(iii)) were lower than in FRD (see Figure 10(i)–(iii)) and FKS (see Figure 11(i)–(iii)), the percentage of change in viscous dissipation with respect to cell size was higher in Gyroid than in the FRD and FKS, confirming that cell size variation influences the performance of Gyroid more strongly than the other two topologies.

Table 3.

Total dissipation values of Gyroid, FRD, and FKS for different cell sizes and thicknesses.

Topology	Cell size (mm)	Thickness (mm)	Porosity (%)	Resonant frequency (Hz)	Total dissipation (W/m³)
Gyroid	3	15	80	4200	0.04332
Gyroid	5	15	80	4200	0.02253
Gyroid	7	15	80	4200	0.01199
FRD	3	15	80	4000	0.06933
FRD	5	15	80	4000	0.04397
FRD	7	15	80	4000	0.02725
FKS	3	15	80	4000	0.0562
FKS	5	15	80	4000	0.03301
FKS	7	15	80	4000	0.01888
Gyroid	5	10	80	6200	0.02425
Gyroid	5	15	80	4200	0.02253
Gyroid	5	20	80	3200	0.0206
Gyroid	5	30	80	2500	0.01695
FRD	5	10	80	5800	0.05389
FRD	5	15	80	4000	0.04397
FRD	5	20	80	3000	0.0380
FRD	5	30	80	2000	0.02211
FKS	5	10	80	6000	0.03816
FKS	5	15	80	4200	0.03301
FKS	5	20	80	3200	0.03175
FKS	5	30	80	2200	0.02534

Figure 9.

(a) Volumetric viscous dissipation of Gyroid of 5 mm cell size, 80% porosity, and 15 mm thickness with the cut box zoomed for viscous dissipation at different cell sizes; (i) 3 mm, (ii) 5 mm, (iii) 7 mm, and different porosities; (iv) 70%, (v) 80%, (vi) 90%, at their respective frequency of maximum absorption.

Figure 10.

(a) Volumetric viscous dissipation of FRD of 5 mm cell size, 80% porosity, and 15 mm thickness with the cut box zoomed for viscous dissipation at different cell sizes; (i) 3 mm, (ii) 5 mm, (iii) 7 mm, and different porosities; (iv) 70%, (v) 80%, (vi) 90%, at their respective frequency of maximum absorption.

Figure 11.

Volumetric viscous dissipation of FKS of 5 mm cell size, 80% porosity, and 15 mm thickness with the cut box zoomed for viscous dissipation at different cell sizes; (i) 3 mm, (ii) 5 mm, (iii) 7 mm, and different porosities; (iv) 70%, (v) 80%, (vi) 90%, at their respective frequency of maximum absorption.

The total dissipation values for the topologies with different thicknesses at the resonant frequency are provided in Table 3. For the Gyroid topology, the total dissipation values for sample thicknesses of 10 mm and 30 mm were 0.02425 W/m³ at 6200 Hz and 0.01695 W/m³ at 2500 Hz, respectively. For the FRD topology, the values were 0.05389 W/m³ at 5800 Hz and 0.02211 W/m³ at 2000 Hz. For FKS, they were 0.03816 W/m³ at 6000 Hz and 0.02534 W/m³ at 2200 Hz. The variation in total dissipation across different sample thicknesses showed a percentage difference of approximately 30% for the Gyroid, 58% for FRD, and 35% for FKS. This observation aligns well with the feature importance rankings obtained from the permutation analysis, further validating that thickness plays a more critical role in FRD and FKS than the Gyroid governing sound absorption performance.

It is evident that variations in porosity have little to no significant impact on the dissipation values, with their influence being considerably less pronounced compared to the effects of cell size and thickness. This observation is supported by the trends shown in the Figures 9(iv)–(vi)–11(iv)–(vi) and aligns consistently with the permutation analysis results.

Conclusions

A deep learning-based convolutional neural network model was developed to predict the sound absorption behavior of sheet-based TPMS lattice metamaterials, specifically Gyroid, FRD, and FKS configurations. Histograms and kernel density estimation plots revealed a non-uniform distribution of the output feature (SAC), highlighting the effectiveness of the CNN model in learning from asymmetric and sparse data distributions. The model effectively captured both the localized and frequency-dependent acoustic behavior. It was observed that smaller cell sizes and higher frequencies enhanced sound absorption, while porosity had a relatively minor effect, as indicated by the correlation heat map analysis. Overall, the CNN outperformed both RFA and XGBoost in terms of accuracy and generalization. Its parameter-sharing mechanism, ability to autonomously extract spatial features, and localized processing contributed to its low error rates and efficiency. The CNN achieved prediction errors below 5% compared to cross-validated and experimental values for all TPMS topologies. Results from feature importance analysis show that the frequency was identified as the most influential factor, accounting for over 50% of the impact on SAC. Sample thickness and cell size had a moderate effect (10–20%), while porosity contributed the least. Among the topologies, cell size had a greater impact on SAC in the Gyroid structure, whereas sample thickness was more influential in FRD and FKS. These findings were further supported by dissipation analysis obtained from the numerical simulations. Overall, the proposed CNN model provides accurate predictions of the sound absorption behavior of TPMS topologies while substantially reducing computational time and resources. Furthermore, the use of permutation-based feature importance analysis enables the identification of the most influential parameters affecting acoustic performance, facilitating targeted design optimization. Together, these approaches support the development of next-generation acoustic metamaterials with enhanced absorption characteristics.

A primary limitation of this study is the relatively small size of the numerical dataset, as generating each data point required substantial computational resources and time-intensive simulations. To address this, future efforts could focus on developing analytical or semi-analytical models capable of estimating the sound absorption coefficient for complex TPMS architectures, thereby enabling data generation with reduced computational cost. Additionally, inverse design frameworks using CNN could be employed to automatically generate optimal TPMS configurations based on target acoustic responses. Moreover, incorporating the topology type as an explicit input feature could further improve the predictive capability of the model and support its application across a broader spectrum of metamaterial architectures.

Supplemental Material

Supplemental Material - Prediction of sound absorption in TPMS-based acoustic metamaterials using convolutional neural networks and machine-learning-based feature importance analysis

Supplemental Material for Prediction of sound absorption in TPMS-based acoustic metamaterials using convolutional neural networks and machine-learning-based feature importance analysis by Vignesh Sekar, Lavanya Govindarajan, Kamal K. Sirivuri, Wesley J. Cantwell, Kin Liao, Nicolas Ravaud, Benoit Berton, Pierre-Marie Jacquart, and Rashid K. Abu Al-Rub in Journal of Low Frequency Noise, Vibration and Active Control

Footnotes

Acknowledgments

The authors acknowledge the fruitful discussions with Professors Sami Muhaidat and Omar Alhussein on the feature importance assessment within the machine learning framework. Furthermore, the authors acknowledge the financial support provided by Dassault Aviation under grant number 8434000495. RK Abu Al-Rub further acknowledges the financial support provided by Khalifa University under award number 8474000163.

ORCID iD

Vignesh Sekar

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Dassault Aviation [grant number 8434000495]; Khalifa University [grant number 8474000163].

Declaration of conflicting interests

No potential conflict of interest was reported by the authors.

Supplemental material

Supplemental material for this article is available online.

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