Uncertainty quantification of compressive stress response in expanded polystyrene foams using evidential neural networks

Abstract

This study investigates the application of Deep Evidential Regression in shallow feed-forward neural networks to model and quantify the compressive stress response of expanded polystyrene foam. This foam material, widely utilized for impact protection and packaging, exhibits distinct mechanical behavior characterized by elasticity, plateau, and densification stages during compressive loading. This research adopts a data-driven approach, leveraging artificial neural networks enhanced with evidential learning to predict the distribution of stress responses, thereby addressing both aleatoric and epistemic uncertainties. The methodology involves organizing stress-strain data into training, validation, and test sets, adding noise to simulate real-world conditions, and training models with evidential layers. Results demonstrate that the proposed models maintain high predictive accuracy, with coefficients of determination exceeding 0.90 for noisy test data and above 0.99 for noise-free data. The evidential regression models also provide robust uncertainty quantification, essential for applications where data quality varies. This study’s findings highlight the efficiency and effectiveness of Deep Evidential Regression in enhancing the reliability of stress-strain predictions for EPS foam, offering significant potential for broader application to similar foam materials.

Keywords

expanded polystyrene foams compressive stress response uncertainty quantification neural networks cellular foams

Introduction

Expanded polystyrene foam (EPS) is a material widely used in the design of protective devices against impact and shock in helmet design.^1–3 Depending on the protective application, one of the variables selected for this material is its initial density, which is a parameter related to the material’s performance when undergoing compressive deformation.⁴

The mechanical behavior of EPS foam under compression, which hereafter will simply be referred to as mechanical response, is generally characterized by three clearly defined stages on the stress-strain curve: elasticity, plateau, and densification. According to theory and experience,⁵ of these three stages, the plateau is the most important for the design of protective devices, as it is the stage that presents, mathematically, the largest area under the stress-strain curve during compression, which represents most of the energy absorption during the deformation of such materials. The derivation of deformation energy, and energetic parameters that are useful for selecting and designing a material to absorb energy due to compression, are obtained from the stress-strain curve of the material.⁶ This curve, in turn, is acquired during experimental compression tests on EPS foam. For this reason, modeling this curve is not only fundamental for the scientific understanding of this material, but also relevant and important for activities involving inference and prediction of the structural and mechanical response of protective devices made with EPS foams.

Modeling efforts of the mechanical response to compression of a polymeric foam of the EPS type are well documented in foundational literature.⁵ One of the most successful approaches has been related to hyperelastic models.^7,8 Depending on the analysis, the applications of these models have become increasingly specialized. In foams, these models are essential for capturing their unique compressive behavior. Foams can undergo very large strains (e.g., 60–80% compression) and still recover. Hyperelastic models are used in Finite Element Analysis to accurately simulate this large deformation, allowing engineers to predict foam performance in applications like cushioning, impact absorption, and protective packaging. Examples of these models are the hyperfoam model and the Blatz-Ko model.⁹

A limitation of hyperelastic models used to model polymer foams is that they fit only one material curve and are only capable of, for example, simulating or reproducing the behavior of a material within the data range of that curve, and cannot incorporate multiple input variables beyond strain (for instance, loading rates for compression, temperature or initial density of a foam). It is also fair to say that without these descriptions, many of the more sophisticated models in Computer-Aided Engineering, such as finite element approaches, would not be feasible, as efforts have also been made to connect the mathematics of a hyperelastic or viscoelastic model with numerical models,¹⁰ hence their importance and relevance. Examples of successful cases on the application of hyperelastic models for foam materials can be consulted, for example, in the work of Palta et al.¹¹ or Forero Rueda et al.¹²

An approach and an alternative to the problem of non-generalization of traditional hyperelastic or viscoelastic descriptions for modeling the stress-strain curve is the use of artificial neural network models. This is a data-driven approach that involves using experimental data harvested through tests and, subsequently, trials to train neural network models to approximate the mechanical response of a material.¹³ Although a data-driven approach is not limited solely to the use of artificial neural networks, as other machine learning models can be used, it is true that, for the case of compression response of a polymeric foam, these types of models have been validated in the past as successful and accurate.^14,15 Nevertheless, so far, an intrinsic limitation of the data-driven approach is that the resulting models do not have the capacity to quantify the uncertainty in the data, and the construction of several of these models (usually dictated by principles such as the central limit theorem) is required to understand the models’ uncertainty but through the individual predictions of each one of them.

This work presents advancements in the construction of artificial neural network models to address the mechanical response to compression of EPS foam. A data-driven approach is employed to construct regressive models of the Evidential Learning type,¹⁶ which, instead of approximating a scalar in their prediction, estimate the possible distribution of such a response.

The Deep Evidential Regression approach has been proved successfully in a wide range of applications. For instance, for robust and reliable classification models based on neural networks in health care applications.^17,18 Furthermore, in the case of materials science, the approach has been used for prediction and the discovery of molecular properties.¹⁹ This approach has been recognized as a framework among the current advanced uncertainty quantification approaches.²⁰ In the case of its application for quantifying uncertainty in stress response in materials, the approach has used for thermoplastic elastomers.²¹ Thus, the materials and methods section introduce the basics of shallow neural networks; subsequently, the theory of Evidential Deep Learning is introduced. Then, the conditions and considerations of the dataset used to build the presented models are outlined, as well as the conditions of the architectures of the models and the hyperparameters for their training. Finally, results, conclusions, and discussion regarding their relevance and utility are presented.

Materials and methods

Basics of regressive artificial neural networks

Given a structured dataset $S$ that contains $N$ inputs $x_{i}$ and targets $y_{i}$ ,

S = {(x_{i}, y_{i})}_{i = 1}^{N},

(1)

the objective of a neural network model is to determine the functional mapping

f_{A N N} : R^{D} \to R

that associates inputs with targets. For regression tasks, a model predicts a quantity, such as a single scalar value

y_{i}

, using an input feature vector

x_{i}

with

D

dimensions:

x_{i} = [\begin{array}{l} x_{i}^{(1)} \\ x_{i}^{(2)} \\ \begin{array}{l} ⋮ \\ x_{i}^{(D)} \end{array} \end{array}]

(2)

In its fundamental architecture, an artificial neural network (ANN) consists of units called artificial neurons arranged into layers. The input layer ingests data from the feature vector $x_{i}$ , while the output layer produces the predicted scalar ${\hat{y}}_{i}$ . The hidden layers, comprised of hidden neurons, sequentially process activations to yield ${\hat{y}}_{i}$ at the output. The activation of a single hidden neuron is given by the following expression:

a_{l}^{(k)} = g_{l} (z_{l}^{(k)}),

(3)

where

z_{l}^{(k)}

is an internal sum that can be obtained as follows:

z_{l}^{(k)} = \sum_{j = 1}^{s i z e_{l - 1}} w_{l, k}^{(j)} a_{l - 1}^{(j)} + b_{l, k},

(4)

where

s i z e_{l - 1}

is the number of units in the previous layer

l - 1

a_{l - 1}^{(j)}

refers to the activation of the neuron also in the previous layer,

b_{l, k}

stores the biases for each of the neurons,

g_{l} ()

is termed neuron’s activation function, and

w_{l, k}^{(j)}

represent the weights of neurons in a current layer with respect to the neurons in a previous layer.

An ANN undergoes training as optimization algorithms adjust the weights and biases within its architecture. The gradient descent algorithm optimizes these weights and biases relative to a loss function $L$ through iterative updates:

w_{l, k}^{(j)} \leftarrow w_{l, k}^{(j)} - \frac{l_{r}}{N} \sum_{i = 1}^{N} {(\frac{\partial L}{\partial w_{l, k}^{(j)}})}_{i},

(5)

and,

b_{l, k} \leftarrow b_{l, k} - \frac{l_{r}}{N} \sum_{i = 1}^{N} {(\frac{\partial L}{\partial b_{l, k}})}_{i},

(6)

where

l_{r}

is a learning-rate hyperparameter that updates the weights and biases during such iterations. Readers seeking deeper mathematical insights into ANNs may consult Aggarwal²² and Kneusel.²³

Although traditional feed-forward neural networks excel in regression tasks, they cannot quantify prediction uncertainty. This issue can be mitigated by integrating Deep Evidential Regression theory, which enhances traditional architectures to model the distribution of the target value $\hat{y}$ .

Evidential Deep Learning

Deep Evidential Regression¹⁵ allows for learning the parameters of a distribution, such as mean and variance, thereby enabling the quantification of epistemic and aleatoric uncertainties in an ANN’s prediction. The aleatoric uncertainty captures the inherent noise in the data while the epistemic uncertainty represents the uncertainty in the model’s parameters. This section presents the basics of Deep Evidential Regression and its application in enhancing the uncertainty estimation capabilities of neural networks in regression problems. Thus, the targets $y_{i}$ in the dataset $S$ are assumed to follow an independent and identically distributed (i.i.d.) Gaussian distribution with an unknown mean and variance:

(y_{1}, \dots, y_{N}) \sim N (μ, σ^{2}),

(7)

where

μ

and

σ^{2}

are parameters of a Gaussian distribution and a Normal Inverse-Gamma distribution (NIG) respectively¹⁵:

μ_{i} \sim N (γ_{i}, σ_{i}^{2}, υ_{i}^{- 1}), σ_{i}^{2} \sim Γ_{i}^{- 1} (α_{i}, β_{i}),

(8)

where

γ_{i}

υ_{i}^{- 1}

α_{i}

β_{i}

are the hyperparameters of the NIG distribution. Hence,

m_{i} = [\begin{array}{l} γ_{i} \\ υ_{i} \\ \begin{array}{l} α_{i} \\ β_{i} \end{array} \end{array}] f o r : γ_{i} \in R, υ_{i} > 0, α_{i} > 1, and β_{i} > 0

(9)

Neural Networks that use the Deep Evidential Regression approach try to estimate the posterior distribution $q (μ, σ^{2}) = p (μ, σ^{2} | y_{1}, \dots, y_{N})$ , which can be factorized as follows:

q (μ, σ^{2}) = q (μ) q (σ^{2}),

(10)

hence, the approximation can be given in form of a Gaussian conjugate prior and a NIG distribution:

p (\underset{θ}{\underset{⏟}{μ, σ^{2}}} | \underset{m}{\underset{⏟}{γ, ν, α, β}}) = \frac{β^{α \sqrt{ν}}}{Γ (α) \sqrt{2 π σ^{2}}} {(\frac{1}{σ^{2}})}^{α + 1} \exp {- \frac{2 β + ν {(γ - μ)}^{2}}{2 σ^{2}}} .

(11)

The NIG distribution in this evidential regression approach is seen as a “higher-order evidential distribution on top of the unknown lower-order likelihood distribution from which observations are drawn.”¹⁵ Therefore, Deep Evidential Regression estimates both the aleatoric and the epistemic uncertainties in a model. Thus, under such assumptions, the aleatoric uncertainty, prediction, and the epistemic uncertainty are computed as follows:

E [μ] = γ, prediction

(12)

E [σ^{2}] = \frac{β}{α - 1}, aleatoric

(13)

Var [μ] = \frac{β}{υ (α - 1)}, epistemic

(14)

The Deep Evidential Regression approach offers a powerful framework to quantify uncertainties in neural network predictions. By training a model to output the parameters of an evidential distribution, this technique enables the direct calculation of uncertainties through the analytic computation of the maximum likelihood Gaussian. This eliminates the need for time-consuming repeated inference or training sampling procedures.

Under the Deep Evidential Regression framework, the total loss, where a training algorithm adjusts the weights of an ANN, comprises two loss terms:

L_{i}^{N L L} (w) = \frac{1}{2} \log (\frac{π}{υ}) - α \log (Ω) + (α + 1 / 2) \log ({(y_{i} - γ)}^{2} υ + Ω) + \log (\frac{Γ (α)}{Γ (α + 1 / 2)})

(15)

L_{i}^{R} = | y_{i} - γ | \cdot (2 υ + α),

(16)

the

L_{i}^{N L L}

loss accounts for the Negative Log-Likelihood of an ANN model evidence, which integrates Bayesian probability theory, and

L_{i}^{R}

serves as a regularization term to control the increase in uncertainty for incorrect predictions. Thus, the overall loss is then expressed as follows:

L_{i} (w) = L_{i}^{N L L} (w) + λ L_{i}^{R} (w),

(17)

where

λ

is a hyperparameter balancing uncertainty with model fit.

In essence, an ANN employing Deep Evidential Regression learns an evidential distribution, characterized by parameters $m$ , and yields four outputs for each observation $\hat{y}$ , diverging from the single output typical in conventional neural networks. This model thus produces an evidential distribution $m_{i} = f (x_{i}; w)$ . Therefore, in this work, a regressive shallow neural network equipped with an evidential regression output layer is used to model and quantify uncertainty in the mechanical response of an EPS foam with different initial densities.

EPS compressive stress data

Figure 1 shows a typical compressive stress-strain curve of an EPS foam. In this figure, the previously mentioned stress stages are shown. This type of curve is obtained from experimental testing on material samples defined and specified in standards. The data used for this work was obtained from compressive experimental tests on hot-wire cut EPS white cubic specimens with a size of 100 +/− 1 mm in a universal testing machine INSTRON 8872, equipped with a Dynacell load cell with a data acquisition rate of 500 Hz, and following the criteria given in the ASTM D1621 standard (examples of the EPS specimens are shown in Figure 1(b)). The EPS material was obtained in three densities: 8.5 kg/m³, 12 kg/m³, and 24 kg/m³. Such material was studied because is widely used within the Mexican region and Central America to manufacture packaging products for protective applications. As such, the material was obtained from the same supplier, Poliespuma del Bajio S.A, currently based in the Mexican state of Guanajuato.

The initial dataset for this study was derived from compressive tests on the EPS material. The experimental design, specified in Table 1, involved three material densities and three loading rates. Four replicates were performed for each of these nine experimental combinations, yielding a total of thirty-six (36) stress-strain curves. This dataset, used to train the neural networks, comprises 83,335 stress-strain data points. This dataset is mathematically described as follows:

S_{E P S} = {(\underset{x_{i}^{⊤}}{\underset{⏟}{{[ρ_{i}, V_{i}, s_{i}, ε_{i}]}^{⊤}}}, \underset{y_{i}}{\underset{⏟}{σ_{i}^{c}}})}_{i = 1}^{N},

(18)

where

ρ_{i}

refers to the initial density of the EPS foam,

V_{i}

correspond to the loading rate of the applied load,

s_{i} \in {0, 1}

is a switch characteristic that describes the loading/unloading of the material (1 for loading and 0 for unloading) and ε_i refers to the engineering strain.

Table 1.

EPS experimental data factors and their levels.

Factor	Level
Factor	Low	Mid	High
Density (kg/m³)	8.5	12	24
Loading rate (mm/s)	0.164	1.640	16.400

This initial dataset was divided into three sub-sets: (i) a test set with 15,318 data points, corresponding to a total of four stress-strain curves of the 12 kg/m³ and 1.640 mm/s experimental combination, (ii) a training set with 49,653 data points, and (iii) a validation set comprising 18,364 data points. On this, it is important to mention that both the training and validation datasets correspond to a total of thirty stress-strain curves. The maximum value of strain for all the tests was 90% of the cubic EPS specimens. Additionally, Gaussian noise was added to the targets of each of the above-described datasets as is described in Table 2. The purpose of introducing noise to the data was to mimic real-world conditions, improving the models’ ability to generalize and perform reliably. On this, the maximum value of 2 adheres to the maximum expected variance in the response of the studied material when tested multiple times without modifying its conditions. The lower values are zero, to test the networks models at no noise and half of the maximum to the effect of change in such parameter.

Table 2.

Gaussian noise parameter values.

Dataset	Gaussian noise parameters
Dataset	$μ_{d a t a}$ (MPa)	$σ_{d a t a}$ (MPa)
Training	0	0
		0.025
		0.05
Validation	0	0
		0.025
		0.05
Test	0	0
		0.025
		0.05

An example of the entire dataset with and without noise is shown in Figure 2. This figure presents the comparison between the original data and the data with added noise corresponding to a mean of µ_data = 0 MPa and a standard deviation of $σ_{d a t a} = 0.025$ MPa. Here, the experimental effects of initial density and loading rate factors can also be directly observed: an increase in their levels tends to generate a higher compressive stress response in the material. Ideally, a neural-network-based model should capture such behavior in a material.

It is important to highlight that the data shown in Figure 2(e) corresponds to the test data that will be used to test the models in this work. As such, this data was not used for training or validation of any model presented here.

The above-described data is used to train shallow evidential regressive neural networks as is specified in the following section.

Method

To model the stress-strain data of the EPS material and quantify the associated uncertainty, both in the models themselves (epistemic) and in the data (aleatoric), the following method was implemented:

1. First, compressive stress-strain data were organized into three sets: (a) training, (b) validation, and (c) test. The primary objective was to use these sets to train, validate, and subsequently test the models with new data. The number of data points reserved for each set was specified in the previous section. The training and validation stress-strain curves were chosen randomly to train and validate the neural networks. This was carried out using a Python program.

2. Second, noise, as previously described, was added to the datasets using a Python script. It is important to highlight that this method does not exclude working with data that may inherently present non-simulated noise. In fact, the main objective of this study is to work with data under such conditions.

3. Third, three feed-forward neural network architectures equipped with an evidential layer at their output are proposed. These models were trained for the different noise scenarios as described in Table 2. Additionally, Table 3 shows the main chosen architecture for these networks, which was based on previous architectures used for the same task on polymer foams. The evidential layer is used as described in the work by Amini et al.,¹⁹ wherein individual predictions of the neural networks provide a distribution rather than a deterministic scalar value, in this case for the mechanical stress response of the analyzed foam material.

4. Finally, error metrics for the testing dataset were addressed. The evaluation metrics used were the Normalized Mean Absolute Difference (NMAD) and the Coefficient of Determination (R²):

NMAD = 100 \cdot \frac{〈 | e - p | 〉}{〈 | e | 〉},

(19)

R^{2} = 1 - \frac{\sum_{i = 1}^{N} {(e_{i} - p_{i})}^{2}}{\sum_{i = 1}^{N} {(e_{i} - \bar{p})}^{2}},

(20)

where

e

represents the vector of noisy experimental data,

p

contains the mean values from the mean evidential regression prediction,

〈 \cdot 〉

denotes the average of the given vector, and

\bar{p}

is the mean of the elements in

p

. For these metrics, the ideal model would have an NMAD of zero and an R² of one.

Table 3.

Specifications for the evidential neural networks.

Component	Specification
Model architecture	Sequential
Hidden layers	Two fully connected layers with 16 and 8 hyperbolic tangent neurons
Output layer	Dense normal gamma
Loss function	Evidential regression loss, regularizer term λ = 0.01
Optimizer	RMSProp
Optimizer hyperparameters	ℓ_r = 0.001, momentum = 0.97, ρ = 0.97
Training epochs	3000
Callbacks	TensorFlow ModelCheckpoint

Regarding the above method, it is worth mentioning that the TensorFlow library was used to implement the Evidential Deep Learning output layers of the models.²⁴ Additionally, a normalization procedure was applied to each element of the i-th input vector $x_{i}^{(j)}$ of the datasets as follows:

x_{i, scaled}^{(j)} = \frac{x_{i}^{(j)} - \min (x^{(j)})}{\max (x^{(j)}) - \min (x^{(j)})} \cdot (x_{\max} - x_{\min}) + x_{\min},

(21)

where

x_{i}^{(j)}

is the j-th element of the i-th input vector

x_{i}

\min (x^{(j)})

is the minimum value of the j-th element across all input vectors in the training dataset,

\max (x^{(j)})

is the maximum value of the j-th element across all input vectors in the training dataset, and

x_{\min}

and

x_{\max}

are the desired range for the scaled values (in this case, between 0 and 1).

The steps of the method are general and can potentially be applied to other foam-like materials. The primary objective of this study, however, is to obtain a model capable of learning the distribution of the compressive stress response of EPS foam material to quantify epistemic and aleatoric uncertainty. This enhances a model’s capacity to predict stress under uncertainties within the data.

Results and discussions

Figure 3 shows the training and validation loss curves of the three models trained with different noise values. Further analysis and discussion of this figure are provided below.

Figure 4 shows the predictions of different models trained with varying noise levels in comparison to test data. As observed, the three models accurately predict the stress-strain curve for a material with a density of 12 kg/m³, compressed at a loading rate of 1.640 mm/s. Key observations include: (i) all three models address the elastic, plateau, and densification stress stages in the material, (ii) they capture the loading and unloading steps, (iii) the models exhibit viscoplastic behavior, as residual strain is addressed after the unloading step, and (iv) the uncertainty bands provided by the evidential predictions account for the noise in the data (noisy data points could fall within a band of 3 ± MPa as can be observed in Figure 4(c)). These results emphasize empirical adequacy and the good predictive capabilities of the ANNs for the stress response of EPS material.

Table 4 shows the performance of the three trained models under different conditions on the test data. This table quantifies the results shown in Figure 5, using mean predictions from the models and data from equations (19) and (20). The table indicates values above 0.90 for the coefficient of determination for the test set, both with and without Gaussian noise. However, the NMAD metric increases with higher levels of two factors: (a) noise in the test data, and (b) models trained with such noise. This pattern occurs because the mean predictions of the evidential models are directly compared with noisy data. Nevertheless, the results in Figure 4 demonstrate the capabilities of an evidential regressive model: their output can quantify aleatoric uncertainty with the σ parameter.

Table 4.

Mean predictive results of models on test data for different noise conditions.

Models’ training noise level (σ_data)	Test condition	R²	NMAD
0	Without noise	0.9962	3.45
0.025	With noise without noise	0.9785	11.56
0.025	With noise without noise	0.9975	3.28
0.050	With noise without noise	0.9089	24.31
0.050	With noise without noise	0.9973	3.43

The results in Figure 4 demonstrate the capabilities of an evidential regressive model: their output can quantify aleatoric uncertainty with the σ parameter. In this case, noise falls within the predictive uncertainty bands of ±3 standard deviations. This ability to account for uncertainty is crucial in applications where data quality cannot always be guaranteed, providing a more comprehensive understanding of the model’s reliability under different conditions.

Focusing on the noise-free performance metrics in Table 4 and the results in Figure 6, it is observed that adding noise to the training data marginally improves performance on the original test data compared to a model trained without noise. This is also evident in Figure 3: the more noise is added to the training data, the less instability observed in the training curves. Although the injection of noise in training data is known to benefit generalization, accuracy, and speed of convergence in neural networks for various tasks, including regression,^25–27 this study does not delve into this aspect. In the engineering praxis, the improvement obtained by adding noise might be seen as marginal, since models with a coefficient of determination above 0.99 are considered good predictors for stress-strain data in EPS material.

Figure 6 shows the mean predictions of the models with respect to the test data. A high correlation is observed for the three models, as most of the data centers around the ideal predictive line. However, some deviations are noted at high experimental and predictive values (above 0.97 MPa). These deviations are due to differences between the peak experimental data and peak mean predictions at a strain of 90%, as also observed in Figure 5. Nevertheless, the uncertainty is well addressed and can be quantified by the model, as shown in Figure 4, where the uncertainty bands cover data in these regions.

Considering previous results, how the presented models compare with previous efforts on modeling the stress-strain response of an EPS material? In the article by Rodríguez-Sánchez and Plascencia-Mora,¹³ a frequentist approach was used to address model uncertainty: 31 models were trained, and their combined predictions were compared to ground-truth data. Thereafter, the best model was chosen based on the distribution of error metrics. This approach, although reliable, requires time and a considerable number of computational resources, as statistical inferences and hypotheses must be validated on the distribution of error performance across the trained models. Moreover, the best-reported model in that study attained a coefficient of determination of 0.9983, which, compared to the best model reported in Table 4, is marginally better but requires more training attempts to achieve. In contrast, only one evidential shallow neural network presented in this study was run for each of the noise scenarios reported. Thus, the present approach could improve efficiency and, in a single run, can address both aleatoric and epistemic uncertainty.

Regarding the use of data-driven approaches powered by neural networks to model stress-strain curves or to model parameters of materials, efforts in the literature have focused on finding deterministic models for such purposes. However, outside the present study, no other works report the implementation of Deep Evidential Regression to quantify uncertainty for the stress-strain prediction of mechanical responses in materials. Hence, it is recommended to implement Deep Evidential Regression in similar foam-like materials as analyzed in this study.

Table 5 presents the data for the maximum standard deviations found in the experimental stress-strain curves. As can be seen, except for the condition of a density of 24 kg/m³ at a loading rate of 1640 mm/min, the maximum standard deviation found was less than 0.01 MPa. This means that trained models with a simulated noise of 0.05 MPa can address the uncertainty of the actual experimental data; this is relevant, because laboratories that cannot afford to have high fidelity testing equipment could quantify and model the uncertainty of noisy data by employing the presented method in this contribution.

Table 5.

Maximum standard deviation values obtained from EPS stress-strain curves at different density and loading rate conditions.

Factors		Maximum standard deviation (MPa)
Density (kg/m³)	Loading rate (mm/min)	Maximum standard deviation (MPa)
8.5	0.164	0.006703
8.5	1.64	0.001172
8.5	16.4	0.002185
12	0.164	0.000979
12	1.64	0.005352
12	16.4	∼0.0001
24	0.164	∼0.0001
24	1.64	0.048421
24	16.4	∼0.0001

Note. These values were obtained from four replicates from each combination of experimental factors.

The novelty of this contribution compared to previous work lies in the fact that Evidential models do not require an ensemble of models to quantify uncertainty in the stress-strain response of EPS foam. Furthermore, given that energy parameters such as energy absorption or efficiency in these materials are derived from this type of mechanical response, the presented approach could, in principle, be useful for quantifying and modeling uncertainty in energy properties regarding their maximum values (note that many foam-type materials are chosen for protective purposes based on their energy absorption efficiency and ideality). This is outside the scope of this work but is recommended as a topic for future research.

Finally, it is interesting to note that, apart from one previous study,²¹ the literature does not actually contain an application of evidential neural networks for modeling the mechanical response of materials using the formulation of Amini et al.,¹⁶ the closest studies in neural networks for non-foam materials are those reported by Linka et al.,^28,29 which use Constitutive Neural Networks³⁰ and Physics Informed Neural networks³¹ coupled with Bayesian inference to quantify uncertainty in a material’s response. This indicates that the proposed approach in this contribution is certainly novel and may find greater application in more complex neural network architectures, or application for different materials.

Conclusions

This article presented the application of the Deep Evidential Regression approach on shallow feedforward neural networks to predict and quantify uncertainty in the compressive stress response of EPS foam. Based on the above results and discussions, the following conclusions are drawn:

1. The implementation of evidential regression in shallow neural networks that model compressive stress in EPS foam helps in quantifying uncertainty when stress-strain data presents noise. This improvement does not affect performance in the model and increases predictive capabilities, as values of the coefficient of determination, even with noise in the training data, are above 0.90 when models are compared with test data with noise; furthermore, for test data with no noise, this value is above 0.99.

2. Predictive uncertainty bands in the models, obtained at a level of µ = 0, and a standard deviation of 3 ± σ, covered noise in the data for the three models analyzed. As such, a neural network equipped with an evidential layer can address noisy scenarios for stress-strain data and predicting stress response robustly.

3. Furthermore, when obtaining the maximum experimental standard deviation from four replicates of each of the conditions tested for the material, it was observed that a model trained with a 0.05 MPa of noise can quantify and address the variation of the stress response of the EPS.

4. The neural network models trained with an evidential-based approach can: (i) predict and simulate the three stress-strain stages in EPS material, (ii) address viscoplastic behavior and model residual strain, and (iii) quantify aleatoric and epistemic uncertainty in data.

Interestingly, it was observed that adding noise to the training and validation sets marginally increases the capabilities of a model. However, because this phenomenon is not fully addressed in this study, it is recommended to further investigate its relevance to modeling compressive stress responses in foam materials in future works.

Since energy absorption is an important and relevant parameter for the study of EPS foams, future studies are suggested where the presented method of Evidential Neural Networks is used to obtain and address uncertainty in this type of quantity, both for this class of foams and others (e.g., polypropylene or biofoams).

Footnotes

Acknowledgements

The author would like to thank Dr Héctor Plascencia-Mora for the support provided in facilitating the materials and equipment necessary to carry out the experimental tests from which the data supporting this article were obtained.

ORCID iD

Alejandro E. Rodríguez-Sánchez

Funding

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

Declaration of conflicting interests

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

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