Deep Learning-Based Porosity Prediction of Concrete Under Freeze–Heaving Conditions Using Strain Fields

Abstract

Freeze-induced damage in concrete is governed by complex interactions between pore-scale phase transition and macroscopic mechanical response, while the underlying pore structure is typically difficult to observe directly. This study proposes an integrated framework for porosity inversion in concrete under freeze–heaving conditions, combining mechanical modeling, finite element simulation, and deep learning. A mechanics-based model is first developed to describe frost-heaving behavior in porous concrete, accounting for elastoplastic deformation of the matrix and partial volumetric expansion induced by pore water freezing. Based on this formulation, a parametric finite element model with randomly distributed pores is constructed to generate datasets linking pore characteristics to full-field deformation responses. Building upon these physics-consistent data, a deep learning framework is established to reconstruct pore distribution directly from three-component strain fields. The model employs a Vision Transformer backbone to capture global deformation patterns and incorporates a Kolmogorov–Arnold Network-based nonlinear mapping to enhance representation of the highly nonlinear inverse relationship. The results demonstrate that the proposed approach achieves accurate pore reconstruction and porosity prediction with stable convergence and satisfactory generalization performance across different porosity levels. The study provides a physically interpretable and computationally efficient pathway for linking deformation fields to internal pore structure, offering new potential for non-destructive characterization and durability assessment of concrete in cold-region environments.

1. Introduction

The internal pore structure of concrete plays a fundamental role in governing its mechanical performance and durability, particularly under freeze–heaving conditions [1]. As a typical heterogeneous porous material, concrete contains a complex system of voids and microcracks, whose geometric characteristics—such as porosity, pore size distribution, and spatial connectivity—directly influence the evolution of internal stresses and damage [2]. Under freezing conditions, the phase transition of pore water induces volumetric expansion and generates internal pressure, which may lead to microcrack initiation and progressive deterioration of the material [3,4]. Therefore, establishing a quantitative relationship between pore structure and freeze-induced mechanical response is essential for understanding and predicting freeze–heaving damage.

From a mechanistic perspective, freeze-induced damage in concrete originates from the interaction between pore-scale phase transition and the surrounding solid skeleton [5,6]. Classical theories, including the hydraulic pressure theory and osmotic pressure theory, have provided fundamental explanations for frost damage by attributing it to water–ice phase transformation and moisture migration processes [7]. Subsequent developments in poromechanics and thermodynamic modeling further incorporated the coupling effects of temperature, pore pressure, and material deformation, enabling more comprehensive descriptions of freeze–heaving processes [8,9]. Despite these advances, most existing models rely on simplified assumptions regarding pore geometry or treat pore structure in an averaged sense [10], making it difficult to explicitly capture the influence of heterogeneous pore distributions on mechanical response.

To overcome these limitations, numerical simulation has been widely employed to investigate freeze–heaving behavior in concrete [11,12]. Finite element methods and related computational approaches allow for the incorporation of multi-physics coupling, including heat transfer, moisture migration, and mechanical deformation [13,14,15,16]. In particular, recent studies have introduced mesoscale heterogeneity into numerical models to account for the influence of aggregates, pores, and interfacial transition zones on freeze–heaving damage evolution [17,18]. In addition, peridynamic formulations provide an alternative framework for modeling crack initiation and propagation without predefined discontinuities [19], offering advantages in capturing damage evolution under freeze–heaving loading [20,21]. However, these forward simulation approaches typically require detailed information about internal pore structure as input, which is difficult to obtain in practical engineering.

In contrast, experimental techniques such as X-ray computed tomography and microscopic imaging can provide direct observations of pore structure and damage evolution [22]. Nevertheless, these methods are often limited by high cost, restricted sample size, and challenges in capturing real-time internal processes during freezing. More importantly, both experimental and numerical approaches predominantly follow a forward-analysis paradigm, in which material response is predicted based on known microstructural information. The inverse problem—inferring pore structure from observable mechanical or deformation fields—remains insufficiently explored.

Recent advances in data-driven methods offer new opportunities for addressing such inverse problems [22,23,24,25]. Deep learning techniques have demonstrated strong capability in extracting complex nonlinear relationships from high-dimensional data [26,27,28,29] and have been successfully applied to image-based microstructure characterization and damage assessment in concrete [30,31,32,33]. However, existing studies primarily focus on identifying or segmenting pore structures from imaging data, rather than reconstructing internal structural features from mechanical response fields [34]. In particular, the potential of displacement or strain fields as carriers of microstructural information has not been fully utilized.

A key challenge therefore lies in establishing a physically consistent and computationally efficient framework that links freeze-induced mechanical response to underlying pore structure, enabling reliable inversion of porosity and pore distribution. Such a framework requires not only an accurate description of freeze–heaving mechanics but also a systematic strategy for generating high-quality datasets and a robust model for solving the associated inverse problem.

To address this challenge, the present study develops an integrated framework that combines mechanical modeling, finite element simulation, and deep learning-based inversion for concrete under freeze–heaving conditions. First, a mechanics-based model is established to describe the frost-induced deformation and internal stress evolution associated with pore water phase transition. Based on this formulation, a parametric finite element model is constructed to simulate the coupled response of heterogeneous concrete with varying pore structures, generating a dataset that captures the relationship between internal strain fields and pore characteristics. Building upon these data, a deep learning model is developed to perform porosity inversion directly from strain-field patterns, thereby bypassing the need for explicit microstructural observation. By integrating physical modeling and data-driven inference, the proposed approach provides a new pathway for linking macroscopic deformation fields to microscopic pore structure, offering a physically interpretable and computationally efficient tool for assessing freeze–heaving damage in concrete.

2. Models for Frost Heaving in Porous Concrete

2.1. Theoretical Model

Concrete is a typical porous medium containing a large number of randomly distributed spherical pores. Consider a spherical pore of radius a embedded in a concrete matrix subjected to freezing-induced internal pressure p. To account for the interaction between the pore and the surrounding medium, the matrix is modeled as a finite spherical domain with outer radius R, which represents the effective interaction volume associated with the pore. The outer boundary is assumed to be traction-free.

The surrounding concrete is modeled as an ideal elastoplastic material, characterized by Young’s modulus E, Poisson’s ratio ν, and tensile yield strength ${\sigma }_t$. The ice phase inside the pore is assumed to be linearly elastic with bulk modulus K_i. The equivalent outer radius R is determined from the porosity n and a characteristic length scale $R_0$ as follows:

$$R={({\displaystyle \frac{a^3}{n}}+R_0^3)}^{1/3}$$

(1)

Here, R₀ represents a characteristic interaction length of the surrounding matrix, reflecting the finite spatial extent over which a pore interacts mechanically with its neighboring pores. This length scale arises from the heterogeneous pore distribution in concrete and cannot be captured by a purely local geometric relation based solely on porosity.

The dimensionless confinement ratio λ, defined as the ratio of the pore radius to the equivalent outer radius, is expressed as

$$\lambda ={\displaystyle \frac{a}{R}}$$

(2)

Under internal pressure p, the surrounding matrix may undergo elastic–plastic deformation. When yielding occurs, a plastic zone develops in the region (a ≤ r ≤ c), where c is the plastic radius (depicted in Figure 1).

The radial and circumferential stresses are denoted by σ_r and σ_θ, respectively, and r is the radial coordinate. The equilibrium equation for spherical symmetry is

$${\displaystyle \frac{d{\sigma }_r}{dr}}+{\displaystyle \frac{2}{r}}({\sigma }_r-{\sigma }_{\theta })=0$$

(3)

and ${\sigma }_t$ in the yield condition is

$${\sigma }_{\theta }-{\sigma }_r={\sigma }_t$$

(4)

Substituting Equation (4) into Equation (3) yields.

Integrating over the plastic region with the boundary condition σ_r(a) = −p, one obtains

$${\sigma }_r(r)=-p+2{\sigma }_t\ln ({\displaystyle \frac{r}{a}})$$

(5)

$${\sigma }_{\theta }(r)={\sigma }_r(r)+{\sigma }_t$$

(6)

In the elastic region c ≤ r ≤ R, the displacement field is

$$u(r)=Ar+{\displaystyle \frac{B}{r^2}}$$

(7)

where A and B are integration constants arising from the general solution of the spherically symmetric elastic problem and are determined by the boundary and interface conditions.

The radial stress is

$${\sigma }_r^{(e)}(r)=(3{\lambda }_e+2\mu )A-{\displaystyle \frac{4\mu B}{r^3}}$$

(8)

where λ_e and μ are Lamé constants. Applying the traction-free boundary condition σ_r^(e)(R) = 0 and the yield condition at r = c yields the radial stress at the elastic–plastic interface:

$${\sigma }_r^{(e)}(c)=-{\displaystyle \frac{2}{3}}{\sigma }_t(1-{\displaystyle \frac{c^3}{R^3}})$$

(9)

Enforcing stress continuity at r = c yields the pressure–plastic radius relation. Solving the elastoplastic problem yields the pressure–plastic radius relation:

$$p=2{\sigma }_t\ln ({\displaystyle \frac{c}{a}})+{\displaystyle \frac{2}{3}}{\sigma }_t(1-{\displaystyle \frac{c^3}{R^3}})$$

(10)

The radial displacement at the cavity surface is expressed as

$$u_r(a)=u_e(a,c)+u_p(a,c)$$

(11)

where u_e and u_p denote the elastic and plastic contributions, respectively.

The compatibility condition associated with the volumetric expansion during freezing is

$$u_r(a)={\displaystyle \frac{1}{3}}\chi a$$

(12)

which gives

$$u_e(a,c)+u_p(a,c)={\displaystyle \frac{1}{3}}\chi a$$

(13)

where χ is the volumetric expansion.

Equations (10) and (13) define a coupled nonlinear system for p and c.

For complete freezing, the volumetric expansion is approximately

$${\chi }_0=0.09$$

(14)

Substituting Equation (14) into Equation (13) shows that no admissible solution exists for typical material parameters. This indicates that the assumption of complete confinement is not physically consistent for porous concrete.

To resolve this inconsistency, an effective freezing coefficient is introduced:

$${\chi }_g={\chi }_{0}g(a)$$

(15)

where g(a) ∈ (0,1] represents the fraction of volumetric expansion contributing to pressure buildup.

The introduction of ${\chi }_g$ reflects the fact that the freezing process in porous concrete does not occur in a closed system. During freezing, part of the pore water can migrate out of the pore before fully participating in the phase transition. As a result, only a portion of the theoretical volumetric expansion contributes to internal pressure. The retained fraction depends strongly on pore size. Smaller pores exhibit stronger adsorption and capillary effects, which inhibit water migration and lead to larger effective expansion. In contrast, larger pores allow more efficient drainage, resulting in smaller effective expansion.

In addition, freezing is a transient process governed by heat conduction. Smaller pores tend to freeze rapidly, leaving insufficient time for water migration, whereas larger pores experience longer freezing durations, which further enhance drainage.

Based on these considerations, the effective function is taken as

$$g(a)={\displaystyle \frac{1}{1+{\alpha }_d{(a/R)}^k}}$$

(16)

where k is a dimensionless parameter, and ${\alpha }_d$ denotes a drainage attenuation coefficient characterizing the efficiency of water migration during freezing. The term ${(a/R)}^k$ represents a dimensionless measure of pore size relative to the interaction domain, capturing the scale-dependent drainage effect. As the pore size increases, drainage becomes more effective, leading to a reduction in the effective freezing expansion ratio. In the limiting case of negligible drainage (${\alpha }_d$ tending to 0), one obtains g(a) tending to 1, corresponding to a fully confined freezing process. Conversely, for large pores or strong drainage (${\alpha }_d$ ≫ 1), the effective expansion is significantly reduced.

It should be noted that Equation (16) is a phenomenological but physically constrained attenuation function rather than a universal material law. The function satisfies the basic requirements of the drainage-controlled freezing process, 0 < g(a) ≤ 1, g(a) → 1, when drainage is negligible, and g(a) decreases with increasing relative pore size a/R₀ or increasing drainage capacity. The parameter χ₀ is determined by the theoretical volumetric expansion of water during complete freezing and is taken as 0.09 in this study. The coefficient α_d characterizes the effective drainage capacity during freezing, including the combined influence of pore connectivity, permeability, freezing duration, and water migration resistance. A larger αd corresponds to stronger drainage and thus a smaller fraction of volumetric expansion contributing to pressure buildup. The exponent k controls the sensitivity of the attenuation function to the relative pore size a/R₀.

When experimental or high-fidelity numerical data are available, α_d and k can be determined by inverse calibration. For example, if the effective expansion ratio χ_g is known for different pore sizes, Equation (16) can be rewritten as Equation (17).

$$\ln \left({\displaystyle \frac{{\chi }_0}{{\chi }_g}}-1\right)=\ln {\alpha }_d+k\ln \left(a/R_0\right)$$

(17)

so that α_d and k can be obtained by regression. Alternatively, they can be identified by minimizing the discrepancy between measured and predicted frost-heaving pressures (Equation (18)).

$$\underset{{\alpha }_d,k}{\min }{{\displaystyle {\sum }_j\left[p_{\mathrm{mod}el}\left(a_j;{\alpha }_d,k\right)-p_{\exp }\left(a_j\right)\right]}}^2$$

(18)

In the present study, because systematic experimental pressure–pore-size data are not yet available, α_d is varied over representative values to describe different drainage scenarios, and k = 2 is adopted to represent a nonlinear pore-size-dependent drainage response. Therefore, the selected parameters are used for mechanism analysis and dataset generation rather than as universal constants.

At the fully plastic state c = R, the critical expansion is defined as

$${\chi }_{cr}(a)={\displaystyle \frac{{\left.3u_r(a)\right|}_{c=R}}{a}}$$

(19)

Thus, the effective expansion is given by

$${\chi }_{eff}=\min ({\chi }_g(a),{\chi }_{cr}(a))$$

(20)

For a given pore radius a, the frost-heaving pressure is obtained by solving Equations (10) and (13) (see Algorithm 1).

Algorithm 1. Computational procedure for evaluating frost-heaving pressure

Compute R and λ From Equations (1) and (2) Compute χg(a) From Equation (15) Compute χeff from Equation (20) Define Residual $$F\left(c\right)=u_e\left(a,c\right)+u_p\left(a,c\right)-{\displaystyle \frac{1}{3}}{\chi }_{eff}a$$ Solve F(c) = 0, c∈[a, R] Compute p From Equation (10)

The proposed formulation establishes a unified and mechanics-consistent framework linking pore size, nonlocal confinement, fluid transport, and frost-heaving pressure in porous concrete. This formulation provides a mechanistic basis for interpreting experimentally observed frost-heaving pressures, which are typically much lower than those predicted by complete freezing assumptions.

2.2. Model Validation and Mechanism Analysis

2.2.1. Parameter Setting

To evaluate the physical consistency and predictive capability of the proposed frost-heaving model, a series of parametric analyses are conducted. The material parameters are selected to represent typical concrete subjected to freeze–heaving conditions. The tensile strength of the matrix is taken as σ_t ≈ 3–5 MPa, which corresponds to a critical expansion ratio on the order of χ_cr ≈ 0.04–0.055 as obtained from the elastoplastic solution at the fully plastic state. The dimensionless pore-size parameter is defined as a/R₀, with 0 < a/R₀ < 1 ensuring a physically admissible interaction domain, where R₀ denotes the characteristic interaction length controlling pore-scale confinement and drainage. For each prescribed value of a/R₀, the corresponding outer radius R used in the elastoplastic boundary-value problem is determined through Equation (1). Therefore, the parametric analysis is fully consistent with the theoretical formulation in Section 2.1. The theoretical volumetric expansion associated with complete freezing is taken as χ₀ = 0.09. To account for drainage effects, the attenuation coefficient is varied as α_d = 20, 40, 80, 120, and the exponent is taken as k = 2, which provides sufficient sensitivity to capture the size-dependent drainage behavior.

These values are selected to cover different drainage intensities from relatively weak to strong attenuation, rather than to represent a unique calibrated material parameter. In practical applications, α_d and k should be calibrated using experimental frost-heaving pressure, pore-size distribution, or full-field strain measurements.

It should be noted that χ₀ corresponds to a fully confined freezing condition. Experimental studies have shown that such conditions may lead to extremely high freezing pressures on the order of several hundred MPa, whereas significantly lower effective pressures are typically observed in porous concrete due to drainage and incomplete freezing [35,36].

2.2.2. Effect of Drainage on Effective Expansion

Based on the above pore-size-dependent drainage mechanism, a phenomenological attenuation function is introduced to describe the fraction of freezing-induced volumetric expansion that effectively contributes to pressure buildup. This form is motivated by poromechanical descriptions of freezing porous media, where ice formation induces pressure buildup while excess liquid water can be expelled from the freezing region, and by commonly used closed-form hydraulic attenuation functions for pore-size-dependent transport behavior in porous media [7,37,38]. The drainage-controlled expansion ratio is defined as

$${\chi }_g={\displaystyle \frac{{\chi }_0}{1+{\alpha }_d{\left(a/R_0\right)}^k}}$$

(21)

Figure 2 illustrates the variation in χ_g with respect to the dimensionless pore size a/R₀. It is observed that χ_g decreases monotonically with increasing a/R₀, indicating that larger pores exhibit stronger drainage effects. For small pores, χ_g approaches the theoretical expansion χ₀, implying that the freezing process is nearly confined, and most of the volumetric expansion contributes to pressure buildup. In contrast, for larger pores or higher values of α_d, the effective expansion is significantly reduced due to enhanced fluid migration. This behavior is consistent with the physical mechanism of freezing in porous media: small pores restrict water migration due to capillary and adsorption effects, while larger pores allow more efficient drainage, thereby reducing the effective expansion.

2.2.3. Competition Between Elastoplastic Admissibility and Drainage

The freezing-induced expansion cannot be arbitrarily large. For a given pore size, the elastoplastic boundary value problem described in Section 2.1 admits a solution only within a limited range of volumetric expansion.

The critical expansion χ_cr is defined from the elastoplastic solution at the fully plastic state (c = R). It represents the maximum admissible expansion that can be sustained by the surrounding matrix. The effective expansion is therefore determined by ${\chi }_{eff}=\min ({\chi }_g,{\chi }_{cr})$. It should be emphasized that this relation does not represent an artificial truncation but rather reflects the intrinsic admissibility constraint of the elastoplastic problem.

Figure 3 compares χ_g and χ_cr, from which two distinct regimes can be identified:

• Elastoplastic-controlled regime (χ_g > χ_cr)
  The imposed expansion exceeds the admissible limit, and the response is governed by the mechanical strength of the matrix.

• Drainage-controlled regime (χ_g < χ_cr)
  The effective expansion is sufficiently reduced by drainage, and the elastoplastic constraint is not activated.

The transition between these regimes demonstrates the interplay between material resistance and pore-scale transport processes. Such a competition is also reflected in experimental observations, where the freezing pressure does not continuously increase with decreasing temperature but instead exhibits a saturation behavior due to mechanical and transport limitations [35].

2.2.4. Resulting Frost-Heaving Pressure

The frost-heaving pressure is obtained by solving the coupled nonlinear system defined by Equations (10) and (13), using the effective expansion χ_eff. Figure 4 presents the variation in the predicted pressure with respect to pore size.

The results show that the predicted pressure remains within the range p ≈ 3–14 MPa, which is consistent with reported values for concrete under freeze–heaving conditions. In particular, measurements in saline environments and partially saturated systems typically indicate pressure levels on the order of several to several tens of MPa [35]. For small pores, the pressure approaches an upper bound governed by χ_cr, reflecting the limitation imposed by the tensile strength of the matrix. For larger pores, the pressure decreases significantly due to enhanced drainage, indicating that only a fraction of the theoretical volumetric expansion contributes to stress buildup. To further illustrate the necessity of the proposed formulation, the closed-system assumption (χ = χ₀) is also examined. Under this assumption, the predicted pressure reaches unrealistically high values on the order of hundreds of MPa. It is important to note that such high pressures are not unrealistic but correspond to fully constrained freezing conditions. Experimental studies have reported that freezing expansion pressure can reach approximately 300–400 MPa at low temperatures (below −30 °C) in confined systems [35,36]. Therefore, the discrepancy between the theoretical upper bound and the predicted pressure range (3–14 MPa) is not a deficiency of the model but rather reflects the essential role of drainage and partial freezing in realistic porous media.

The proposed model provides a unified framework for describing frost-heaving behavior in porous concrete by incorporating three key mechanisms: the thermodynamic driving expansion (χ₀), the drainage-induced attenuation (χ_g), and the elastoplastic admissibility constraint (χ_cr). Experimental observations indicate that freezing-induced pressure may span several orders of magnitude, ranging from a few MPa at the initial freezing stage to several hundred MPa under fully constrained conditions [35,36]. The present model successfully reconciles these seemingly contradictory observations. The high-pressure regime corresponds to the theoretical upper bound associated with full confinement, whereas the much lower pressure levels observed in concrete arise naturally from the combined effects of drainage and mechanical constraints. The parameter α_d characterizes the efficiency of water migration during freezing and governs the transition from confinement-dominated to drainage-dominated behavior. Importantly, it does not directly modify the pressure–expansion relationship but determines the effective expansion entering the elastoplastic problem.

The results demonstrate that the realistic pressure level arises from the combined effect of drainage and elastoplastic constraints. Neither mechanism alone is sufficient to explain the observed behavior. Overall, the model successfully captures the transition from strength-controlled to transport-controlled regimes and provides a physically consistent explanation for the relatively low frost-heaving pressures observed in concrete.

2.3. Finite Element Models of Concrete Plates

To investigate the macroscopic response induced by pore-scale frost heaving, finite element models of concrete plates with randomly distributed pores are constructed. The simulations aim to establish a quantitative link between pore distribution, porosity level, and the resulting stress and deformation fields and to provide a dataset for subsequent data-driven inverse analysis.

A square concrete plate with dimensions 10 mm × 10 mm is considered. The pore structure is explicitly represented by circular voids randomly distributed within the domain. Five porosity levels are investigated, namely 8%, 10%, 12%, 14%, and 16%. For each porosity level, 100 independent realizations of pore distributions are generated, resulting in a total of 500 numerical samples. Compared with the initial dataset containing 100 samples, the dataset was expanded to 500 samples to improve the diversity of pore distributions and to provide a more reliable basis for model training and testing. It should also be noted that each sample contains full-field strain information and a pixel-level pore mask with a resolution of 224 × 224, providing dense spatial supervision rather than a single scalar label. The pore diameters are randomly sampled within the range of 0.15 mm to 0.5 mm, with an average diameter of approximately 0.3 mm. The number of pores in each realization is determined to satisfy the prescribed porosity. To minimize boundary effects, a geometric constraint is imposed such that the minimum distance between any pore and the plate boundary is no less than 0.5 mm. This ensures that the stress and deformation fields around pores are not artificially influenced by boundary proximity. The computational domain is discretized using a structured mesh. Each edge of the plate is divided into 224 segments, resulting in a sufficiently fine resolution to capture stress concentrations and deformation gradients induced by pore-scale effects. The mesh density is selected to ensure numerical convergence and to accurately resolve the local mechanical response near pore boundaries, where strong stress localization is expected.

The concrete matrix is modeled as an elastoplastic material, consistent with the theoretical formulation in Section 2.1. The material behavior is characterized by Young’s modulus E, Poisson’s ratio ν, and tensile yield strength σ_t. The frost-heaving effect is incorporated through the effective volumetric expansion χ_eff(a), defined in Equation (20), which depends on pore size via the combined geometric and mechanical constraints. This ensures full consistency between the theoretical model and the numerical implementation. For each pore, the freezing-induced expansion is represented as an equivalent internal loading, which generates stress and deformation in the surrounding matrix. The interaction among multiple pores is naturally captured through the finite element solution.

For each realization, the spatial distributions of displacement and von Mises stress are computed. Representative contour maps of strain fields are extracted, resulting in a dataset of 500 samples corresponding to different pore distributions and porosity levels. These strain-field data are subsequently used to construct a dataset for data-driven modeling (as shown in Figure 5). In particular, the dataset serves as input for a hybrid deep learning framework based on Kolmogorov–Arnold Networks (KANs) [39] and Transformer architectures, with the objective of reconstructing the pore distribution or inferring the porosity of the concrete plate from observed deformation fields. By combining multiple realizations across different porosity levels, the dataset captures both the deterministic influence of porosity and the stochastic variability induced by random pore distributions.

The adopted finite element framework explicitly accounts for pore-scale heterogeneity, elastoplastic material behavior, and frost-induced expansion. Moreover, the generated dataset establishes a direct connection between physics-based simulations and data-driven inverse analysis, providing a foundation for intelligent prediction of pore characteristics from measurable deformation fields.

3. Methodology

3.1. Problem Formulation and Data Representation

The goal of this work is to predict the pore distribution of concrete cross-sections subjected to freeze–heaving conditions based on the three-component strain fields. This is an inverse problem, where the spatial pore distribution is inferred from the deformation response in the form of strain fields. The task is complicated by the fact that the relationship between the strain response and the pore morphology is highly nonlinear and spatially dependent. Pores influence the strain distribution, but the exact relationship is not explicit, which makes the problem challenging for conventional image segmentation or regression approaches. The input strain data is preprocessed to standardize the three strain components. For each strain component, the mean and standard deviation are computed across the training set, and each pixel value is normalized using these statistics.

The use of the three-component strain field is motivated by its direct mechanical connection with the internal pore structure. Under freeze–heaving action, pores act as local sources of deformation disturbance and alter the spatial redistribution of the surrounding matrix. The normal strain components ε_xx and ε_yy reflect the directional expansion or contraction of the concrete section, while the shear strain component ε_xy describes the local distortion and asymmetric deformation induced by heterogeneous pore arrangements. Compared with displacement fields, strain fields are derived from spatial displacement gradients and are therefore more sensitive to local deformation localization, pore-boundary effects, and interactions among neighboring pores. In the two-dimensional setting adopted in this study, ε_xx, ε_yy, and ε_xy constitute the independent in-plane strain components, providing a mechanically complete description of the local deformation state. Therefore, the combined three-component strain field can serve as an informative indicator for reconstructing the underlying pore distribution.

From a mechanics viewpoint, the strain field can be regarded as an indirect projection of the pore geometry through equilibrium and deformation compatibility. Each pore introduces a local geometric discontinuity and a freezing-induced expansion source, which disturbs the surrounding matrix and produces spatially correlated strain perturbations. The location of a pore is reflected by local strain concentration and strain-gradient variation, whereas its size, spacing, and neighboring interactions affect the intensity, spatial extent, and coupling pattern of the strain response. Therefore, the full-field strain maps contain not only local pore-boundary information but also information on nonlocal interactions among multiple pores. It should be noted that this does not imply a mathematically unique inversion under arbitrary conditions. Rather, under the controlled simulation setting and porosity range considered in this study, the three-component full-field strain response provides sufficient pore-sensitive information for statistically reliable pore reconstruction.

The dataset is divided into training and testing subsets, with the training set used to optimize the model parameters and the test set used to evaluate the generalization capability of the trained model [40]. The images are organized into a 3D tensor for each sample, and the corresponding pore mask is represented as a binary image, where pore locations are marked as 1 and non-pore regions as 0.

3.2. Backbone Architecture of the Proposed Model

Figure 6 illustrates the overall architecture of the proposed enhanced Vision Transformer (ViT) [41] for pore prediction. The framework mainly consists of four components: patch embedding [42], context-anchored attention-based feature refinement [43], transformer-based global feature extraction, and progressive upsampling for pore map reconstruction [44]. Specifically, the input image is first divided into flattened patches and projected into token embeddings. To improve the representation of local structural context, a context-anchored attention mechanism is introduced before the main transformer encoder. The refined tokens are then processed by multiple transformer blocks to capture long-range dependencies and global spatial correlations. In addition, the conventional nonlinear mapping module is enhanced using KANs [45], which improves the expressive capability of the network for complex pore morphology learning. Finally, the encoded features are reshaped and progressively upsampled to generate the predicted pore distribution map.

3.2.1. Vision Transformer Backbone

In the present study, the task of the network is to infer the pore distribution of a concrete cross-section from its full-field strain response under freeze–heaving conditions. Different from conventional forward prediction problems, the input of the model is not a structural image but a three-channel strain-field image, and the output is the corresponding pore-distribution map. Specifically, the input strain field consists of the three components ε_xx, ε_xy, and ε_yy, which jointly describe the spatial deformation state of the concrete section under freeze–heaving-induced deterioration. Since pore locations affect not only local strain concentration but also the global transmission and redistribution of deformation [44], the inverse mapping from strain fields to pore morphology requires the model to capture both local response features and long-range spatial dependencies. For this reason, a Vision Transformer backbone is adopted to extract global latent representations from the strain-field input for subsequent pore prediction.

Let the input strain-field tensor be denoted by X ∈ ℝ^H×W×3, where the three channels correspond to X_:,:,1 = ε_xx, X_:,:,2 = ε_xy, X_:,:,3 = ε_yy. The target output is the pore-distribution image of the concrete cross-section, written as Y ∈ ℝ^H×W×C, where C denotes the number of output channels. For a binary pore prediction task, C = 1, and each pixel indicates the predicted pore likelihood of the corresponding spatial position.

To enable global interaction over the strain field, the input tensor X is first partitioned into N non-overlapping patches of size P × P, where N = HW/P². The i-th patch is flattened into a vector $x_i^p\in {ℝ}^{3P^2}$ and then projected into a D-dimensional embedding space through a learnable linear transformation:

$$\begin{array}{cc}{z}_i^{(0)}=x_i^{p}E,& i=1,2,\dots ,N\end{array}$$

(22)

where $E\in {ℝ}^{3P^2}$ is the projection matrix. Through this tokenization process, different local strain-response regions are represented in a unified latent space, which makes it possible to establish global interactions among spatially separated deformation patterns related to pore presence.

Because the transformer encoder does not inherently preserve the spatial order of the original strain field, positional information is injected into the token sequence. The initial embedded sequence is thus written as Equation (23) [46].

$$Z^{(0)}=\left[z_1^{(0)};z_2^{(0)};\cdots ;z_N^{(0)}\right]+E_{pos}$$

(23)

where $E_{pos}\in {ℝ}^{N\times D}$ denotes the learnable positional embedding. This step is important in the present inverse problem because similar local strain patterns may correspond to different pore configurations when they appear in different spatial contexts.

The token sequence is then fed into stacked transformer encoder layers. For the input feature sequence Z^(l−1) at the l-th layer, the query Q, key K, and value V matrices are computed as in Equation (24).

$$\begin{array}{ccc}Q=Z^{(l-1)}{W}_Q,& K=Z^{(l-1)}{W}_K,& V=Z^{(l-1)}{W}_V,\end{array}$$

(24)

where W_Q, W_K, and W_V are trainable projection matrices. The attention operation is expressed as Equation (25).

$$\mathrm{Attention}\left(Q,K,V\right)=\mathrm{Soft}\max \left({\displaystyle \frac{Q{K}^{\top }}{\sqrt{d}}}\right)V$$

(25)

where d is the feature dimension of each attention head. By this mechanism, each patch-level strain token is updated through adaptive interaction with all other tokens, allowing the backbone to capture not only local deformation anomalies but also global strain correlation patterns induced by pore distribution.

To improve the diversity of learned dependencies, multi-head self-attention is adopted. The m-th attention head is defined by Equation (26).

$${\mathrm{head}}_m=\mathrm{Attention}\left(Q_m,K_m,V_m\right)$$

(26)

And the outputs of all heads are concatenated and projected by Equation (27).

$$\mathrm{MSA}\left(Z^{\left(l-1\right)}\right)=\mathrm{Concat}\left({\mathrm{head}}_1,{\mathrm{head}}_2,\dots ,{\mathrm{head}}_M\right)W_O$$

(27)

where M is the number of attention heads, and W_O is the output projection matrix. This design enables the network to capture multiple types of pore-related strain dependencies in parallel, such as local strain concentration, directional deformation continuity, and long-range coupling among separated response regions.

Each encoder layer consists of an attention-based interaction stage and a nonlinear feature transformation stage, together with residual connections and layer normalization. The layer-wise update is given by Equations (28) and (29).

$${\tilde{Z}}^{\left(l\right)}=Z^{\left(l-1\right)}+\mathrm{MSA}\left(\mathrm{LN}\left(Z^{\left(l-1\right)}\right)\right)$$

(28)

$$Z^{\left(l\right)}={\tilde{Z}}^{\left(l\right)}+\mathcal{F}\left(\mathrm{LN}\left({\tilde{Z}}^{\left(l\right)}\right)\right)$$

(29)

where LN (⋅) denotes layer normalization, and $\mathcal{F}$ (⋅) denotes the nonlinear mapping module. After passing through multiple encoder layers, the strain-field tokens evolve into a global latent representation that encodes both local deformation signatures and their cross-region interactions, thereby providing the feature basis for reconstructing the pore distribution of the concrete cross-section.

In the standard transformer, $\mathcal{F}$ (⋅) is usually implemented as a feed-forward nonlinear mapping. In this study, this stage is further enhanced by a KAN-based strategy to improve the representation and fitting capability for the inverse mapping from strain fields to pore morphology. The detailed formulation is introduced in Section 3.2.2.

3.2.2. KANs-Enhanced Nonlinear Mapping

Although the transformer backbone can effectively establish long-range interactions among patch-level strain tokens, the inverse mapping from three-component strain fields to pore distribution still requires strong nonlinear feature transformation after attention-based aggregation. This is because the pore signature is not represented by a single strain component or a simple local response pattern. Instead, it is embedded in the coupled spatial variations in ε_xx, ε_xy, and ε_yy, together with their cross-region correlations under freeze–heaving-induced deterioration. In other words, the target pore morphology is implicitly encoded in a highly nonlinear response manifold rather than in a direct pixel-to-pixel correspondence. In essence, the present task involves recovering internal structural information from external mechanical response, which is generally more nonlinear and less explicit than forward field prediction. Therefore, the nonlinear mapping stage inside the encoder plays a crucial role in translating the globally aggregated strain features into pore-relevant latent representations.

In a standard transformer encoder, this stage is usually implemented by a feed-forward network. For an input feature vector z, the conventional nonlinear mapping can be expressed as Equation (30).

$${\mathcal{F}}_{\mathrm{FFN}}\left(z\right)=W_2\sigma \left(W_{1}z+b_1\right)+b_2$$

(30)

where W₁ and W₂ are learnable weight matrices, b₁ and b₂ are bias vectors, and σ(⋅) denotes the activation function. Although this formulation is effective for general-purpose representation learning, its expressive flexibility can be limited when handling the present inverse problem, in which pore distribution must be inferred from complex couplings among multiple strain components, local response anomalies, and nonlocal deformation redistribution.

To enhance the nonlinear approximation capability of the encoder, a KAN-based mapping strategy is introduced to replace or strengthen the conventional feed-forward transformation [47,48]. Instead of relying only on fixed linear projections followed by pointwise activation, KAN constructs the feature transformation through learnable functional mappings, enabling a more adaptive description of complex hidden relationships. In the present task, this property is desirable because the same pore-related structural feature may manifest differently in the three strain fields depending on its surrounding context, while different pore configurations may produce partially similar local strain patterns. A more flexible nonlinear mapping is therefore required to distinguish subtle but physically meaningful variations in the strain-response space.

Accordingly, the nonlinear transformation in the l-th encoder layer is written as Equation (31).

$${\mathcal{F}}_{\mathrm{KAN}}\left(z\right)=\left[{\mathrm{\Phi }}_1\left(z\right),{\mathrm{\Phi }}_2\left(z\right),\dots ,{\mathrm{\Phi }}_{d_{out}}\left(z\right)\right]$$

(31)

where Φ_j (⋅) denotes the j-th learnable nonlinear mapping function. In a component-wise form, each output channel can be expressed as Equation (32).

$${\mathrm{\Phi }}_j\left(z\right)={\displaystyle {\sum }_{i=1}^{d_{in}}{\varphi }_{j,i}\left(z_i\right)}$$

(32)

where ϕ_j,i (⋅) represents a learnable one-dimensional nonlinear function acting on the i-th input component. Compared with the conventional feed-forward structure, this formulation provides a more flexible mechanism for fitting high-order and non-explicit relationships among the encoded strain features.

In the present architecture, the KAN-enhanced nonlinear mapping is not used as an independent feature extractor but serves as a token-wise nonlinear feature translator after attention-based aggregation. The self-attention module first establishes global interactions among strain-field patches, while the KAN module further transforms these globally aggregated tokens into pore-sensitive latent features. This design is important for the present inverse problem because pore locations are not determined by isolated strain magnitudes alone. Similar local strain concentrations may correspond to different pore sizes, pore spacings, or neighboring pore interactions, whereas weak strain disturbances may still indicate pore boundaries. The learnable nonlinear functions in KAN provide adaptive channel-wise transformations of the encoded strain features, allowing the network to distinguish subtle pore-related patterns that may be difficult to separate using fixed activation functions in a conventional feed-forward network. Therefore, the KAN-enhanced mapping acts as a nonlinear bridge between the strain-response representation and the final pore-morphology reconstruction.

With the KAN enhancement, the layer-wise update in the transformer encoder becomes

$$Z^{\left(l\right)}={\tilde{Z}}^{\left(l\right)}+{\mathcal{F}}_{\mathrm{KAN}}\left(\mathrm{LN}\left({\tilde{Z}}^{\left(l\right)}\right)\right)$$

(33)

In this way, the attention mechanism is responsible for capturing the global interaction structure of the strain field, whereas the KAN-enhanced nonlinear mapping further refines the latent representation so that complex pore-related response patterns can be more effectively separated and reconstructed. This design is particularly beneficial for identifying pore distribution from strain fields of porous concrete under freeze–heaving conditions, where the inverse relationship between deformation response and internal pore morphology is strongly nonlinear and spatially coupled.

3.2.3. Context Refinement and Pore Reconstruction

In the context of strain-field-based pore reconstruction, the model’s ability to generate accurate pore distributions from full-field strain responses hinges not only on the global relationships between strain components but also on the subtle local features associated with pore boundaries and small-scale deformation [44]. While the ViT backbone is capable of capturing long-range dependencies and providing high-level global representations, it may still overlook important local structural details. To address this, a lightweight context refinement module is employed to enhance the local feature sensitivity of the encoded strain features, which are critical for accurate pore localization.

To preserve local strain-response cues before global token interaction, a lightweight context refinement operation is applied to the input strain fields, as illustrated in the upper part of Figure 6. Let the three-component strain input be denoted by X ∈ ℝ^H×W×3. A compressed global contextual feature is first obtained by Equation (34).

$$F_g={\mathrm{Conv}}_{1\times 1}\left({\Pi }_{avg}\left(X\right)\right)$$

(34)

where Π_avg (⋅) denotes average pooling. The contextual responses along two strip directions are then aggregated as Equation (35).

$$F_c={\mathrm{DWConv}}_{1\times k}\left(F_g\right)+{\mathrm{DWConv}}_{k\times 1}\left(F_g\right)$$

(35)

And the corresponding attention map is generated by Equation (36).

$$A={\sigma }_s\left({\mathrm{Conv}}_{1\times 1}\left(F_c\right)\right)$$

(36)

where σ_s(⋅) denotes the Sigmoid function. The refined strain input is thus written as Equation (37).

$$X_r=A\odot X$$

(37)

where $\odot$ denotes element-wise multiplication, and A is broadcast along the channel dimension when necessary.

Through this operation, the strain-response regions that are more informative for pore localization are enhanced before entering the transformer backbone. After the ViT encoding and the KAN-enhanced nonlinear mapping, the latent token sequence Z^(l) is converted back to the image space for pore reconstruction, as shown in the lower part of Figure 6. The encoded tokens are first rearranged as Equation (38),

$$F_0=\mathrm{Re}\mathrm{shape}\left(Z^{\left(l\right)}\right)$$

(38)

and then progressively recovered through a series of learnable upsampling modules:

$$\begin{array}{cc}{F}_{s+1}={\mathcal{U}}_s\left(F_s\right),& s=0,1,\dots ,S-1\end{array}$$

(39)

where ${\mathcal{U}}_s$ (⋅) denotes the upsampling function. Based on the final reconstructed feature map, the pore-likelihood field is obtained by Equation (40).

$$\widehat{Y}=\sigma \left({\mathrm{Conv}}_{1\times 1}\left(F_s\right)\right)$$

(40)

The corresponding binary pore map is then determined by thresholding:

$$\widehat{{\rm M}}\left(u,v\right)=\left\{\begin{array}{cc}1,& \widehat{Y}\left(u,v\right)\ge \tau \hfill \\ 0,& \widehat{Y}\left(u,v\right)<\tau \hfill \end{array}\right.$$

(41)

where $\widehat{Y}$ denotes the predicted pore probability field, $\widehat{{\rm M}}$ is the reconstructed pore-distribution map, and τ is the predefined threshold. In this way, the encoded three-component strain fields are finally mapped to the pore morphology of the concrete cross-section.

3.2.4. Training Strategy and Loss Function

For network training, the input to the model is a three-channel strain-field tensor, whereas the target output is a single-channel pore mask. In the implemented dataset pipeline, each sample is represented as X ∈ ℝ^H×W×3 and Y ∈ ℝ^H×W×1, where the three input channels correspond to the strain components ε_xx, ε_xy, and ε_yy, respectively, and the output mask indicates the pore region of the concrete cross-section. Before training, channel-wise statistics were computed from the training set only, and each strain component was globally standardized using the corresponding mean and standard deviation [49,50]. To reduce the influence of extreme values, the normalized strain fields were further clipped to a limited range. This preprocessing strategy helps stabilize optimization and improves the numerical consistency of the three strain channels [51].

The proposed network was initialized from pretrained ViT weights [52]. During weight loading, the classification-related parameters in the original pretrained model were removed, and the remaining parameters were transferred to the present pore-reconstruction framework by non-strict matching [53,54]. This strategy allows the model to inherit generic visual representations from large-scale pretraining while adapting the output head to the strain-field-to-pore inversion task. Training and test samples were organized through separate data loaders, with shuffled mini-batches for training and deterministic ordering for evaluation. In this study, the batch size was set to 6.

To optimize the network, the Smooth L1 loss was adopted as the training objective [55,56]:

$$L_{\mathrm{SmoothL}1}={\displaystyle \frac{1}{N}}{\displaystyle {\sum }_{i=1}^{N}l\left({\widehat{y}}_i-y_i\right)}$$

(42)

where ${\widehat{y}}_i$ and y_i denote the predicted and target pore responses at the i-th pixel, respectively, and l(⋅) is the Smooth L1 penalty. Compared with a purely quadratic loss, this formulation is less sensitive to local outliers while still preserving stable gradient propagation, which is beneficial for the present inverse prediction problem. Although alternative segmentation-oriented losses such as weighted regression loss and BCE–Dice combinations were also explored in the code implementation, the final training reported in this study was conducted using Smooth L1 loss.

The network parameters were optimized using stochastic gradient descent (SGD) with momentum. The initial learning rate was set to 0.01, the momentum coefficient was 0.9, and the weight decay was 5 × 10⁻⁴. The model was trained for 1000 epochs. A cosine-decay learning-rate schedule was employed during the training process, with the minimum learning-rate ratio controlled by l_rf = 0.01 [57,58]. Accordingly, the learning rate at epoch t can be written as

$${\eta }_t={\eta }_0\left[\left({\displaystyle \frac{1+\cos \left(\pi t/T\right)}{2}}\right)\left(1-l_{rf}\right)+l_{rf}\right]$$

(43)

where η₀ is the initial learning rate, T is the total number of epochs, and l_rf is the lower-bound ratio of the cosine schedule. A warm-up schedule was also defined in the implementation, although the effective epoch-wise update in the final loop was governed by the cosine scheduler.

During training and evaluation, the prediction quality was monitored using the loss value together with mean squared error (MSE) [59], root mean squared error (RMSE) [60], mean absolute error (MAE) [61], and the coefficient of determination R² [62,63]. These metrics were computed by comparing the flattened predicted pore field with the flattened target mask over the whole batch or the whole test set. The best model was selected according to the minimum test MSE and saved as the final checkpoint. In addition, intermediate checkpoints were also recorded when the test R² exceeded predefined thresholds, which facilitated the inspection of model evolution during training.

4. Experimental Results and Discussion

4.1. Training Convergence Analysis

Figure 7 compares the test-set performance evolution of the proposed model, its variant without KAN, and three baseline models, including U-Net [46], Res-U-Net, and a CNN encoder–decoder [64]. The MSE and R² curves are shown in Figure 7a and Figure 7b, respectively. Overall, the proposed model achieves the best convergence behavior, with the lowest MSE and the highest R² among all models. Its best test MSE reaches 0.0152, while the corresponding best R² reaches 0.8565. In comparison, the model without KAN obtains a best MSE of 0.0193 and a best R² of 0.8173, indicating that the introduction of KAN further reduces the MSE by approximately 21.2% and improves R² by 0.0392. Compared with the strongest conventional baseline, namely U-Net, the proposed model reduces the MSE from 0.0284 to 0.0152 and improves R² from 0.7317 to 0.8565. Res-U-Net and the CNN encoder–decoder show weaker predictive performance, with best MSE values of 0.0566 and 0.1066 and best R² values of 0.4650 and −0.0073, respectively. These results demonstrate that the proposed architecture provides more accurate pore reconstruction than conventional convolutional baselines, while the comparison with the model without KAN further confirms the contribution of the KAN-enhanced nonlinear mapping module.

The improved convergence behavior can be attributed to the complementary roles of self-attention and KAN-enhanced nonlinear mapping. The self-attention mechanism captures long-range spatial coupling among strain-field patches, but the subsequent nonlinear transformation is still required to convert the aggregated strain features into pore-discriminative representations. In the model without KAN, this transformation is mainly achieved by conventional linear layers and fixed activation functions, which may be less flexible for representing the highly nonlinear inverse relationship between strain redistribution and pore morphology. By introducing learnable nonlinear functions, the KAN-enhanced mapping provides a more adaptive feature transformation and improves the separation of pore-related and non-pore-related latent patterns. This helps stabilize the optimization process and leads to better reconstruction accuracy, as evidenced by the lower test MSE and higher R² compared with the model without KAN.

4.2. Threshold Sensitivity Analysis

Since the proposed network outputs a continuous pore-likelihood field, an additional thresholding step is required to obtain the final binary pore-distribution map. To determine an appropriate threshold for pore reconstruction, a threshold sensitivity analysis was conducted based on four standard segmentation metrics, namely intersection over union (IoU) [65,66], Dice coefficient, precision, and recall. These metrics were computed from the numbers of true positives (TP), false positives (FP), true negatives (TN), and false negatives (FN) [67] and are defined as

$$\mathrm{IoU}={\displaystyle \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}+\mathrm{FN}}}$$

(44)

$$\mathrm{Dice}={\displaystyle \frac{2\mathrm{TP}}{2\mathrm{TP}+\mathrm{FP}+\mathrm{FN}}}$$

(45)

$$\mathrm{Precision}={\displaystyle \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}}}$$

(46)

$$\mathrm{Recall}={\displaystyle \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}}}$$

(47)

The sensitivity of pore segmentation performance to the probability threshold is illustrated in Figure 8. As the threshold increases, precision gradually improves, whereas recall decreases continuously, indicating a clear trade-off between false positive suppression and false negative increase. In contrast, IoU and Dice first increase and then decrease, reaching their peak values at intermediate thresholds. This behavior suggests that excessively low thresholds tend to overestimate pore regions, while overly high thresholds lead to missed pore predictions. A threshold of 0.50 was therefore selected as a balanced operating point, at which IoU and Dice remain close to their optimal values while precision and recall are reasonably well balanced.

It should be noted that the threshold of 0.50 is selected as an appropriate operating point for the present simulation-based dataset rather than a universally fixed value. As shown in Figure 8, IoU and Dice remain relatively stable within an intermediate threshold range of around 0.40–0.60, indicating that the segmentation performance is not highly sensitive to small deviations from 0.50 in the current porosity range. However, the optimal threshold may change when the pore morphology, porosity range, or noise level differs from those of the present dataset. For example, higher noise levels in experimentally measured strain fields may increase false-positive responses and therefore require a higher threshold, whereas datasets containing smaller or less distinct pores may require a lower threshold to avoid missed detections. Therefore, for applications to other datasets, especially experimental strain-field data, the threshold should be recalibrated using a validation set by jointly considering IoU, Dice, precision, recall, and porosity error.

4.3. Quantitative Evaluation of Pore Prediction Performance

To further quantify the pore reconstruction performance, the prediction results were additionally evaluated using accuracy, F1 score, and porosity absolute error. The accuracy is defined as

$$\mathrm{Accuracy}={\displaystyle \frac{\mathrm{TP}+\mathrm{TN}}{\mathrm{TP}+\mathrm{TN}+\mathrm{FP}+\mathrm{FN}}}$$

(48)

which measures the overall proportion of correctly classified pore and non-pore pixels. The F1 score is written as

$$\mathrm{F}1={\displaystyle \frac{2\mathrm{TP}}{2\mathrm{TP}+\mathrm{FP}+\mathrm{FN}}}$$

(49)

which balances precision and recall in binary pore identification. For the present binary prediction task, this metric is equivalent to the Dice coefficient. In addition, the porosity absolute error is defined as

$$e_p=\left|p_{pred}-p_{gt}\right|$$

(50)

where p_pred and p_gt denote the predicted and ground-truth porosity, respectively. This metric reflects the deviation of the reconstructed pore content from the reference value.

Based on the above evaluation metrics, Figure 9 compares the distributions of IoU, accuracy, F1 score, and porosity absolute error between the training and test sets. The mean IoU decreases from 0.8673 on the training set to 0.8550 on the test set, while the mean F1 score decreases from 0.9288 to 0.9217. A similar but less pronounced trend is observed for accuracy, whose mean value slightly drops from 0.9826 to 0.9808. In contrast, the mean porosity absolute error increases only marginally from 0.0038 for the training set to 0.0041 for the test set. Although the test-set performance is consistently lower than that on the training set, the overall shift remains limited, and the distributions still exhibit substantial overlap. This indicates that the proposed model maintains favorable pore prediction performance on unseen samples and demonstrates acceptable generalization capability.

The slight differences between the training and test distributions do not necessarily indicate pronounced overfitting. Instead, they reflect a mild generalization gap caused by the increased uncertainty of pore reconstruction on unseen samples. As shown in Figure 9, the distributions of the training and test metrics still exhibit substantial overlap, and the decreases in the mean values are limited. The mean IoU decreases by 0.0123, the mean F1 score decreases by 0.0071, and the mean accuracy decreases by only 0.0018 from the training set to the test set, while the porosity absolute error increases only slightly. These results suggest that the model does not simply memorize the training samples but retains stable predictive capability on unseen pore configurations. Nevertheless, the slight performance reduction may become more evident for high-porosity samples, where denser pore distributions, narrower inter-pore spacing, and stronger pore interactions make boundary localization more difficult. Therefore, the observed discrepancy is more appropriately interpreted as a limited generalization gap associated with increased morphological complexity, rather than severe overfitting.

Figure 10 provides a comprehensive comparison of segmentation performance and porosity prediction between the training and test sets. Here, a negative Δμ indicates lower performance on the test set than on the training set, whereas a positive value for porosity error indicates a slight increase in prediction error on unseen samples. As shown in Figure 10a, the mean IoU, accuracy, and F1 score decrease from 0.8673, 0.9826, and 0.9288 on the training set to 0.8550, 0.9808, and 0.9217 on the test set, respectively. Despite this reduction, the test-set metrics remain at relatively high levels, indicating that the proposed model preserves favorable segmentation capability on unseen samples. Figure 10b further shows that the mean porosity absolute error increases only slightly from 0.00384 for the training set to 0.00408 for the test set, suggesting stable porosity prediction accuracy.

The generalization gap is summarized in Figure 10c, where the largest decrease is observed for IoU (−0.0123), followed by recall (−0.0080), F1 score (−0.0072), and precision (−0.0063), whereas the drop in accuracy is comparatively small (−0.0018). Meanwhile, the porosity error increases by only 0.0002 from training to testing. These results indicate that the performance degradation on unseen samples remains moderate overall. In addition, Figure 10d shows that the predicted porosity agrees closely with the ground-truth values for both datasets, with most points distributed near the diagonal line. This confirms that the proposed model maintains good calibration and satisfactory generalization performance in porosity-related prediction.

Figure 11 shows the variation in prediction performance with ground-truth porosity for both the training and test sets. As the porosity increases from 0.08 to 0.16, all three metrics exhibit a decreasing trend, indicating that samples with higher porosity are more challenging for accurate prediction. Specifically, the mean IoU on the training set decreases from 0.888 to 0.840, whereas the corresponding test-set value drops from 0.879 to 0.823. Similarly, the mean accuracy decreases from 0.991 to 0.973 for the training set and from 0.990 to 0.969 for the test set. The mean F1 score also declines from 0.941 to 0.913 on the training set and from 0.936 to 0.903 on the test set. Overall, the test-set curves remain consistently below the training-set curves, but both exhibit similar downward trends, suggesting that the influence of porosity on prediction difficulty is captured consistently across datasets. Moreover, the gap between the training and test curves becomes slightly larger at higher porosity levels, particularly for IoU and F1 score, suggesting that highly porous samples impose greater generalization challenges on the model.

4.4. Representative Visualization of Reconstructed Pore Distributions

Figure 12 presents representative visual comparisons of pore prediction results for samples with different porosity levels ranging from 8% to 16%. For all cases, the predicted probability maps clearly highlight the pore regions, and the thresholded predictions reproduce the main spatial distribution patterns of the ground-truth pores. The boundary overlays show generally good agreement between the predicted and reference contours, while the pixel-wise error maps indicate that most pixels are correctly classified. Notably, the residual errors are mainly concentrated around pore boundaries and in locally dense pore regions, rather than in the interiors of pores or the matrix. This suggests that the model captures the overall pore morphology well, whereas slight deviations mainly arise from boundary localization. As the porosity increases, the pore distribution becomes denser and the inter-pore spacing becomes narrower, which makes boundary delineation more challenging and leads to a gradual reduction in prediction accuracy. Quantitatively, the ground-truth porosity increases from 7.99% to 16.02%, whereas the corresponding predicted porosity changes from 7.89% to 15.53%, and the accuracy decreases from 98.7% to 95.5%. These observations indicate that the proposed model maintains robust pore prediction capability over a broad porosity range, although higher-porosity samples impose greater challenges for fine-scale boundary reconstruction.

4.5. Limitations and Future Work

Although the proposed framework demonstrates promising performance for strain-field-based pore reconstruction, several limitations should be acknowledged. First, the dataset used in this study is still limited in scale. A total of 500 numerical samples were generated from five prescribed porosity levels, which is sufficient for demonstrating the feasibility of the proposed inverse framework but may not fully cover the variability in pore morphology in real concrete. In particular, the current dataset mainly considers randomly distributed circular pores within a two-dimensional concrete plate, while actual concrete may contain more complex pore shapes, connected voids, microcracks, aggregates, and interfacial transition zones. Therefore, expanding the dataset to include more diverse pore geometries, porosity ranges, and material heterogeneity will be important for improving model robustness.

Second, training and evaluation in the present study are based on simulation-generated data. The finite element model provides controlled and physically consistent strain fields, which are useful for establishing the initial mapping between pore structure and deformation response. However, simulation-based data inevitably involve idealizations in geometry, material behavior, boundary conditions, and frost-heaving loading. In practical measurements, strain fields may be affected by experimental noise, spatial resolution, boundary uncertainty, and environmental variability. These factors may introduce a domain gap between numerical simulations and real experimental observations.

Third, future work should further validate the proposed method using experimentally measured strain-field data. For example, full-field strain measurements obtained from digital image correlation or other non-contact measurement techniques can be combined with microscopic imaging or X-ray computed tomography to provide reference pore information. Such experimental validation would help assess the transferability of the trained model from simulated data to real concrete specimens. In addition, incorporating noise augmentation, uncertainty quantification, and transfer learning may further improve the reliability of the framework for practical non-destructive characterization of freeze–heaving-damaged concrete.

5. Conclusions

This study proposes a deep learning-based framework for pore distribution prediction in concrete cross-sections subjected to freeze–heaving conditions. The task involves predicting internal pore morphology from strain-field data, which is an inverse problem with complex nonlinear and spatially dependent relationships. The proposed framework leverages a Vision Transformer backbone and KAN-enhanced nonlinear mapping to effectively model these relationships.

• A mechanics-based frost-heaving model was established by considering pore-scale elastoplastic response, drainage-induced attenuation, and mechanical admissibility. The model explains why realistic frost-heaving pressure in porous concrete remains much lower than the fully confined freezing pressure and provides the physical basis for generating strain-field data with randomly distributed pores.
• A strain-field-to-pore inversion framework was constructed using an enhanced Vision Transformer with KAN-enhanced nonlinear mapping. The three strain components, ε_xx, ε_yy, and ε_xy, provide mechanically informative inputs for pore reconstruction, while the combination of self-attention and KAN improves the representation of the nonlinear relationship between strain redistribution and pore morphology.
• The proposed model achieved the best test MSE of 0.0152 and R² of 0.8565, outperforming the model without KAN and the convolutional baselines. Compared with the model without KAN, the proposed model reduced the test MSE from 0.0193 to 0.0152 and improved R² from 0.8173 to 0.8565, confirming the contribution of the KAN-enhanced nonlinear mapping.
• Under the selected threshold of 0.50, the proposed model achieved a mean IoU of 0.8550, accuracy of 0.9808, F1 score of 0.9217, and porosity absolute error of 0.00408 on the test set. The train–test differences remained limited, indicating satisfactory generalization capability. Prediction errors were mainly concentrated near pore boundaries and locally dense pore regions, especially for high-porosity samples.

Future work should focus on expanding the dataset, introducing more realistic pore geometries and material heterogeneity, and validating the proposed framework using experimentally measured strain-field data.