A method and system for predicting performance and optimizing mix proportion of engineered cementitious composites based on physical guided deep learning and multi-objective optimization

Abstract

This invention belongs to the field of engineering cement-based composite materials technology, and discloses a method and system for performance prediction and mix proportion optimization of engineering cement-based composite materials based on physics-guided deep learning and multi-objective optimization. The method includes: acquiring tensile test data of engineering cement-based composite materials; constructing a physics-guided convolutional neural network model PhyResNet; using the physics-guided convolutional neural network model PhyResNet to predict the performance of the tensile test data of engineering cement-based composite materials; and using a multi-objective optimization method based on NSGA-II to optimize the mix proportion of the predicted performance.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of engineered cementitious composites, and particularly relates to an engineered cementitious composite performance prediction and mix proportion optimization method and system based on physical guidance deep learning and multi-objective optimization. BACKGROUND

[0002] Engineered Cementitious Composites (ECC) is a new type of high-performance fiber-reinforced cementitious composite material. It has excellent tensile properties, multiple micro-crack characteristics and remarkable durability in harsh environments, and has attracted much attention. Therefore, ECC has become a very promising concrete material in the application of bridges, tunnels, subways and anti-seismic buildings, which have very high requirements for structural safety and durability of infrastructure.

[0003] Due to the complex material composition and variable characteristics of ECC, it is difficult for traditional regression analysis methods to effectively establish the relationship between the mechanical properties and the mix proportion, and the design method mainly depends on the micro-mechanical theory, which needs to optimize the interaction of fibers, matrix and interface to improve the mechanical properties of the material. ECC usually needs to go through several links to obtain its mechanical properties in several months from the theoretical proposal to performance confirmation.

[0004] Machine Learning (ML) can effectively establish the non-linear mapping of mix proportion-strength under sufficient samples, but shallow models such as decision tree and random forest are extremely sensitive to the high dispersion of ECC ultimate tensile strain and experimental noise, and are prone to overfitting, with low reliability of engineering extrapolation. Deep learning has been rapidly applied in ECC mechanical performance prediction due to its multi-layer nonlinear transformation capability. The ability of BP network to capture non-linear mapping has been verified; Shi et al. further used BPNN to predict compressive strength, tensile strength and elastic modulus simultaneously, and verified the model generalization performance; Malik et al. used multi-layer ANN to reconstruct the full tensile stress-strain curve of ECC, and verified the advantage of deep network in mining complex constitutive relationship. However, deep models rely on a large amount of high-quality data to suppress overfitting, Huynh et al. pointed out that small samples significantly weaken the generalization ability, and how to maintain high precision and robustness in the case of data scarcity is still a problem; in addition, the network does not embed the physical constraints of fracture energy and pseudo-strain hardening, and the extrapolation results often violate the laws of mechanics, and there is still a lack of robust and explainable prediction mechanism for the ultimate tensile strain with greater dispersion.

[0005] In recent years, physics-guided AI methods have made significant progress, becoming a key path to improve model interpretability, generalization ability, and engineering applicability. The Physical Information Neural Networks (PINNs) proposed by Raissi et al., by embedding the residuals of partial differential equations (PDEs) into the loss function, achieve accurate modeling of complex physical processes under small sample or no data conditions, significantly outperforming traditional pure data-driven models. Building upon this, Zhang et al. developed RheologyNet, embedding the rheological PDEs of cement-based materials into the network structure, simultaneously improving thixotropic prediction accuracy and significantly reducing the computational cost of finite element methods, demonstrating the powerful potential of PINNs in micro-macro coupled modeling of materials. Furthermore, the Physically Supervised Ensemble Learning (PELM) model proposed by Yu et al. effectively alleviates the overfitting problem of traditional machine learning models in the recognition of failure modes of reinforced concrete columns by introducing physical constraints on the hyperparameter optimization process. Han et al., in predicting the cracking performance of recycled aggregate concrete, employed physical mechanisms to guide feature selection, significantly enhancing the physical consistency and predictive reliability of the model. These studies demonstrate that physics-guided AI methods not only inherit the nonlinear fitting capabilities of data-driven models, but also significantly improve the engineering reliability and generalization performance of models by integrating mechanical principles, constitutive relations, and experimental knowledge.

[0006] Existing data-driven models are prone to overfitting when predicting the ultimate tensile strain of highly discrete, small samples, and lack physical constraints, resulting in predictions that violate the principles of fracture mechanics. At the same time, the black-box nature of the model hinders the optimization of mix proportions. Summary of the Invention

[0007] To address the problems existing in the prior art, this invention provides a method and system for performance prediction and mix proportion optimization of engineering cement-based composite materials based on physics-guided deep learning and multi-objective optimization. A physics-guided convolutional neural network model, PhyResNet, is constructed, and ECC micromechanical constraints (fracture energy, pseudo-strain hardening) are embedded into the CNN loss function to achieve high-precision, robust, and interpretable predictions under small sample conditions.

[0008] To achieve the above objectives, the present invention provides the following solution:

[0009] A method for performance prediction and mix proportion optimization of engineering cement-based composite materials based on physics-guided deep learning and multi-objective optimization, the method comprising:

[0010] Obtain tensile test data for engineering cement-based composite materials;

[0011] Construct a physics-guided convolutional neural network model, PhyResNet;

[0012] The PhyResNet model, a physics-guided convolutional neural network, is used to predict the performance of tensile test data of engineering cement-based composite materials.

[0013] The predicted performance was optimized using a multi-objective optimization method based on NSGA-II.

[0014] Preferably, the PhyResNet physical-guided convolutional neural network model organically integrates theoretical models and data-driven models;

[0015] The theoretical models include: tensile strength model and ultimate tensile strain model;

[0016] The data-driven model is a data-driven prediction model based on convolutional neural networks (CNNs).

[0017] The preferred expression for the tensile strength model is:

[0018] ;

[0019] ;

[0020] ;

[0021] ;

[0022] Where UTS represents the tensile strength of the composite material, Indicates matrix strength. , , These represent fiber volume fraction, length, and diameter, respectively, with AR indicating the fiber aspect ratio. The nonlinear coefficient characterizes the change in load transfer efficiency with aspect ratio; The dynamic coefficient is a nonlinear correction factor related to the matrix composition; , , These are empirical parameters obtained through fitting via material experiments. For interface coefficients, For the weight vector, is the input variable, and b is the bias term.

[0023] The preferred expression for the ultimate tensile strain model is:

[0024] ;

[0025] Where W represents the amount of water used. , , These represent fiber volume, fiber length, and fiber diameter, respectively; A, B, C, D, and E are empirical parameters obtained through material testing.

[0026] Preferably, data-driven prediction models based on convolutional neural networks (CNNs) include:

[0027] A rectified linear unit is typically used as the activation function after a convolutional layer.

[0028] A pooling layer is introduced to compress the spatial dimension of the feature map through downsampling operations;

[0029] The max pooling strategy is adopted, that is, the maximum value is selected from each local region as the feature representation of that region;

[0030] At the back end of the network, the features extracted through multiple convolutions and pooling are flattened into a one-dimensional vector and input into a fully connected layer to achieve the synthesis and mapping of high-level features.

[0031] In the regression task, the last fully connected layer is set as a single output node, and continuous predicted values ​​are output through an activation function to achieve quantitative prediction of the mechanical properties of engineering cement-based composite materials.

[0032] The present invention also provides a system for performance prediction and mix proportion optimization of engineering cement-based composite materials based on physics-guided deep learning and multi-objective optimization. The system is used to implement the aforementioned method and includes: an acquisition module, a construction module, a prediction module, and an optimization module.

[0033] The acquisition module is used to acquire tensile test data of engineering cement-based composite materials;

[0034] The building module is used to build the PhyResNet convolutional neural network model based on physical guidance;

[0035] The prediction module is used to predict the performance of tensile test data of engineering cement-based composite materials using the PhyResNet physics-guided convolutional neural network model.

[0036] The optimization module is used to optimize the mix ratio of the predicted performance using a multi-objective optimization method based on NSGA-II.

[0037] Preferably, the PhyResNet physical-guided convolutional neural network model organically integrates theoretical models and data-driven models;

[0038] The theoretical models include: tensile strength model and ultimate tensile strain model;

[0039] The data-driven model is a data-driven prediction model based on convolutional neural networks (CNNs).

[0040] The preferred expression for the tensile strength model is:

[0041] ;

[0042] ;

[0043] ;

[0044] ;

[0045] Where UTS represents the tensile strength of the composite material, Indicates matrix strength. , , These represent fiber volume fraction, length, and diameter, respectively, with AR indicating the fiber aspect ratio. The nonlinear coefficient characterizes the change in load transfer efficiency with aspect ratio; The dynamic coefficient is a nonlinear correction factor related to the matrix composition; , , These are empirical parameters obtained through fitting via material experiments. For interface coefficients, For the weight vector, is the input variable, and b is the bias term.

[0046] The preferred expression for the ultimate tensile strain model is:

[0047] ;

[0048] Where W represents the amount of water used. , , These represent fiber volume, fiber length, and fiber diameter, respectively; A, B, C, D, and E are empirical parameters obtained through material testing.

[0049] Preferably, data-driven prediction models based on convolutional neural networks (CNNs) include:

[0050] A rectified linear unit is typically used as the activation function after a convolutional layer.

[0051] A pooling layer is introduced to compress the spatial dimension of the feature map through downsampling operations;

[0052] The max pooling strategy is adopted, that is, the maximum value is selected from each local region as the feature representation of that region;

[0053] At the back end of the network, the features extracted through multiple convolutions and pooling are flattened into a one-dimensional vector and input into a fully connected layer to achieve the synthesis and mapping of high-level features.

[0054] In the regression task, the last fully connected layer is set as a single output node, and continuous predicted values ​​are output through an activation function to achieve quantitative prediction of the mechanical properties of engineering cement-based composite materials.

[0055] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0056] This invention proposes a physics-guided PhyResNet framework that embeds ECC micromechanical constraints into the CNN loss function, enabling effective and rapid convergence with a small number of samples while maintaining high accuracy. Combined with interpretability analysis, it reveals the coupling effect mechanism between water-glue ratio and fiber volume fraction, achieving high accuracy, strong robustness, and interpretable prediction under small data conditions. Attached Figure Description

[0057] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0058] Figure 1 This is a schematic diagram illustrating the distribution of features in the dataset according to an embodiment of the present invention;

[0059] Figure 2 This is a correlation heatmap of features in the dataset of an embodiment of the present invention;

[0060] Figure 3 This is a comparison chart of ablation experiments of the model in the embodiments of the present invention, wherein (a) the average value of the UTS nine-fold cross-validation is... Figure (b) shows that the highest cross-validation result for UTS (90% cross-validation) is... Figure; (c) UTSn ninefold cross-validation average Figure (d) shows the highest cross-validation result for UTSn (90% cross-validation). picture;

[0061] Figure 4 The bar chart shows the performance comparison of embodiments of the present invention, where (a) tensile strength and (b) ultimate tensile strain.

[0062] Figure 5 The test sets for each model under different small sample ratios in the embodiments of the present invention A bar chart, where (a) tensile strength; (b) ultimate tensile strain;

[0063] Figure 6This is the test set for each model under different noise injection ratios in the embodiments of the present invention. A bar chart, where (a) tensile strength; (b) ultimate tensile strain;

[0064] Figure 7 The diagram shows the SHAP analysis of an embodiment of the present invention, where (a) tensile strength and (b) ultimate tensile strain.

[0065] Figure 8 This is a Layercam visualization diagram of an embodiment of the present invention, wherein (a) tensile strength; (b) ultimate tensile strain;

[0066] Figure 9 This is a schematic diagram of a method for predicting the performance and optimizing the mix proportion of engineering cement-based composite materials based on physics-guided deep learning and multi-objective optimization, according to an embodiment of the present invention. Detailed Implementation

[0067] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0068] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0069] Example 1

[0070] Engineering cement-based composites (ECCs) have shown broad application prospects in critical infrastructure such as bridges, tunnels, and earthquake-resistant buildings due to their high ductility and multi-crack characteristics. However, their mechanical properties are affected by the coupling of multiple parameters in the mix proportion, making it difficult for traditional regression methods to establish accurate mappings. Furthermore, purely data-driven machine learning models are prone to overfitting when faced with small samples and highly discrete ultimate tensile strains, and lack physical constraints, leading to predictions that violate the laws of fracture mechanics. To address these issues, such as... Figure 9 As shown, this invention provides a method for performance prediction and mix proportion optimization of engineering cement-based composite materials based on physics-guided deep learning and multi-objective optimization. The method includes:

[0071] Obtain tensile test data for engineering cement-based composite materials;

[0072] Construct a physics-guided convolutional neural network model, PhyResNet;

[0073] The PhyResNet model, a physics-guided convolutional neural network, is used to predict the performance of tensile test data of engineering cement-based composite materials.

[0074] The predicted performance was optimized using a multi-objective optimization method based on NSGA-II.

[0075] The specific implementation process is as follows:

[0076] To improve the physical consistency and generalization ability of prediction models under conditions of data scarcity, this invention first constructs two theoretical models based on physical mechanisms to predict the tensile strength (UTS) and ultimate tensile strain (UTSn) of composite materials, respectively. By combining the microscopic reaction mechanism of cement-based composite materials with fiber bridging behavior, these theoretical models provide physical prior constraints for subsequent data-driven networks, thereby improving the reliability and interpretability of predictions.

[0077] Tensile Strength Model: Matrix strength is fundamental to the tensile properties of composite materials. This invention proposes a modified water-cement ratio formula based on Abram's law to more accurately predict the matrix strength of multi-component cementitious systems. The calculation formula can be expressed as:

[0078] ;

[0079] in, The matrix strength is indicated by W, the amount of water used is indicated by C, FA, GGBFS, LP, and SF, which represent the amounts of cement, fly ash, blast furnace slag, lime powder, and silica fume, respectively. , , , represents the equivalence coefficients for cement, fly ash, blast furnace slag, and lime powder, respectively, which are empirical parameters obtained through material testing and fitting. , These are empirical parameters obtained through material testing. Unlike the traditional water-cement ratio formula, this model introduces the concept of equivalent gelation amount, which can more realistically characterize the degree of hydration reaction in multi-component systems, thereby improving the accuracy of strength prediction and providing reliable input for subsequent fiber bridging stress models.

[0080] Subsequently, a fiber bridging stress model is introduced, considering the reinforcing effect of fiber volume fraction and fiber aspect ratio on tensile strength. The calculation formula can be expressed as:

[0081] ;

[0082] ;

[0083] ;

[0084] ;

[0085] in, Indicates matrix strength. , , These represent fiber volume fraction, length, and diameter, respectively. AR represents the fiber aspect ratio, defined as... ; The nonlinear coefficient characterizes the change in load transfer efficiency with aspect ratio; The interface coefficient reflects the significant effect of interface roughness on the tensile strength of the interface. The dynamic coefficient is a nonlinear correction factor related to the matrix composition; , , These are empirical parameters obtained through fitting via material experiments; For the weight vector, is the input variable, and b is the bias term.

[0086] Ultimate Tensile Strain Model: The second theoretical model is used to predict the ultimate tensile strain UTSn. Based on the UTS prediction model, this model establishes a functional relationship between UTSn and UTS, explicitly considering the combined effects of water-cement ratio, fiber volume fraction, and fiber aspect ratio on ductility. The model ensures that the predicted ultimate tensile strain value increases reasonably with increasing fiber content and exhibits a saturation trend in the high-content range, thus consistent with the actual strain hardening and high ductility characteristics of PE-ECC. The calculation formula can be expressed as:

[0087] ;

[0088] Where W represents the amount of water used. , , , representing fiber volume, fiber length, and fiber diameter, respectively; A, B, C, D, and E are empirical parameters obtained through material testing. This model not only establishes a quantitative relationship between UTS and UTSn, explicitly considering the influence of water-cement ratio, fiber volume fraction, and aspect ratio on ultimate tensile strain, but also reveals the coupling effect of material design parameters on toughness indices, providing theoretical support for performance-driven ECC optimization design.

[0089] The two theoretical models mentioned above not only provide physical priors for subsequent data-driven models, but also serve as physical constraints in the loss function to correct the prediction results of deep learning models.

[0090] After completing the physical theory modeling, this invention uses a convolutional neural network (CNN) as the basic data-driven prediction model to fully explore the potential nonlinear relationship between material mix proportions and mechanical properties.

[0091] Convolutional Neural Networks (CNNs) are a type of feedforward neural network widely used for feature extraction and pattern recognition tasks on structured data. Compared to traditional manual feature engineering, CNNs automatically extract multi-level features through end-to-end learning, effectively avoiding the subjectivity and limitations of manual feature selection. Their unique local connectivity and weight-sharing mechanisms not only significantly reduce the number of network parameters and model complexity but also enhance generalization ability, thus maintaining good predictive performance even with limited data.

[0092] A typical CNN architecture usually consists of an input layer, convolutional layers, non-linear activation layers, pooling layers, and fully connected layers, stacked in multiple layers to form a deep network. The input layer receives raw data and represents it as a two-dimensional or multi-dimensional matrix, facilitating subsequent local feature extraction. The convolutional layer, as the core of the CNN, extracts local features and captures spatial correlations between variables by sliding convolutional kernels (filters) across the input matrix, effectively representing the potential patterns between input and output. Each convolutional kernel shares weights across the entire input space, significantly reducing the number of parameters and improving training efficiency.

[0093] To enhance the nonlinear expressive power of the network, a nonlinear activation function is typically applied after the convolutional layer. This invention selects the Rectified Linear Unit (ReLU) as the primary activation function, which effectively alleviates the vanishing gradient problem and reduces computational overhead while maintaining model expressive power. The subsequent pooling layer compresses the spatial dimension of the feature map through downsampling, preserving the most representative features while reducing redundant information and computational burden. This invention employs a max pooling strategy, selecting the maximum value from each local region as the feature representation of that region, thereby improving the model's translation invariance and noise resistance.

[0094] At the back end of the network, the features extracted through multiple convolutions and pooling are flattened into one-dimensional vectors and input into a fully connected layer to achieve the synthesis and mapping of high-level features. In the regression task of this invention, the last fully connected layer is set as a single output node and outputs continuous predicted values ​​through an appropriate activation function, thereby achieving quantitative prediction of ECC mechanical performance.

[0095] In summary, CNNs, through the synergistic effect of multi-layer convolution, non-linear activation, and pooling, gradually extract high-level feature representations from the original input, laying a solid feature foundation for subsequent hybrid modeling that incorporates physical constraints.

[0096] While purely data-driven models can learn complex nonlinear relationships, they are prone to overfitting under small sample conditions and may produce physically unreasonable predictions. Therefore, this invention proposes a convolutional neural network (PhyResNet) that incorporates physical prior constraints, organically integrating theoretical models with data-driven learning.

[0097] The data-driven branch, a core component of the PhyResNet model, plays a crucial role in automatically learning complex nonlinear relationships from raw feature data. This branch employs a deep convolutional neural network architecture, using multi-level feature extraction and transformation to achieve high-precision prediction of the tensile strength of engineering cement-based composite materials. Its mathematical expression is as follows:

[0098] ;

[0099] in, For input variables, These are the trainable parameters for the CNN branches. Specific network results include:

[0100] 1. Multi-scale feature extraction layer: 1-5 variable convolutional layers are used, each layer includes: (1). Convolution operation: ;in, Let be the convolution kernel weight matrix of the l-th layer. For bias terms, This represents the convolution operation. The output is the activation value of the previous layer. The convolution kernel parameters are automatically determined through Bayesian optimization: the number of filters ∈ [4, 64] and the kernel size ∈ [1, 5], ensuring that the model can adapt to feature patterns of different complexities.

[0101] (2). Nonlinear activation function: A parameterized activation mechanism is adopted, and the ReLU or LeakyReLU function is adaptively selected according to the data characteristics.

[0102] (3) Adaptive Pooling Strategy: After the activation function, max pooling can be optionally performed to reduce the dimensionality of the feature map and increase the translation invariance and robustness of the model. The pooling size is chosen between 2 and 3, and the same padding is used to maintain the feature map size.

[0103] 2. Global Feature Fusion Layer: Global average pooling or flattening operations are used to convert spatiotemporal features into feature vectors of fixed dimensions.

[0104] 3. Fully connected regression layer: Nonlinear transformation is performed through 0-3 fully connected layers to finally output the predicted value.

[0105] The loss function of this model consists of two parts: one part is the prediction error based on the data. The other part is the physical consistency regularization term. This is used to constrain the prediction results from deviating from the reasonable range of the theoretical model. The comprehensive loss function can be expressed as:

[0106] ;

[0107] in, Represents the overall loss function. The mean squared error (MSE) represents the difference between the model's predicted values ​​and the actual experimental values. The mean squared error (MSE) represents the difference between the theoretical and model predictions from a physical model. It is a scalar hyperparameter, ranging from 0 to 1, used to balance data loss terms. and physical loss item The contribution of this paper to the total loss. =0.6.

[0108] Explainable AI Techniques: Deep learning models are often considered "black boxes," making it difficult to explain their decision-making processes. To improve the transparency and engineering usability of models, this invention introduces two complementary explainable artificial intelligence techniques.

[0109] First, to gain a deeper understanding of the influence mechanism of various input features on the tensile properties of engineering cement-based composite materials, such as... Figure 7 As shown, this invention introduces the SHAP (SHapley Additive Explanations) interpretable artificial intelligence method to establish a complete quantitative evaluation system for feature importance. This framework, based on Shapley value theory in cooperative game theory, can accurately quantify the marginal contribution of each input feature to the model's prediction results.

[0110] For each sample's predicted value, SHAP interprets it using the following additive feature attribute model: ;

[0111] Where g is the explanatory model, ∈{0,1}, M represents a binary vector indicating whether a feature exists (1 indicates existence, 0 indicates non-existence), and M is the number of features. ∈R is the Shapley value of feature i, representing the contribution of feature i to the prediction. It is the baseline prediction of the model when there are no feature inputs (usually the average prediction of the training set).

[0112] The formula for calculating the Shapley value is as follows:

[0113] ;

[0114] in, F It is the set of all features. S yes F The middle does not contain features i A subset of. f ( S ) indicates that only a subset of features is used. S The model's predicted value. This formula takes into account features. i Marginal contributions across all possible subsets of features are calculated and then weighted and averaged.

[0115] Secondly, to gain a deeper understanding of the internal decision-making mechanism and feature learning process of the PhyResNet model, this invention employs the Layer-wise Class Activation Mapping (LayerCAM) method to visualize and analyze the activation responses of CNN convolutional layers. LayerCAM generates a high-resolution attention map by calculating the weighted sum of feature gradients at each spatial location, accurately revealing the model's areas of interest across different feature dimensions.

[0116] LayerCAM is an improvement on Gradient Weighted Class Activation Mapping (Grad-CAM), which improves visualization accuracy by calculating activation maps in layers. For target class c, the weight of the k-th feature map at position (i,j) is calculated as follows:

[0117] ;

[0118] in, This indicates the importance of the k-th feature map in predicting category c. This represents the model's predicted score for category c. This represents the activation intensity of the neuron at spatial location (i,j) in the k-th feature map. Z represents the sensitivity of the model output to the activation of the feature map, Z represents the spatial size of the feature map (height × width), and i and j represent the row and column positions in the feature map.

[0119] Through these two methods, researchers can not only verify whether the model conforms to engineering understanding, but also provide theoretical reference for the material optimization design of PE-ECC.

[0120] Non-dominated Sorting Genetic Algorithm II (NSGA-II): To achieve efficient solutions to complex multi-objective optimization problems, this invention introduces the Non-dominated Sorting Genetic Algorithm II (NSGA-II). This algorithm is a multi-objective optimization algorithm based on Pareto-optimal set theory, capable of achieving dynamic balance among multiple conflicting objective functions. Unlike traditional single-objective optimization methods, NSGA-II, through a genetic algorithm framework, retains multiple non-dominated solutions while conducting a global search, thereby obtaining a set of optimal solutions that balance different objectives.

[0121]

[0122]

[0123]

[0124] in, A vector of decision variables; Let k be the objective function; Inequality constraints; and This represents the upper and lower bounds of the variable.

[0125] From a theoretical perspective, the core idea of ​​NSGA-II is to construct a fitness evaluation system for multi-objective problems through a dual selection mechanism of non-dominated sorting and crowding distance. First, the algorithm divides the population into several levels of Pareto fronts based on non-dominated relationships. Individuals in the first level are not dominated by any other individuals and represent the optimal solution set; individuals in the second level are dominated only by the first level, and so on. Second, within the same Pareto level, the algorithm measures the density of solutions in the solution space by calculating the crowding distance of each individual, thereby maintaining the diversity of the solution set. Individuals with larger crowding distances represent sparse positions in the objective space and therefore have a higher retention priority during the selection process, avoiding excessive clustering of solutions and premature convergence.

[0126] ;

[0127] in, Indicates the crowding distance of an individual in the target space; and This represents the target values ​​of the adjacent individuals of individual i after sorting along the k-th objective function; and represents the maximum and minimum values ​​of the i-th objective in the current population; m represents the number of objective functions.

[0128] While the traditional NSGA algorithm can achieve multi-objective optimization, its high computational complexity (O(MN³)), uneven solution set distribution, and low accuracy make it difficult to meet the needs of complex engineering optimization tasks. To overcome these limitations, NSGA-II introduces an elitism strategy to enhance global convergence performance. This strategy merges parent and child solutions in each generation, using non-dominated sorting to select the top N high-quality solutions to form a new population. This ensures that high-quality solutions are preserved during iterations and are not lost due to random selection. This mechanism significantly improves the algorithm's stability and search efficiency, enabling it to accelerate convergence to the global optimum Pareto front while maintaining solution diversity.

[0129] Furthermore, NSGA-II introduces a crowded comparison operator, which comprehensively considers both non-dominated level and crowding distance during population evolution, achieving a balanced selection of the solution set. This operator prioritizes retaining individuals with lower non-dominated levels (i.e., better) during the selection operation, and within the same level, tends to select individuals with larger crowding distances to maintain a uniform distribution of solutions. Through this mechanism, NSGA-II can generate uniformly distributed and highly accurate Pareto solution sets under complex design variables and multiple constraints, fully reflecting the actual needs of multi-objective trade-offs in engineering design.

[0130] Hyperparameter Optimization: To improve the prediction accuracy and generalization performance of the proposed PhyResNet physical guided convolutional neural network, this invention employs Bayesian Optimization (BO) to systematically search for the network's key hyperparameters. Compared to traditional grid search or random search, Bayesian Optimization can approximate the global optimum with fewer iterations, significantly improving the efficiency of hyperparameter tuning. It is particularly suitable for deep learning models with high training costs and large search space dimensions.

[0131] The core idea of ​​Bayesian optimization is to treat the objective function as a black box and model its uncertainty through a prior distribution. In this invention, it is assumed that the objective function follows a Gaussian Process (GP) distribution. GP has good expressive power and uncertainty quantification capabilities, and can update the posterior distribution based on existing sampling points in each iteration, thereby dynamically adjusting the search strategy.

[0132] ;

[0133] Where m(x) represents the mean function, which is usually set to a constant or zero;k (x,x′) represents the covariance function (kernel function), which measures the similarity between input points.

[0134] The design of the acquisition function is a crucial step in Bayesian optimization, determining the next sampling location in the parameter space. The acquisition function needs to balance two objectives: "exploration" and "exploration." The former involves a local search near the currently known optimal solution to refine the optimal solution; the latter involves sampling in unexplored regions to avoid getting trapped in local optima. In this invention, the Upper Confidence Bound (UCB) is chosen as the primary acquisition function. By balancing the prediction mean and prediction variance, the UCB guides the algorithm to explore potential high-performance regions while controlling uncertainty, thereby accelerating convergence and improving search efficiency.

[0135] ;

[0136] in, These are time-dependent exploration parameters.

[0137] In its implementation, this invention uses key hyperparameters such as learning rate, number of convolutional kernels, kernel size, batch size, and regularization coefficient as search variables, and sets an appropriate search range. Each set of candidate hyperparameters is evaluated using K-fold cross-validation, with mean squared error (MSE) or coefficient of determination (R²) as the objective function, ensuring that the finally selected hyperparameters perform excellently on both the training and validation sets.

[0138] Through the Bayesian optimization strategy described above, this invention can obtain near-global optimal hyperparameter combinations with limited computational resources, effectively improving the robustness and generalization ability of the PhyResNet model in the ECC mechanical performance prediction task.

[0139] Evaluation metrics: Model performance is comprehensively evaluated using multiple metrics. First, the coefficient of determination is used. The goodness of fit between predicted and measured values ​​is measured, and the prediction accuracy is evaluated using root mean square error (RMSE) and mean square error (MSE). Furthermore, this invention evaluates the robustness and stability of the model under small sample sizes and noisy conditions through experimental designs with different proportions of training data and varying levels of noise, providing a reference for practical engineering applications.

[0140] ;

[0141] ;

[0142] ;

[0143] ;

[0144] ;

[0145] Where n represents the total number of data points; i represents the i-th data point, which is the index; This represents the actual observed value of the x variable at the i-th data point; This represents the actual observed value of the y variable at the i-th data point; This represents the arithmetic mean (average) of all observations of variable x. This represents the arithmetic mean (average) of all observations of the variable y. y represents the predicted value of the variable y for the i-th data point; m represents the total number of data points.

[0146] Example 2

[0147] Developing accurate and reliable predictive models requires, first and foremost, constructing a high-quality and representative dataset, which is the foundation and key step of the entire research. The dataset used in this example comes from 154 sets of experimental data on polyethylene fiber-reinforced ECC (PE-ECC) from published literature, ensuring broad sample coverage and scientific rigor. These data are all from uniaxial tensile tests conducted after 28 days of standard curing, and the main mechanical properties measured include tensile strength (UTS) and ultimate tensile strain (UTSn), which can comprehensively characterize the deformation and failure characteristics of ECC materials under tensile stress.

[0148] The dataset contains 12 features: 10 input features and 2 output targets. The input features and mix proportion features are related to fiber properties, specifically the mass ratio of cement to binder (C / B), fly ash to binder (FA / B), blast furnace slag to binder (GGBFS / B), lime powder to binder (LP / B), silica fume to binder (SF / B), sand to binder (S / B), water to binder (W / B), and fiber volume fraction (…). ), fiber length ( ). Fiber diameter ( To ensure the systematicity and comparability of the database, this invention standardized the experimental mix proportions based on the equivalent mass of cementitious materials (B), where B is the sum of cement (C), fly ash (FA), blast furnace slag (GGBFS), lime powder (LP), and silica fume (SF). The database covers typical ECC mix proportion systems, including cement-based cementitious systems, sand, water, and polyethylene (PE) fibers, comprehensively reflecting the influence of different material compositions on mechanical properties. Table 1 summarizes the statistical information and numerical ranges of each feature in the dataset, revealing significant statistical differences between features. Figure 1 Histograms further illustrate the distribution of each feature, providing an intuitive visual overview of the overall data distribution.

[0149] Table 1 Statistical details of features in the dataset

[0150]

[0151] To comprehensively understand the relationship between input features and output targets, this invention performs statistical analysis on the database based on the Pearson correlation coefficient and generates heatmaps of the correlation coefficients between pairs of features, such as... Figure 2 As shown. The range of Pearson correlation coefficient values ​​is... The larger the absolute value, the stronger the linear relationship between the variables: a positive value indicates a positive correlation, a negative value indicates a negative correlation, and a value close to 0 indicates a lack of obvious linear relationship between the variables.

[0152] from Figure 2 It can be seen that the water-cement ratio (W / B) is significantly negatively correlated with the tensile strength (UTS), with a correlation coefficient of [value missing]. The results indicate that as the water-cement ratio increases, the matrix strength decreases, leading to a drop in the tensile strength of the sample. This result is consistent with experimental conclusions in multiple literature studies, verifying the reliability of the database. Furthermore, the blast furnace slag content ratio (GGBFS / B) shows a moderate positive correlation with UTS (Ultra-Strength Tolerance), with a correlation coefficient of approximately 0.54, indicating that appropriate incorporation of blast furnace slag can improve the hydration degree and density of the cement-based system, thereby enhancing the material strength.

[0153] In the correlation analysis of ultimate tensile strain (UTSn), fiber parameters played a crucial role. Fiber volume fraction (Vf) and fiber length (Lf) were positively correlated with UTSn, with correlation coefficients of 0.37 and 0.41, respectively, indicating that increasing fiber content and selecting longer fibers both contribute to improving the material's tensile deformation capacity. In contrast, silica fume ratio (SF / B) and fiber diameter (Df) showed a weak negative correlation with UTSn, with correlation coefficients of [missing data]. and This indicates that excessive silica fume content may lead to increased matrix brittleness, while coarser fibers may reduce the number density of bridging cracks, thus hindering strain hardening behavior.

[0154] Overall, the Pearson heatmap reveals that the water-cement ratio and slag content are the main factors affecting tensile strength, while fiber parameters are the key factors controlling the ultimate tensile strain. This analysis not only verifies the physical mechanism assumptions proposed by the theoretical model but also provides data support for feature selection and model interpretation in the subsequent PhyResNet model.

[0155] Before training a deep learning model, the raw data must be properly preprocessed to reduce the impact of different units and value ranges on model training and ensure the comparability of features. This invention employs the Min–Max Normalization method to scale all input features, and its mathematical expression is as follows:

[0156]

[0157] Where x is the original feature value, and These are the minimum and maximum values ​​of the feature in the entire dataset, respectively. These are the normalized feature values. After this processing, all input features are mapped to the interval [0,1].

[0158] It is worth noting that the same normalization parameter (i.e., calculated based on the training set) is used in both model training and testing. and This preprocessing method ensures the consistency of feature distribution and avoids data leakage and the introduction of bias. Through this method, the features of the dataset are effectively standardized, laying a solid foundation for the stable training of the subsequent PhyResNet model.

[0159] To make full use of the limited dataset and improve the generalization ability of the model, this invention adopts a 9-fold cross-validation (CV) strategy combined with an early stopping mechanism to train and evaluate the performance of the PhyResNet model.

[0160] First, the original dataset is randomly divided into two parts: 90% as the cross-validation set (CV set) and 10% as the independent test set. Then, the CV set is further divided into nine equal subsets (folds). In each iteration, eight folds are selected as the training set, and the remaining fold is used as the validation set to evaluate the generalization performance of the current training epoch. This process is repeated nine times, ensuring that each fold participates in the performance evaluation as a validation set, thereby minimizing the impact of randomness caused by data partitioning.

[0161] During each cross-validation training process, the hyperparameters are dynamically searched using Bayesian optimization to ensure that the model achieves a near-optimal structural configuration under different training folds. After each training iteration, a loss function incorporating physical loss is calculated. The model weights corresponding to the best performance are recorded as the primary performance metric. After nine training iterations, the average of the best model performance obtained from the nine-fold cross-validation is taken as the overall generalization performance of the model. Finally, the model is validated on an independent test set to evaluate its predictive ability on unseen data.

[0162] This embodiment will demonstrate the experimental results of the proposed CNN+PHY model in the PE-ECC mechanical property prediction task and discuss the results in depth. First, the effectiveness of the physical prior is verified through ablation experiments; then, the performance of this method is compared with existing mainstream machine learning and deep learning models, demonstrating its advantages in accuracy and generalization; next, robustness tests are conducted, including two scenarios: few-shot learning and noise interference, to evaluate the reliability of the model under real-world complex conditions; finally, interpretability analysis is used to reveal the model's decision-making mechanism, verifying its physical consistency and engineering usability.

[0163] To thoroughly evaluate the specific role of the physical guidance mechanism in model performance, this invention designed a systematic ablation experiment for two prediction tasks: tensile strength (UTS) and ultimate tensile strain (UTSn). Three types of models were set up in the experiment: (1) a pure physics-based model, based on empirical formulas established according to fracture energy and crack propagation theory; (2) a pure data-driven model (CNN), relying on deep convolutional structures to automatically extract features and perform regression prediction; and (3) a physics-guided convolutional neural network model (CNN+PHY), introducing micromechanical constraints, including fracture energy conservation and pseudo-strain hardening conditions, into the CNN loss function to achieve synergistic optimization of data learning and physical laws. All experiments adopted a nine-fold cross-validation strategy, recording the R², RMSE, and MAE indices for each fold, and reporting the average value and the best single-fold result to ensure a comprehensive evaluation of the model's prediction accuracy, stability, and generalization ability.

[0164] First, the overall results show significant differences in predictive performance among the three types of models. The pure physical model achieved high accuracy in the UTS prediction task (R²=0.821 for training set, R²=0.796 for test set), indicating its good theoretical effectiveness in capturing the main trends in the tensile strength of materials. However, when used for UTSn prediction, the R² for training set was only 0.324, and the R² for test set was 0.565, showing a significant performance decline. The fundamental reason for this difference is that the physical model of UTSn is built on the basis of the UTS model, aiming to explicitly consider the combined effects of water-cement ratio, fiber volume fraction, and fiber aspect ratio on ductility by establishing a functional relationship between UTSn and UTS. Since the model's structural description of the ductility mechanism relies on intermediate variables from the UTS prediction, its capture of nonlinear responses and interface slip behavior in the high-strain stage is limited, resulting in relatively low overall prediction accuracy. This phenomenon reveals the limitations of physical models that rely solely on empirical formulas in the applicability of multi-factor coupling and high-dimensional nonlinear characteristic fields.

[0165] In contrast, pure CNN models exhibit stronger expressive power in data fitting, capturing complex nonlinear relationships between high-dimensional features during training. However, under conditions of small sample sizes and highly discrete datasets, their prediction results show significant fluctuations, with the R² of some test folds being significantly lower than that of the training folds, indicating a degree of overfitting. Simultaneously, their RMSE and MAE indices are relatively high, revealing that networks lacking physical constraints are prone to learning non-physical mappings when faced with experimental noise and physical outliers.

[0166] In contrast, the CNN+PHY hybrid model exhibited the best performance and highest stability across all prediction tasks. For both UTS and UTSn tasks, its average test set R² was significantly higher than that of pure CNN and pure physics models, with substantial reductions in RMSE and MAE. More importantly, the CNN+PHY model showed the smallest fluctuation in its nine-fold cross-validation results, indicating that it not only demonstrated high accuracy on average but also maintained consistent prediction stability across different data partitions. This result clearly demonstrates that embedding physical constraints into deep networks can effectively improve the model's physical consistency and generalization robustness.

[0167] Figure 3 This paper presents a performance comparison of the CNN and CNN+PHY models for the UTS and UTSn tasks. For both UTS and UTSn, the CNN+PHY model significantly outperforms the pure CNN in prediction performance. Firstly, in terms of accuracy, the average test set R of CNN+PHY is significantly higher. 2 It shows a significant improvement over pure CNN, with the highest R value on the test set. 2There was also a significant improvement, indicating that the physical guidance term can effectively help the model learn a feature space that is more consistent with reality and reduce the risk of overfitting. Secondly, in terms of error performance, the RMSE and MAE of CNN+PHY are significantly lower than those of pure CNN, and the results of the nine-fold cross-validation fluctuate less, indicating that its prediction results are more stable and reliable.

[0168] From a mechanistic perspective, the advantages of CNN+PHY are mainly reflected in the following three aspects: (1) Physical constraints guide feature learning: Physical consistency constraints are introduced into the loss function, so that the model can fit the experimental data and maintain the consistency of physical laws during the optimization process, thereby avoiding learning physically unreasonable solutions; (2) Improve feature space distribution: Physical priors promote the clustering of latent space features, making the distribution of samples under the same physical mechanism more compact in the feature space, and improving the separability of regression tasks; (3) Enhance generalization ability and robustness: Physical regularization suppresses the model's excessive dependence on noise features, so that it can still maintain strong generalization ability under small samples and highly discrete data.

[0169] To systematically evaluate the predictive performance of the proposed CNN+PHY model, this invention compares it with several common machine learning and deep learning models, including traditional Random Forest, k-Nearest Neighbors (KNN), Support Vector Regression (SVR), Ridge Regression, CatBoost, XGBoost, and pure CNN models without physical priors. All models employ the same dataset splitting and nine-fold cross-validation strategy to ensure fairness in the comparison.

[0170] Figure 4 A bar chart comparing the CNN+PHY model with other models is presented. The results show that the pure CNN model demonstrates superior prediction accuracy compared to traditional machine learning models, especially in the task of predicting tensile strength (UTS). The CNN model achieves higher accuracy on the test set R... 2 The performance is significantly higher than that of random forest, KNN, and linear regression models. This indicates that convolutional neural networks have a natural advantage in handling the complex nonlinear relationship between material composition and mechanical properties, and can automatically extract high-order features and capture local patterns. However, pure CNN models are prone to overfitting in small sample scenarios, and their test set performance fluctuates greatly, resulting in insufficient generalization ability, which is particularly evident in the ultimate tensile strain (UTSn) prediction task.

[0171] In contrast, the CNN+PHY model achieved state-of-the-art results on both UTS and UTSn prediction tasks, with its highest R-value on the test set. 2The scores reached 0.938 and 0.79 respectively, significantly outperforming pure CNNs and other machine learning models. The fundamental reason for this performance improvement lies in the fact that CNN+PHY embeds physical priors into the deep learning process. By regularizing the loss function, it ensures that the model follows physical mechanisms during optimization, thereby reducing the risk of learning physically inconsistent data. This physical consistency constraint effectively alleviates the common "black box" problem in deep learning models, making the prediction results more interpretable and reliable.

[0172] Robustness is a crucial indicator of a model's reliability and stability in real-world engineering applications, especially in materials research and development scenarios where small sample sizes and high-noise data are almost unavoidable. This invention systematically evaluates the robustness of the CNN+PHY model through two types of experiments: small sample tests and noise tests. It also conducts a comprehensive comparison with various comparative models, including CNN, Random Forest, KNN, CatBoost, SVR, Ridge, and XGBoost. The results show that CNN+PHY exhibits significantly better robustness than other models under various adverse conditions, which is attributed to the unique role of its physical information fusion mechanism in model learning.

[0173] To simulate the scenario of scarce sample size in the early stages of materials research and development, the dataset was divided into training and test sets according to 90% and 10% respectively. In the training set, subsets of 90%, 70%, 50%, and 30% were selected for training. Each subset was randomly sampled 5 times to eliminate the influence of chance. The average value was taken as the model performance under the condition of that subset.

[0174] Depend on Figure 5 It can be observed that the prediction accuracy of all models decreases significantly with the reduction of training sample size, which is consistent with the expectations of statistical learning theory. However, the performance degradation of CNN+PHY is much less than that of other comparative models. Under extreme conditions (only 30% of the samples), CNN+PHY's prediction accuracy on the UTS test set R is significantly lower. 2 It remains at 0.740, while the test set R for other models... 2 The values ​​generally decreased to the 0.244–0.664 range, approaching the prediction failure threshold; in UTSn prediction, the CNN+PHY test set R... 2 It can still maintain =0.629, demonstrating excellent robustness.

[0175] This superior performance with small sample sizes stems from the physical information fusion mechanism of CNN+PHY. On one hand, physical constraints act as a strong regularizer in the loss function, effectively suppressing overfitting of the model to a limited number of samples. On the other hand, prior physical knowledge provides the model with additional information when data is scarce, enabling it to learn more generalizable feature representations within a physically plausible framework. In other words, the learning process of CNN+PHY is upgraded from a purely "data-driven" model to a fusion of "physics-driven + data-driven" approaches, greatly reducing the model's dependence on the amount of data. This characteristic is particularly crucial in the early stages of materials research and development where only a limited amount of experimental data is available, providing a practical and feasible technical path for achieving low-cost, high-efficiency formulation optimization.

[0176] In the noise robustness test, the dataset was first divided into a 90% training set and a 10% test set. Zero-mean Gaussian noise with a standard deviation of 10%, 20%, 30%, 40%, and 50% of the label standard deviation was injected into the labels of the training set to simulate measurement errors and system disturbances that may occur in actual experiments. The test set was kept clean and no noise was injected.

[0177] Depend on Figure 6 It can be observed that the overall performance of all models decreases with increasing noise level, but the performance decline of CNN+PHY is significantly more gradual, indicating that it has the lowest sensitivity to noise perturbations. At the highest noise level (50%), CNN+PHY performs best on the test set R in UTS predictions. 2 It can still maintain a value of 0.687, and the test set R predicted by UTSn is... 2 It remained at 0.430, while the performance of other traditional machine learning models was even closer to failure.

[0178] The superior noise resistance of CNN+PHY can be attributed to two main reasons: First, the physics loss acts as a "noise filter" during the optimization process. When there are anomalous samples in the training data that severely violate physical laws, these samples will cause a large physics loss. During gradient descent, the model will reduce the weight of these samples to avoid interfering with the overall parameter learning. Second, the physical equations serve as strong prior anchors, providing the model with a reference framework that it always follows during the learning process. Even if some data points are noisy, the model is still "guided" by reliable physical signals, thus maintaining a stable learning trajectory and preventing predictions from deviating from physical plausibility.

[0179] In summary, CNN+PHY maintains high prediction accuracy and stability despite the dual challenges of small sample sizes and high noise levels, significantly outperforming purely data-driven deep learning and traditional machine learning methods. This result fully demonstrates that embedding physical knowledge into data-driven models not only improves model accuracy but also greatly enhances their robustness and engineering usability, providing crucial support for the practical implementation of artificial intelligence models in the field of materials science.

[0180] To further reveal the decision-making logic of the CNN+PH model and verify whether its predictions conform to the fundamental laws of materials science, this invention conducts a systematic interpretability analysis of the model from both global and local perspectives. Specifically, SHAP analysis is used to quantify the contribution of features in the prediction results, revealing the model's global decision-making basis; simultaneously, LayerCAM visualization technology is used to capture the feature-focused regions of the convolutional layers, thus intuitively demonstrating what the model "focuses on" when making predictions. This complementary analysis from macro to micro not only improves the model's transparency but also provides strong support for its trust and application in engineering practice.

[0181] SHAP is a game theory-based interpretation method that fairly allocates the marginal contribution of each input feature to the model output. This invention obtains the global importance ranking of each feature in predicted tensile strength (UTS) and ultimate tensile strain (UTSn) by calculating the average absolute value of the SHAP values ​​for all samples.

[0182] Depend on Figure 8 It can be observed that the characteristic importance of UTS is in the order of W > VOF > GGBFS > LOF, where VOF, GGBFS, and LOF have significant positive contributions to strength, while W shows a strong negative correlation. This conclusion is highly consistent with the mechanism of materials science: a higher VOF indicates that more fibers participate in load bridging per unit crack cross section, significantly improving strength; GGBFS generates additional CSH gel through the volcanic ash reaction, which can refine the matrix pore structure and improve overall compactness; an increase in LOF means a larger fiber-matrix interface contact area, a more significant mechanical anchoring effect, and enhanced interfacial adhesion. According to Abram's law, the higher the water-cement ratio (W / B), the lower the matrix strength; therefore, the negative correlation of W is consistent with the expectations of classical theory.

[0183] For UTSn, the feature importance ranking is VOF > LOF > W > DOF, where VOF and LOF both have a significant positive effect on strain capacity, while DOF has a negative correlation. This indicates that increasing fiber volume fraction and length can increase the number of crack bridging and fiber slip distance, thus enhancing the strain hardening effect; while reducing fiber diameter can increase the aspect ratio, which is beneficial to improving toughness. These results show that the CNN+PHY model not only captures the statistical patterns in the data but also learns and internalizes the physical laws closely related to material properties.

[0184] To verify how much the CNN+PHY model focuses on key features during prediction, LayerCAM technology was further used to visualize the responses of the convolutional layers. LayerCAM can reveal the input feature regions that the convolutional layers focus on when the model makes a specific prediction, thus supplementing the numerical interpretation of SHAP.

[0185] Depend on Figure 8 It can be observed that the first convolutional layer in both UTS and UTSn primarily focuses on key input features such as W and VOF, accurately identifying the variables that have the most significant impact on intensity, demonstrating the model's initial ability to filter original features. As features are passed layer by layer, the attention of the second convolutional layer begins to be distributed on other features, reflecting the network's higher-level abstraction and selection of inputs after nonlinear transformation and feature combination. This process shows that the CNN+PHY model does not simply rely on a single feature, but rather captures the complex interaction relationships between features through multi-layer representation learning, thereby achieving more accurate predictions.

[0186] By employing two complementary interpretability techniques, SHAP and LayerCAM, this invention provides an in-depth analysis of the CNN+PHY model from both global and local perspectives. The results consistently demonstrate that its decision-making mechanism is transparent and reasonable. The feature importance ranking upon which the model relies is highly consistent with known physical mechanisms, proving that this model is not only a data-driven statistical tool but also a "grey box" model capable of absorbing and utilizing physical laws.

[0187] This interpretability analysis provides strong theoretical support for the high-precision predictions and superior generalization performance of the CNN+PHY model, significantly increasing researchers' and engineers' confidence in the prediction results and eliminating concerns about the "black box" nature of traditional deep learning models. This provides a solid theoretical and methodological foundation for the practical deployment of this model in future materials design, performance prediction, and engineering optimization.

[0188] In composite material design, mix design optimization must consider not only mechanical properties but also economic efficiency and feasibility. Therefore, to achieve optimal performance and cost, this invention constructs a multi-objective optimization model with tensile strength (UTS), ultimate tensile strain (UTSn), and material cost as the main objectives. This model uses a trained CNN+PHY prediction model as the core for performance prediction and employs a Non-dominated Sorting Genetic Algorithm II (NSGA-II) for solution.

[0189] NSGA-II is a multi-objective evolutionary algorithm based on Pareto optimality theory, capable of rapidly approximating non-dominated solutions while maintaining population diversity. Unlike traditional single-objective optimization methods, NSGA-II achieves a balance between tensile strength, ductility, and cost in a multi-objective space by introducing crowding distance and an elitism strategy. The algorithm mainly consists of five stages: individual initialization, fitness calculation, non-dominated sorting, crossover and mutation operations, and iterative updates. Upon meeting the convergence criteria, the algorithm outputs a Pareto front solution set, providing various performance-cost balance schemes for mix design.

[0190] Furthermore, to ensure the optimization results meet engineering feasibility, this invention introduces multiple constraints during the algorithm's solution process, such as material proportions, cost limits, and mechanical performance ranges. These constraints collectively construct a constrained optimization space, ensuring that the generated solution is not only theoretically optimal but also meets the requirements of actual production and construction.

[0191] In the multi-objective optimization framework, the outputs (UTS and UTSn) of the performance prediction model are used as the fitness function of NSGA-II. Simultaneously, to reflect the economic impact of material proportions, a cost function based on unit price calculation is incorporated. The unit prices of each component are as follows:

[0192] C: $0.11 / kg, FA: $0.046 / kg, GGBFS: $0.1 / kg, LP: $0.122 / kg, SF: $0.5 / kg, S: $0.014 / kg, W: $0.0005 / kg, PE: $11.0 / kg.

[0193] By simultaneously minimizing cost and maximizing UTS and UTSn, this invention achieves a trade-off between performance and economy. The objective function for optimization can be expressed as:

[0194]

[0195]

[0196]

[0197] To ensure the practical feasibility of the obtained solution set, the constraints of the objective function are set as follows:

[0198]

[0199]

[0200]

[0201] The range was set based on existing ECC experimental literature and structural performance requirements, ensuring both the high ductility of the material and controlling its economic cost. Finally, the optimization model sought the Pareto optimal balance between UTS, UTSn, and Cost under constraints.

[0202] To ensure the search process is conducted within a reasonable mix proportion range, all parameters of the mixed materials are subject to both physical and empirical constraints. Specifically:

[0203]

[0204] These constraints ensure the physical rationality and engineering feasibility of the material proportions, preventing the algorithm from generating unfeasible design schemes.

[0205] Once the fitness function and constraints are fully defined, the NSGA-II algorithm is used for multi-objective optimization. The main parameters of the algorithm are set as follows: 60 iterations, 50 population size, 0.8 crossover probability, and 0.05 mutation probability. As the iterations proceed, the individual solutions continuously evolve, gradually approaching the Pareto optimal front.

[0206] The resulting Pareto solution set reflects the optimal trade-off between cost, tensile strength, and ultimate tensile strain. Compared to single-objective optimization, this method provides multiple performance-cost balanced design schemes, offering a basis for decision-making based on different engineering needs (such as prioritizing high ductility or low cost). Notably, the high-precision performance evaluation provided by the CNN+PHY model in the prediction phase gives NSGA-II stronger convergence and solution stability in the search space.

[0207] This invention proposes and verifies a physics-guided interpretable convolutional neural network framework (hereinafter referred to as CNN+PHY) for the prediction and optimization design of tensile strength (UTS) and ultimate tensile strain (UTSn) of polyethylene fiber-reinforced engineering cementitious composites (PE-ECC). The study first establishes two types of physical theoretical models: a matrix strength model based on a modified Abram's law and incorporating an equivalent cementitious amount (Beq), and a fiber bridging stress and ultimate strain model based on matrix strength. Subsequently, using a one-dimensional convolutional neural network as the backbone, a composite loss L combining data-driven loss (MSE) and a physical consistency term is designed. total The method employs Bayesian optimization to automatically optimize hyperparameters. Experiments on an integrated PE-ECC dataset (154 sets) validated the effectiveness of the method through 9-fold cross-validation, early stopping strategy, and system ablation, comparison, small-sample, and noise robustness tests. Results show that compared to purely data-driven CNNs and various traditional machine learning models, CNN+PHY significantly improves prediction accuracy, stability, and robustness (maintaining high R-values ​​on the test set even under extreme small-sample and high-noise conditions). 2 Furthermore, interpretability analysis of SHAP and LayerCAM further demonstrates that the features learned by the model are highly consistent with experimental / physical laws. Based on the model's predictions, this invention further introduces Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) to construct a multi-objective hybrid design framework with tensile strength (UTS), ultimate tensile strain (UTSn), and material cost as optimization objectives. Overall, the introduction of physical priors serves both as a strong regularizer to suppress overfitting and as a reliable prior anchor to compensate for insufficient data, thus transforming "pure data-driven" into a fusion learning paradigm of "physical-driven + data-driven." This provides a practical and engineering-valued solution for common small-sample, high-cost scenarios in materials research and development.

[0208] 1. After a systematic comparison with a series of mainstream machine learning models (including Random Forest, KNN, SVR, Ridge, CatBoost, XGBoost, and pure CNN), the results show that the CNN+PHY model exhibits the best performance in predicting both UTS and UTSn. This model achieves higher performance under nine-fold cross-validation. With lower RMSE / MAE, it significantly outperforms other data-driven methods. It's worth noting that while pure CNNs already demonstrate good predictive performance, the introduction of physical constraints further enhances both prediction accuracy and generalization ability through CNN+PHY. This clearly demonstrates that physical information not only supplements the deficiencies of data features but also plays a stabilizing and standardizing role in the deep learning modeling process.

[0209] 2. Further robustness analysis shows that CNN+PHY maintains a significant advantage under small sample size and noisy conditions. In small sample size tests, when the training set size is reduced to 30%, the UTS prediction of the CNN+PH model remains at 0.740, while most other models drop to the 0.244–0.664 range, nearing the failure threshold. In noisy tests, when the labels are injected with 50% Gaussian noise, the UTS prediction of CNN+PHY still reaches 0.687, while the pure CNN drops sharply to 0.291. These results indicate that the physical prior plays a crucial role in resisting overfitting and data perturbation, enabling CNN+PH to maintain excellent robustness in complex and uncertain experimental environments.

[0210] 3. Regarding interpretability analysis, the internal mechanism of CNN+PHY was validated both quantitatively and visually. Correlation analysis and SHAP results showed that the water-to-binder ratio (W / B) was strongly negatively correlated with tensile strength (-0.73), while GGBFS / B was significantly positively correlated (0.54); fiber volume fraction (V... f ) and fiber length (L f The SF / B and fiber diameter (Df) contribute positively to the ultimate tensile strain (correlation coefficients of 0.37 and 0.41, respectively), while silica fume (SF / B) and fiber diameter (Df) show weak negative effects. Meanwhile, LayerCAM visualization results show that the regions of interest in the convolutional layers closely match these physical laws. These results not only demonstrate that CNN+PHY is not a "black box" but also strengthen its credibility in engineering applications.

[0211] 4. The superior performance of CNN+PHY across multiple scenarios is primarily attributed to its unique physical information fusion mechanism. On one hand, the physical loss acts as a "strong regularizer" during training, effectively suppressing the model's tendency to overfit to noise and outliers. On the other hand, the physical equations provide a stable prior anchor, guiding the model to follow physical laws during optimization, maintaining reasonable predictions even with scarce or noisy data. This dual paradigm of "physics-driven + data-driven" makes the model's learning process closer to the behavior of real materials, thereby improving the reliability and interpretability of predictions.

[0212] 5. Building upon the model's predictions, this invention further introduces the Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) to construct a multi-objective hybrid design framework with tensile strength (UTS), ultimate tensile strain (UTSn), and material cost as optimization objectives. By using the prediction results of CNN+PHY as the fitness function input into the multi-objective optimization model, synergistic optimization of performance and economy is achieved.

[0213] 6. In summary, the CNN+PHY model outperforms existing mainstream methods in terms of accuracy, robustness, and interpretability, significantly reducing its reliance on large-scale experimental data, making it particularly suitable for data-scarce scenarios in the early stages of materials research and development. This research provides a practical path for the deep integration of artificial intelligence and materials mechanics mechanisms, and is expected to promote the efficient design and optimization of cement-based composite materials such as ECC. However, this invention also has certain limitations: the physical model still relies on empirical formulas, and its generalization ability in extreme material systems needs further verification; the scale and diversity of the dataset also need to be expanded to cover a wider range of experimental conditions. Future work can combine multimodal experimental data, uncertainty quantification methods, and active learning strategies to further improve the reliability and adaptability of the model in industrial applications.

[0214] Example 3

[0215] The present invention also provides a system for performance prediction and mix proportion optimization of engineering cement-based composite materials based on physics-guided deep learning and multi-objective optimization. The system is used to implement the method described in Embodiment 1. The system includes: an acquisition module, a construction module, a prediction module, and an optimization module.

[0216] The acquisition module is used to acquire tensile test data of engineering cement-based composite materials;

[0217] The building blocks are used to construct the PhyResNet convolutional neural network model based on physical guidance.

[0218] The prediction module is used to predict the performance of tensile test data of engineering cement-based composite materials using the PhyResNet physics-guided convolutional neural network model.

[0219] The optimization module is used to optimize the mix ratio of the predicted performance using a multi-objective optimization method based on NSGA-II.

[0220] In this embodiment, the PhyResNet physical-guided convolutional neural network model organically integrates theoretical and data-driven models;

[0221] The theoretical models include: tensile strength model and ultimate tensile strain model;

[0222] The data-driven model is a data-driven prediction model based on convolutional neural networks (CNNs).

[0223] In this embodiment, the expression for the tensile strength model is:

[0224] ;

[0225] ;

[0226] ;

[0227] ;

[0228] Where UTS represents the tensile strength of the composite material, Indicates matrix strength. , , These represent fiber volume fraction, length, and diameter, respectively, with AR indicating the fiber aspect ratio. The nonlinear coefficient characterizes the change in load transfer efficiency with aspect ratio; The dynamic coefficient is a nonlinear correction factor related to the matrix composition; , , These are empirical parameters obtained through fitting via material experiments. For interface coefficients, For the weight vector, is the input variable, and b is the bias term.

[0229] In this embodiment, the expression for the ultimate tensile strain model is:

[0230] ;

[0231] Where W represents the amount of water used. , , These represent fiber volume, fiber length, and fiber diameter, respectively; A, B, C, D, and E are empirical parameters obtained through material testing.

[0232] In this embodiment, the data-driven prediction model based on a convolutional neural network (CNN) includes:

[0233] A rectified linear unit is typically used as the activation function after a convolutional layer.

[0234] A pooling layer is introduced to compress the spatial dimension of the feature map through downsampling operations;

[0235] The max pooling strategy is adopted, that is, the maximum value is selected from each local region as the feature representation of that region;

[0236] At the back end of the network, the features extracted through multiple convolutions and pooling are flattened into a one-dimensional vector and input into a fully connected layer to achieve the synthesis and mapping of high-level features.

[0237] In the regression task, the last fully connected layer is set as a single output node, and continuous predicted values ​​are output through an activation function to achieve quantitative prediction of the mechanical properties of engineering cement-based composite materials.

[0238] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for performance prediction and mix proportion optimization of engineering cement-based composite materials based on physics-guided deep learning and multi-objective optimization, characterized in that, The method includes: Obtain tensile test data for engineering cement-based composite materials; Construct a physics-guided convolutional neural network model, PhyResNet; The PhyResNet model, a physics-guided convolutional neural network, is used to predict the performance of tensile test data of engineering cement-based composite materials. The predicted performance was optimized using a multi-objective optimization method based on NSGA-II. PhyResNet, a physics-guided convolutional neural network model, organically integrates theoretical and data-driven models. The theoretical models include: tensile strength model and ultimate tensile strain model; The data-driven model is a data-driven prediction model based on convolutional neural networks (CNNs). The expression for the tensile strength model is: ； ； ； ； Where UTS represents the tensile strength of the composite material, Indicates matrix strength. , , These represent fiber volume, fiber length, and fiber diameter, respectively; AR represents the fiber aspect ratio. The nonlinear coefficient characterizes the change in load transfer efficiency with aspect ratio; The dynamic coefficient is a nonlinear correction factor related to the matrix composition; , , These are empirical parameters obtained through fitting via material experiments. For interface coefficients, For the weight vector, b is the input variable, and b is the bias term; The expression for the ultimate tensile strain model is: ； Where W represents the amount of water used, and A, B, C, D, and E are empirical parameters obtained through material testing.

2. The method according to claim 1, characterized in that, Data-driven prediction models based on Convolutional Neural Networks (CNNs) include: A rectified linear unit is applied after the convolutional layer as the activation function; A pooling layer is introduced to compress the spatial dimension of the feature map through downsampling operations; The max pooling strategy is adopted, that is, the maximum value is selected from each local region as the feature representation of that region; At the back end of the network, the features extracted through multiple convolutions and pooling are flattened into a one-dimensional vector and input into a fully connected layer to achieve the synthesis and mapping of high-level features. In the regression task, the last fully connected layer is set as a single output node, and continuous predicted values ​​are output through an activation function to achieve quantitative prediction of the mechanical properties of engineering cement-based composite materials.

3. A system for performance prediction and mix proportion optimization of engineering cement-based composite materials based on physics-guided deep learning and multi-objective optimization, the system being used to implement the method described in any one of claims 1-2, characterized in that, The system includes: an acquisition module, a construction module, a prediction module, and an optimization module; The acquisition module is used to acquire tensile test data of engineering cement-based composite materials; The building module is used to build the PhyResNet convolutional neural network model based on physical guidance; The prediction module is used to predict the performance of tensile test data of engineering cement-based composite materials using the PhyResNet physics-guided convolutional neural network model. The optimization module is used to optimize the mix ratio of the predicted performance using a multi-objective optimization method based on NSGA-II.

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