Mechanical Property Prediction of Wood Using a Backpropagation Neural Network Optimized by Adaptive Fractional-Order Particle Swarm Algorithm

Abstract

This study proposes a novel LK-BP-AFPSO model for the nondestructive evaluation of wood mechanical properties, combining a backpropagation neural network (BP) with adaptive fractional-order particle swarm optimization (AFPSO) and Liang–Kleeman (LK) information flow theory. The model accurately predicts four key mechanical properties—longitudinal tensile strength (SPG), modulus of elasticity (MOE), bending strength (MOR), and longitudinal compressive strength (CSP)—using only nondestructive physical features. Tested across diverse wood types (fast-growing YKS, red-heart CSH/XXH, and iron-heart XXT), the framework demonstrates strong generalizability, achieving an average prediction accuracy (R²) of 0.986 and reducing mean absolute error (MAE) by 23.7% compared to conventional methods. A critical innovation is the integration of LK causal analysis, which quantifies feature–target relationships via information flow metrics, effectively eliminating 29.5% of spurious correlations inherent in traditional feature selection (e.g., PCA). Experimental results confirm the model’s robustness, particularly for heartwood variants, while its adaptive fractional-order optimization accelerates convergence by 2.1× relative to standard PSO. This work provides a reliable, interpretable tool for wood quality assessment, with direct implications for grading systems and processing optimization in the forestry industry.

1. Introduction

Wood, as a green and renewable material, is widely used in construction, furniture, and structural engineering. Its mechanical properties directly determine the safety and reliability of its engineering application [1]. However, conventional methods for evaluating wood mechanical properties are often destructive, labor-intensive, and costly, making it difficult to conduct rapid, large-scale assessments. Therefore, developing an efficient and nondestructive evaluation (NDE) method for assessing the mechanical properties of wood holds significant practical value [2].

In recent years, artificial intelligence (AI) technologies have been increasingly applied in materials science [3]. Among them, the backpropagation neural network (BPNN) has shown great potential in property prediction due to its powerful nonlinear modeling capability [4]. However, traditional BP models often suffer from issues such as becoming trapped in local minima and slow convergence, limiting their effectiveness in high-precision prediction tasks. To overcome these drawbacks, this study integrates an adaptive fractional-order particle swarm optimization algorithm (AFPSO) to optimize the parameters of the BP model, resulting in the LK-BP-AFPSO model. This optimization significantly enhances the model’s prediction accuracy and generalization capability.

Moreover, due to the heterogeneous and complex internal structure of wood, identifying the most influential variables from a limited set of physical features remains a critical challenge for accurate prediction [5]. To address this, the Liang–Kleeman (LK) information flow theory is introduced to compute the quantitative causal influence between multiple features and multiple target variables. LK information flow can quantitatively describe the strength of causal relationships, and has the characteristics of simple and fast calculation, which is conducive to real-time prediction and improves the prediction efficiency of the model. Rooted in physical first principles, LK information flow offers an effective measure of causal strength with high computational efficiency. By embedding LK-based feature importance ranking into the LK-BP-AFPSO framework, the overall prediction performance is further improved.

This study addresses three critical limitations in current wood property prediction research: (1) conventional gradient-optimized neural networks struggle with the highly nonlinear relationships between wood’s physical and mechanical characteristics, often yielding suboptimal solutions; (2) existing PCA and correlation-based methods cannot adequately distinguish causal features from spurious correlations, particularly in heterogeneous materials like red-heart- and iron-heartwood; (3) current evaluation frameworks typically focus on single wood species without accounting for the substantial variations between fast-growing and dense heartwood types, severely limiting practical applications. This study focuses on four types of wood—fast-growing (YKS), red-heart (CSH and XXH), and iron-heart (XXT)—to evaluate the effectiveness of the proposed method in predicting four mechanical properties: longitudinal tensile strength (SPG), modulus of elasticity in bending (MOE), bending strength (MOR), and longitudinal compressive strength (CSP). The input feature set includes basic wood density, air-dry shrinkage rates (radial, tangential, and volumetric), and oven-dry shrinkage rates (radial, tangential, and volumetric). The experimental results show that the proposed method achieves prediction accuracy up to 90%, demonstrating its feasibility for the nondestructive evaluation of wood mechanical properties. These findings provide valuable insights for selecting high-quality timber and optimizing processing techniques, thereby extending the service life of functional wood and promoting sustainable development in the wood industry.

The main contributions of this paper are summarized as follows:

(1)

A BP neural network model optimized using the adaptive fractional-order particle swarm optimization algorithm (AFPSO) is proposed, which significantly improves parameter optimization. This method enhances prediction accuracy, accelerates convergence, and avoids local optima—common limitations of traditional BP models.

(2)

The LK information flow theory is innovatively applied to wood property prediction, enabling feature importance ranking based on causal influence. And it is proved that its influence on the prediction accuracy of the model is far better than principal component analysis (PCA).

(3)

A unified evaluation framework is established for different wood types (YKS, CSH, XXH, and XXT), and multiple intelligent optimization algorithms (AFPSO, PSO, GWO, WOA, FA, and DE) are systematically compared within the BP framework. The results confirm that the proposed LK-BP-AFPSO model outperforms alternatives in both accuracy and stability across datasets, showing strong generalizability.

This study aims to develop a hybrid intelligent model (LK-BP-AFPSO) based on adaptive fractional-order particle swarm optimization (AFPSO) and LK information flow theory to achieve the high-precision nondestructive prediction of wood mechanical properties. The core objective is to address three critical challenges in current wood testing by integrating causal feature selection with intelligent optimization algorithms: the tendency of traditional neural networks to converge to local optima, the inability of conventional feature selection methods to distinguish causal relationships, and insufficient generalization capability across different wood species.

Specific tasks include (1) constructing an AFPSO-optimized BP neural network to enhance parameter search efficiency through fractional-order dynamic adjustment mechanisms; (2) applying LK information flow theory to quantify the causal influence between wood physical characteristics and mechanical properties (MOE, MOR, etc.), establishing an interpretable feature screening system; and (3) developing a unified evaluation framework to systematically validate the model’s predictive stability across heterogeneous wood types including fast-growing wood (YKS), red-heartwood (CSH/XXH), and iron-heartwood (XXT), ultimately forming an intelligent decision-making system to support wood grading and processing optimization.

2. Related Works

The Liang–Kleeman (LK) information flow technique has been widely adopted in causal analysis. Without requiring any prior knowledge, it quantitatively characterizes the causal effect between two time series by evaluating the amount of information transferred from one to the other within a unit time interval [6]. Currently, there are three main approaches for analyzing correlations in wood properties: correlation-based methods, data-driven models, and multiscale causal modeling. In the literature [7,8], correlation-based techniques were employed to explore differences in physical and mechanical properties among various eucalyptus and Chinese fir species. Existing approaches in material property prediction exhibit several limitations that our LK-BP-AFPSO framework specifically addresses. While CNN-RNN combination shows effective feature fusion for plant identification, it lacks (a) the adaptive fractional-order optimization crucial for handling wood’s nonlinear viscoelasticity, and (b) causal feature selection mechanisms [9]. The deep learning model in the literature [10] for semiconductor screening achieves high bandgap accuracy but depends on static network architectures without our proposed AFPSO’s dynamic parameter adjustment capability—a key factor in overcoming traditional BP networks’ local optima traps. The GCNPCA method in the literature [11], though innovative for miRNA-disease prediction, demonstrates inferior feature selection compared to our LK information flow theory, particularly in quantifying causal relationships between wood features and mechanical properties. Reference [12]’s CNN-based wood identification shares our domain focus but fails to provide (1) the comprehensive algorithmic comparison framework we established for wood subtypes, and (2) quantitative evidence of model generalizability across wood varieties. Most notably, reference [13]’s machine learning approach for particleboard prediction relies on conventional interval reasoning methods, whereas our integrated causal–AFPSO demonstrates markedly superior performance in comparative pilot tests using similar datasets. The evident performance gap highlights the distinct advantages of our fractional-order particle swarm dynamics, particularly in handling the complex time-dependent characteristics of wood’s acoustic-emission signals.

In recent years, various machine learning and intelligent optimization strategies have been developed for nondestructive prediction of thermally modified wood. For instance, reference [14] introduced a BP neural network optimized using a Tent-map-enhanced sparrow search algorithm (TSSA), improving convergence and prediction accuracy. Building on this, reference [15] proposed a novel gray wolf optimization strategy (NAGGWO) by integrating chaotic parameter mapping (CPM), nonlinear control parameters, and adaptive grouping mechanisms to enhance the stability and search performance of BP networks. In reference [16], support vector regression (SVR) models were used to predict MOE, MOR, and shear strength from wood color parameters following thermal treatment, with coefficients of determination reaching up to 0.903—demonstrating the potential of color-based nondestructive evaluation. In reference [17], a linear regression model incorporating longitudinal natural frequency and structural parameters was developed, effectively improving the prediction accuracy of MOE and MOR in components with variable cross-sections. Reference [18] proposed a hybrid model integrating LK information flow with tracheid morphological features to quantify the causal influence of microstructural traits on mechanical properties. Reference [19] combined Lamb wave propagation velocity with group velocity processing techniques to evaluate MOE and MOR under varying moisture content, validating its sensitivity to material condition and effectiveness in nondestructive testing. In reference [20], CNNs were applied to extract anatomical features from cross-sectional images, and the incorporation of density information significantly improved prediction accuracy, underscoring the importance of combining structural and density data. In reference [21], K-S tests were used to analyze the distribution of density and dynamic modulus of elasticity in fast-growing fir, and a shear analogy method was employed to predict the performance of CLT components—highlighting the beneficial effect of high-grade timber on structural strength.

Despite the significant advancements made in applying intelligent optimization algorithms (e.g., TSSA, GWO, and IDBO) and machine learning techniques (e.g., SVR, CNN, and BP neural networks) to the prediction of mechanical properties in thermally modified or other wood types, particularly in enhancing convergence speed, prediction accuracy, and feature extraction, this study makes the following unique contributions:

(1)

From a model optimization perspective, we innovatively incorporate an adaptive fractional-order particle swarm optimization (AFPSO) algorithm to optimize the weights and thresholds of BP neural networks. This approach, leveraging a fractional-order update mechanism and dynamic inertia adjustment strategy, enhances the network’s global search capability and robustness.

(2)

To comprehensively validate model performance, we construct both a LK-BP-AFPSO model and a multiple linear regression (MLR) baseline model, allowing comparison across nonlinear and linear paradigms.

(3)

On the data front, we gather thermally treated Chinese fir samples (CSH2, YKS2, XXH2, and XXT2) from various regions and treatment conditions, significantly improving the model’s generalizability and practical applicability.

(4)

Lastly, the proposed hybrid framework integrates LK information flow with the AFPSO algorithm, enabling interpretable modeling by quantifying the causal strength between wood physical attributes and mechanical performance. The causally relevant features are then used to guide AFPSO in optimizing neural network parameters, culminating in a high-efficiency predictive model with strong interpretability and robustness.

3. Related Theories

3.1. Liang–Kleeman Information Flow and Causal Directed Graph

For the specific theory of this method, please refer to reference [18].

3.2. LK-LK-BP-AFPSO Model

The proposed LK-LK-BP-AFPSO model integrates causal feature selection based on the LK information flow method, a backpropagation neural network (BP), and an adaptive fractional-order particle swarm optimization algorithm (AFPSO). This section provides the theoretical foundation and structural advantages of each component in the hybrid framework.

3.2.1. A. Backpropagation Neural Network (BP)

The BP neural network is a widely used multilayer feedforward network trained via the error backpropagation algorithm. It consists of an input layer, one or more hidden layers, and an output layer, with full connections between neurons in adjacent layers. The network is trained by minimizing the mean square error (MSE) between the predicted and actual outputs using gradient descent. Despite its strong nonlinear approximation capability, the BP model suffers from several limitations, including slow convergence, sensitivity to initial weights, and a tendency to fall into local minima. These shortcomings often necessitate the use of optimization algorithms to enhance its stability and predictive accuracy.

3.2.2. B. Adaptive Fractional-Order Particle Swarm Optimization (AFPSO)

AFPSO is an improved variant of the traditional particle swarm optimization (PSO) algorithm that incorporates fractional calculus and adaptive mechanisms. Unlike classical PSO, which updates particles’ velocities and positions using integer-order difference equations, AFPSO adopts a fractional-order dynamic update strategy. This extension enhances memory effects and global search capabilities by accounting for the historical behavior of particles. Additionally, adaptive adjustment strategies for inertia weights and acceleration coefficients are introduced, allowing the algorithm to balance exploration and exploitation dynamically across iterations. AFPSO has demonstrated superior performance in optimizing complex, high-dimensional nonlinear functions and is particularly effective in fine-tuning the parameters of neural networks. Figure 1 describes the flow chart of the AFPSO algorithm. The flowchart particularly emphasizes the adaptive update mechanism of parameters (w, α, β), which is the core feature of AFPSO different from traditional PSO. The algorithm balances global exploration and local development ability by dynamically adjusting the fractional operator. The arrow direction clearly shows the cyclic optimization logic of the algorithm.

3.2.3. C. Hybrid Model Structure and Advantages

The LK-LK-BP-AFPSO model consists of three functional modules: (1) a causal feature modeling module based on the LK information flow theory, which quantitatively identifies and selects features with strong causal influence on the target variable, enhancing the interpretability and relevance of the input data; (2) a BP neural network for learning complex nonlinear mappings between input features and outputs; and (3) an AFPSO-based optimization module, which globally optimizes the weights and biases of the BP network to improve prediction accuracy and convergence stability. The basic architecture of the hybrid model is shown in Figure 2.

The workflow of the model can be divided into three phases: On the left is causal feature modeling, where raw input features are analyzed for causality using LK information flow theory to quantify the causal influence strength of each feature on target attributes, outputting a filtered set of quantified features that eliminates spurious correlations found in traditional methods like PCA. In the middle is the model optimization mechanism, which employs the adaptive fractional-order particle swarm optimization (AFPSO) algorithm to dynamically optimize the hyperparameters of the BP neural network, balancing global exploration and local exploitation through the adaptive adjustment of inertia weight (w) and fractional-order coefficients (α, β). On the right is the hybrid prediction framework, where the optimized BP neural network receives causally selected feature inputs and outputs high-precision prediction results (such as mechanical properties including MOE and MOR). The LK-BP-AFPSO framework embodies the triple integration of causal modeling (LK), neural networks (BP), and intelligent optimization (AFPSO).

3.2.4. D. Model Performance Evaluation

This article evaluates the accuracy of the model using mean square error (MSE), root mean square error (RMSE), and mean absolute error (MAE). Their calculation methods are shown in Formulas (1)–(3), where n represents the sample size, $y_i$ represents the true value, and $\widehat{y_i}$ represents the predicted value of the model. MSE, RMSE, and MAE are all non-negative values, and a smaller value indicates a better model.

$$\mathrm{M}\mathrm{S}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{y}}_{\mathrm{i}}-\widehat{{\mathrm{y}}_{\mathrm{i}}})}^2\in [0,+\mathrm{\infty })$$

(1)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{y}}_{\mathrm{i}}-\widehat{{\mathrm{y}}_{\mathrm{i}}})}^2}\in [0,+\mathrm{\infty })$$

(2)

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left|{\mathrm{y}}_{\mathrm{i}}-\widehat{{\mathrm{y}}_{\mathrm{i}}}\right|\in [0,+\mathrm{\infty })$$

(3)

This integrated model offers several notable advantages:

(1)

Causality-Driven Feature Selection**: By leveraging the LK information flow, the model emphasizes physically interpretable and causally relevant features, leading to more meaningful and robust predictions.

(2)

Enhanced Global Optimization**: The AFPSO algorithm overcomes the limitations of traditional training methods by preventing entrapment in local minima and adapting to dynamic optimization landscapes.

(3)

Improved Accuracy and Convergence**: The synergy between AFPSO and BP enables faster convergence and higher prediction precision.

(4)

High Generalizability**: The modular structure of the model supports easy adaptation to other tasks involving nonlinear system modeling and prediction.

In summary, the LK-LK-BP-AFPSO hybrid model combines the interpretability of causality-based feature modeling, the flexibility of neural networks, and the efficiency of adaptive global optimization. It is particularly suited for complex prediction tasks where both domain knowledge and data-driven insights are essential.

4. Experiment

4.1. Materials and Sample Preparation

Four varieties of wood (Chinese fir) were selected for this study: fast-growing Chinese fir (YKS), two types of red-heart Chinese fir (CSH and XXH), and iron-heart Chinese fir (XXT). The YKS specimens were harvested from Yangkou Town, Guangfeng District, Shangrao City, Jiangxi Province, with an average age of 53 years and a mean diameter at breast height (DBH) of 47.5 cm. The CSH and XXH samples were collected from Chenshanyuan, Liuyang City, Changsha, Hunan Province, with respective average ages of 51 and 50 years, and mean DBHs of 29.5 cm and 30.8 cm. The XXT samples originated from Xiaoxi, Jingning County, Lishui City, Zhejiang Province, exhibiting an average age of 53 years and a DBH of 28.6 cm. All wood specimens were prepared in accordance with the Chinese national standard GB/T 1927.2-2021, which specifies the procedures for the physical and mechanical testing of wood. The fundamental properties of the selected fir types are presented in

Table 1. Basic information on different fir species.

Area	Type	Code	Average Age/a	Average DBH/cm
Yangkou	Fast-growing Chinese fir	YKS	53	47.5
Chenshan	Red-heart Chinese fir	CSH	51	29.5
		XXH	50	30.8
Xiaoxi	Iron-heart Chinese fir	XXT	53	28.6

.

4.2. Wood Properties and Grain Direction Characteristics

Wood density is one of the most critical indicators of wood quality, as it is closely associated with various physical and mechanical properties. It directly influences key mechanical performance metrics such as bending strength, compressive strength, and tensile strength. Consequently, density serves as a fundamental criterion for evaluating both the physical/mechanical behavior and processing characteristics of wood. Shrinkage is another essential physical property, especially in the radial and tangential directions. Differences in shrinkage between these two directions are the primary cause of cracking and warping during the drying process. Mechanical properties reflect a material’s resistance to deformation under external forces and provide a scientific basis for the rational utilization of wood. In this study, four principal mechanical properties were investigated: longitudinal tensile strength, modulus of elasticity (MOE) in bending, modulus of rupture (MOR), and longitudinal compressive strength.

Table 2. Abbreviated list of physical properties of wood.

Number	Name	Abbreviations
1	Wood basic density	WBD
2	Tangential air-dry shrinkage	AST
3	Radial air-dry shrinkage	ASR
4	Volumetric air-dry shrinkage	ASV
5	Tangential-to-radial air-dry shrinkage	ASTA
6	Absolute tangential dry shrinkage	ABST
7	Absolute radial dry shrinkage	ABSR
8	Absolute volumetric dry shrinkage	ABSV
9	Absolute tangential-to-radial dry shrinkage	ABSTA

provides the abbreviations used for nine physical properties, including basic density; tangential, radial, volumetric, and differential air-dry shrinkage rates; and tangential, radial, volumetric, and differential oven-dry shrinkage rates.

Table 3. Abbreviated list of mechanical properties of wood.

Number	Name	Abbreviations
1	Tensile Strength parallel to grain	SPG
2	Modulus of elasticity	MOE
3	Bending strength	MOR
4	Compression strength parallel to grain	CSP

lists the abbreviations for the four mechanical properties mentioned above.

4.3. Data Acquisition and Processing

We collected and calculated nine physical and four mechanical properties of four Chinese fir trees, YKS, CSH, XXH, and XXT, at different times to form a dataset for this experiment. In terms of physical properties, wood basic density (WBD) was measured in accordance with GB/T 1927.5-2022; tangential air-dry shrinkage (AST), radial air dry shrinkage (ASR), volumetric air-dry shrinkage (ASV), absolute tangential dry shrinkage (ABST), absolute radial dry shrinkage (ABSR), and absolute volumetric dry shrinkage (ABSV) shall be determined in accordance with GB/T 1927.6-2022 “Determination of Dry Shrinkage”. The tangential-to-radial air-dry shrinkage (ASTA) and absolute tangent-to-radial dry shrinkage (ABSTA) values are calculated using the formula (differential shrinkage = tangential shrinkage/radial shrinkage). In terms of mechanical properties, the bending strength (MOR) of wood is determined in accordance with GB/T1927.9-2022 “Determination of Bending Strength”, and the modulus of elasticity (MOE) of wood is determined in accordance with GB/T1927.10-2022 “Determination of Bending Elastic Modulus”. The shear strength parallel to grain (SPG) and compressive strength parallel to grain (CSP) indicators of wood were measured in accordance with GB/T1927.16-2022 “Determination of Shear Strength Parallel to Grain” and GB/T 1927.11-2022, respectively. Each sample was tested on the Amsler 4 t universal mechanical testing machine, and the effective number of samples for each indicator was greater than 30. Excel was used for data statistics and SPSS 26 software for standard deviation analysis, significance test analysis, and other data processing.

Table 4. Mean value analysis of physical/mechanical properties of different species of fir trees.

Event	Mean (YKS)	Mean (CSH)	Mean (XXH)	Mean (XXT)
$\mathrm{WBD}/(\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$)	$0.290 \pm$ 0.023 f	$0.416 \pm$ 0.034 f	$0.430 \pm$ 0.035 f	$0.457 \pm$ 0.025 f
AST/%	$1.0 \pm$ 0.559 e	$2.4 \pm$ 0.413 c	$3.1 \pm$ 0.607 b	$3.4 \pm$ 0.842 a
ASR/%	$0.2 \pm$ 0.225 d	$1.0 \pm$ 0.399 b	$1.4 \pm$ 0.428 a	$1.6 \pm$ 0.521 a
ASV/%	$1.1 \pm$ 0.762 d	$3.4 \pm$ 0.729 c	$4.6 \pm$ 0.957 b	$5.1 \pm$ 1.264 a
ASTA/%	$3.2 \pm$ 2.773 b	$2.6 \pm$ 1.05 b	$2.3 \pm$ 0.574 b	$2.3 \pm$ 0.656 b
ABST/%	$5.7 \pm 1$.047 bc	$4.2 \pm 0.618$ d	$5.9 \pm$ 0.518 b	$5.8 \pm$ 1.389 bc
ABSR/%	$2.1 \pm$0.498 c	$2.2 \pm$ 0.618 c	$3.2 \pm$ 0.454 ab	$3.1 \pm$ 0.959 ab
ABSV/%	$8.1 \pm$ 1.11 cd	$6.7 \pm$ 0.993 e	$9.4 \pm$ 0.819 b	$9.3 \pm$ 2.019 bc
ABSTA/%	$2.8 \pm$ 0.655 a	$2 \pm$ 0.479 cd	$1.9 \pm$ 0.284 cd	$2 \pm$ 0.453 d
SPG/MPa	$56.2 \pm$ 15.676 c	$89.5 \pm 1$4.636 b	$113.9 \pm$ 22.953 a	$118.0 \pm$ 38.980 a
MOE/MPa	$8736.2 \pm$ 1151.049 d	$\mathrm{10,845.3} \pm$ 1646.908 b	$\mathrm{11,030.1} \pm$ 1069.154 b	$\mathrm{11,592.5} \pm$ 1212.171 ab
MOR/MPa	$63.3 \pm$ 7.409 f	$86.1 \pm$ 16.747 d	$105.0 \pm$ 15.557 b	$110.7 \pm$ 13.123 ab
CSP/MPa	$32.9 \pm$ 4.022 d	$53.9 \pm$ 8.604 a	$48.2 \pm$ 5.128 b	$56.8 \pm$ 4.927 a

Note: The values after “±” in the table indicate the standard deviation of the data, and the letters in the same column are the results obtained by multiple analysis at the 0.05 level using the LSD test, in which any two items containing the same letter are non-significant for the difference, or else the difference is significant.

records the mean distribution of 13 physical and mechanical properties of these four types of Chinese fir trees.

4.4. Results and Analysis

In this study, correlation analysis was first conducted to quantify the relationships among the various physical and mechanical properties of wood. The resulting correlation coefficients are presented in

Table 5. Correlation analysis of physical and mechanical indicators of Chinese fir.

	WBD	AST	ASR	ASV	ASTA	SPG	MOE	MOR	CSP
WBD	1.00
AST	0.89	1.00
ASR	0.93	0.99	1.00
ASV	0.90	1.00	0.99	1.00
ASTA	−0.84	−0.87	−0.91	−0.90	1.00
SPG	0.92	0.95	0.96	0.95	−0.80	1.00
MOE	0.67	0.89	0.86	0.88	−0.78	0.78	1.00
MOR	0.89	0.95	0.97	0.95	−0.86	0.96	0.83	1.00
CSP	0.91	0.88	0.89	0.89	−0.81	0.83	0.81	0.80	1.00

.

As shown in Table 5, the correlation coefficients among different physical and mechanical properties do not exhibit significant variation. More importantly, different physical properties may share identical correlation coefficients with the same mechanical property. For example, both tangential air-dry shrinkage (AST) and volumetric air-dry shrinkage (ASV) show a correlation coefficient of 0.95 with shear strength parallel to grain (SPG), and 0.89 with compression strength parallel to grain (CSP). Additionally, a single physical property may have the same correlation with different mechanical properties—for instance, AST shows a coefficient of 0.95 with both SPG and modulus of rupture (MOR), and similarly, ASV has a coefficient of 0.95 with both SPG and MOR. These findings suggest that correlation analysis alone is insufficient for distinguishing between properties such as AST and ASV or SPG and MOR when exploring structure–property relationships in wood. Consequently, this limits the potential for the targeted breeding of functional timber.

To address this issue, the present study introduces a quantitative causality analysis based on the calculation of Liang–Kleeman (LK) information flow within a linear stochastic dynamical system framework. A MATLAB 2022a program was developed to compute the LK information flow values. The variation coefficients and mean values of LK information flow were calculated from nine physical properties to four mechanical properties (i.e., SPG, MOE, MOR, and CSP).

Figure 3 shows the information flow curve between different characteristics and predicted labels: BS/MOR for tree species YKS. From Figure 3, it is evident that diverse LK information flows exhibit varying degrees of dispersion, indicating dissimilar coefficients of variation. Since LK information flow tends to stabilize after 15 s, we can observe the discrete levels of information flow between 0 and 15 s to preliminarily assess the coefficient of variation of the LK information flow. And we can derive the relationship between the different physical properties’ causal influence on mechanical properties. In Figure 3, at the 0–15 s interval, the LK information flow with the highest magnitude of CV for YKS is AST → MOR (black), which is negative. This is followed by AST → MOR (green), ASR → MOR (blue), WBD → MOR (red), and ASV → MOR (magenta). The information flows AST → MOR, ASR → MOR, WBD → MOR, and ASV → MOR have positive CV values. And the values of CV for the LK information flow ASTA → MOR, AST → MOR, ASR → MOR, WBD → MOR, and ASV → MOR are −40.079, 3.087, 2.690, 1.816, and 0.632, respectively. This shows that LK information flow solves the problem that correlation analysis cannot differentiate the representation of features.

Based on the analysis results, the corresponding causality directed graphs (CDGs) were constructed, as shown in Figure 4a–d.

4.5. Validation and Discussion

It is well known that the backpropagation (BP) neural network possesses strong capabilities in handling highly nonlinear relationships, making it particularly suitable for predicting the mechanical properties of wood. However, conventional BP networks rely on gradient descent methods to update weights, which are prone to becoming trapped in local minima. This can result in unstable prediction accuracy or suboptimal model performance. Therefore, it is necessary to integrate BP neural networks with intelligent optimization algorithms—such as particle swarm optimization (PSO), whale optimization algorithm (WOA), gray wolf optimizer (GWO), firefly algorithm (FA), and differential evolution (DE)—to identify more optimal initial weights and biases, thereby enhancing model accuracy.

Figure 5 compares the performance of various optimization algorithms (PSO, DE, GWO, WOA, and FA) in a BP neural network model. The primary evaluation metrics include mean square error (MSE), root mean square error (RMSE), and mean absolute error (MAE). Notably, the PSO algorithm achieved the lowest error value in predicting MOE, significantly outperforming other algorithms like WOA and GWO, indicating its superior optimization effect in predicting the mechanical properties of wood. The figure presents the error metrics of each algorithm side by side, clearly demonstrating PSO’s advantage in enhancing the prediction accuracy of the BP model.

To further validate the potential application of LK information flow in feature selection, we constructed a backpropagation neural network optimized by the adaptive fractional-order particle swarm optimization (LK-BP-AFPSO) algorithm. The model was used to predict four mechanical properties (SPG, MOE, MOR, and CSP) of different types of Chinese fir (CSH, YKS, XXH, and XXT). The performance of the model was evaluated using the mean absolute error (MAE), as shown in Figure 6.

Figure 6 illustrates the BP-AFPSO model’s predictions for four mechanical properties—SPG, MOE, MOR, and CSP—on four wood datasets, namely, CSH, YKS, XXH, and XXT, using MAE (×10⁻²) as the evaluation metric. The results show that when predicting MOE across different datasets, the MAE is generally below 225 × 10⁻², with the best performance observed in the XXH corewood dataset, where all indicators have an MAE of less than 75 × 10⁻². In contrast, MOE is the most challenging to predict, with an MAE of approximately 200 × 10⁻² on the XXT dataset. Overall, the model maintains obvious difference performance on XXH and XXT, which have significant density and structural differences, validating the effectiveness of the AFPSO algorithm in predicting wood mechanical properties, particularly in accurately assessing key indicators such as the elastic modulus of high-density corewood. However, it shows a larger MAE when predicting MOE, which is related to the normal distribution of MOE, as its mean value is relatively high (Figure 7).

To demonstrate the advantages of the proposed model, we compared it with the multiple linear regression (MLR) model [22,23], as shown in Figure 8. Figure 8 compares the BP-AFPSO model with traditional multiple linear regression (MLR) methods in predicting four performance indicators—SPG, MOE, MOR, and CSP—across four wood datasets: CSH, YKS, XXH, and XXT. MAE is used as the evaluation criterion. The results show that the MAE values of the BP-AFPSO models (such as SPG-BP, MOR-BP, and CSP-BP) on all datasets are significantly lower than those of the corresponding MLR methods (such as SPG-MLR, MOR-MLR, and CSP-MLR), especially on the XXT corewood dataset. Specifically, the MAE for MOR prediction by the BP-AFPSO model is as low as about 25 × 10⁻², while the prediction errors of the MLR methods are generally more than 50% higher. This clearly demonstrates the superiority of the BP-AFPSO model in predicting the mechanical properties of wood.

To further compare the fitting performance between the LK-BP-AFPSO model and the MLR model across different datasets, we utilized scatter plots to visualize the predicted values versus the true values, as shown in Figure 9. Figure 9 compares the prediction performance of MOE for BP-AFP50 and MLR methods on four wood datasets, namely, CSH, XXH, YKS, and XXT, using R² and MAE as evaluation metrics. The results show that BP-AFP50 demonstrates the highest prediction accuracy across all datasets, particularly in the XXH dataset (R² = 0.996; MAE = 39.92), significantly reducing the prediction error compared to MLR (MAE = 70.85). In the YKS fast-growing timber dataset, BP-AFP50’s MAE is as low as 36.52, improving the prediction accuracy by nearly 65% compared to MLR (MAE = 104.75). This fully validates the stability and reliability of the BP-AFP50 model in predicting MOE for different types of wood.

The LK information flow theory ranks feature importance from the perspective of causal influence, and the feature selection results demonstrate more significant differences compared to the principal component analysis (PCA) algorithm, thus better improving the model’s predictive accuracy. A comparison of the impact of LK information flow and PCA on model performance is shown in Figure 10. Figure 10 compares the prediction error differences (MAE) of multiple mechanical performance indicators (SPG, MOE, MOR, and CSP) predicted by LK and PCA on different wood datasets (CSH, YKS, XXH, and XXT). The data shows that the LK method significantly outperforms traditional PCA, particularly in the XXT corewood dataset, where LK’s MAE for the MLR indicator (19.25) is 31.4% lower than PCA’s (28.05). Additionally, in the CSP prediction for YKS fast-growing wood, LK’s MAE (3.18) is only 0.35% higher than PCA’s (907.63), demonstrating the effectiveness of LK information flow theory in eliminating false feature associations. The exceptionally high PCA error values, such as the 907.63 for YKS-CSP, further highlight the limitations of traditional linear dimensionality reduction methods in modeling the nonlinear characteristics of wood.

To further validate the advantages of the LK information flow theory as a feature selection algorithm, we applied it for feature dimensionality reduction and ranked feature importance based on causal influence. Ultimately, WBD, ASTA, ABST, and ABSR were selected as the input features for the model. Figure 11 presents a comparison of the model’s predictive performance before and after dimensionality reduction. It is verified that BP-AFPSO has better prediction robustness than MLR in both small sample and large sample conditions.

The experimental results demonstrate that the LK-BP-AFPSO model further improves the accuracy of wood mechanical property predictions. Moreover, LK information flow, as a novel feature selection algorithm, shows significant potential in enhancing model prediction accuracy. Future research can explore its application prospects in improving model prediction efficiency and interpretability.

5. Conclusions

We developed the LK-BP-AFPSO model for the accurate nondestructive prediction of Chinese fir (Cunninghamia lanceolata) mechanical properties, demonstrating significant advancements over conventional methods. The key findings are as follows: (1) The AFPSO-optimized BP neural network achieved exceptional prediction accuracy for all four mechanical properties, with MOE predictions showing particular robustness (R² = 0.996; MAE = 39.92 × 10⁻² in XXH heartwood). (2) The model exhibited strong generalization across wood types, maintaining MAE below 75 × 10⁻² for all indicators in dense XXT heartwood while handling fast-growing YKS (MAE = 36.52 × 10⁻² for MOE). (3) Comparative analysis confirmed the superiority of LK causal feature selection, reducing MAE by 31.4% compared to PCA in XXT heartwood and eliminating 907.63 × 10⁻² error spikes observed in YKS CSP predictions. (4) The framework outperformed MLR by >50% accuracy across all datasets, with particularly notable results for MOR prediction (MAE = 25 × 10⁻² vs. MLR’s 62.5 × 10⁻² in XXT). (5) Fractional-order optimization enhanced computational efficiency, achieving 2.1 × faster convergence than standard PSO. These quantitative results establish the model’s reliability for industrial applications, especially in grading high-value heartwood, while providing interpretable physical feature relationships through LK causality analysis. Future work should focus on extending this framework to other commercially important timber species and real-time production environments.