Reverse Engineering of Laser Welding Process Parameters for Ti6Al4V Alloy Based on Machine Learning

Abstract

The mechanical performance of laser-welded Ti6Al4V alloy joints is governed by multiple process parameters with complex interplay, leading to nonlinear correlations, that complicate the quest for optimal parameters. In this paper, a reverse engineering model for process parameters was developed using backpropagation (BP) neural networks, targeting mechanical properties as the optimization objective for inverse parameter design. The BP neural network was enhanced via differential evolution tuning, achieving significant improvements in both mechanical property prediction and process parameter inversion. The prediction model demonstrated a relative error of approximately 3%, whereas the inverse model exhibited an error of about 6% under varying process conditions. A novel hybrid BP-WC model was then proposed by fusing weight coefficients from both the prediction and inverse models. This model reduced the inverse error of process parameters to 3%, providing a robust framework for efficient parameter optimization in laser welding for Ti6Al4V alloy.

1. Introduction

Ti6Al4V alloy, renowned for its outstanding strength-to-weight ratio, corrosion resistance, and weldability, has been widely employed in the fabrication of aerospace structural components [1]. Laser welding, a technology distinguished by high energy density and precision, plays a crucial role in the fabrication of these components. This is because it can create narrow heat-affected zones and minimize distortion [2]. However, the mechanical performance of laser-welded joints [3,4], encompassing tensile strength [5,6], microhardness [7,8], and fatigue life, is significantly affected by multiple interacting process parameters, such as laser power, welding speed, defocus amount, and wire feeding speed [9,10]. The intricate nonlinear relationship among these parameters and the resulting mechanical properties renders the determination of optimal process conditions through traditional trial-and-error methods arduous. Such methods are not only time-consuming but also material-intensive [11,12].

Forward prediction models have been widely studied in numerous studies to establish correlations between process parameters and weld quality. For example, artificial neural networks (ANNs) [13], especially backpropagation (BP) neural networks, have been utilized to predict microstructural features and mechanical properties based on input parameters [14,15]. Olabi et al. developed an ANN integrated with the Taguchi method to optimize CO₂ laser welding parameters, achieving satisfactory prediction accuracy for weld bead geometry [16]. Wang et al. established a Φ-joint strength relationship and a GA-optimized BP neural network (inputs: penetration/width; outputs: welding parameters) with small errors in magnesium–CFRP riveting–welding bonding, offering a new welding quality control method [17]. However, these studies predominantly focused on the forward mapping from process parameters to outcomes, whereas systematic research on reverse engineering, which involves deriving process parameters from target mechanical properties, remains scarce.

The reverse engineering of process parameters holds significant practical importance for engineering applications, with the aim of establishing a reliable mapping from desired mechanical properties to optimal process settings. Previous endeavors in this field have predominantly relied on response surface methods (RSMs) or simple regression models. These methods often exhibit low accuracy when dealing with highly nonlinear relationships. For example, Kumar et al. established regression models to characterize the relationship between process parameters (laser power, welding speed, and defocused position) and output responses (fusion zone width, heat-affected zone size, and fusion zone area) in fiber laser welding of 5 mm thick Ti-6Al-4V alloy [18]. Yacout et al. conducted a study that employed response surface methodology (RSM) to develop mathematical models for predicting the mechanical properties (ultimate tensile strength (UTS) and hardness (HV)) of friction stir-welded AA6082-T6 joints [19].

The dearth of robust reverse engineering models for laser welding process parameters can be attributed to several challenges [20]. Firstly, the bidirectional nonlinear mapping between parameters and properties gives rise to substantial uncertainty, particularly when multiple parameters interact concurrently. Secondly, the absence of an integrated framework that incorporates microstructural features as intermediate variables impedes the interpretability and accuracy of reverse models. Moreover, the inherent noise in experimental data and the complexity of laser welding physics further exacerbate the challenges in the reverse engineering process. Existing BP models for welding optimization primarily focus on forward mapping (process parameters—mechanical properties) and lack adaptability to noisy experimental data. The BP-WC framework introduces a weight correction (WC) mechanism that dynamically adjusts connection weights based on microstructural feature importance (e.g., martensite length), reducing prediction bias caused by uneven data distribution.

This study aims to address the gaps by developing a BP neural network-based reverse engineering model of laser welding process parameters for Ti6Al4V alloy. The specific objectives are as follows: (1) to construct a bidirectional mapping model that integrates “process parameters–microstructure–mechanical properties” to explore the underlying relationships; (2) to optimize the BP neural network via a differential evolution algorithm to enhance the accuracy of mechanical property prediction and process parameter reverse calculation; (3) to develop a novel BP-WC model based on weight coefficients to reduce reverse engineering errors; and (4) to validate the model through experimental tests for different welding gaps, thereby ensuring its feasibility for aerospace applications. By pursuing these objectives through the established models and experimental validations, this research endeavors to provide an efficient and reliable approach for the reverse engineering design of laser welding parameters.

2. Experimental Methods

2.1. Laser Welding Experiments

Ti6Al4V alloy plates with dimensions of 200 mm × 100 mm × 2 mm were utilized in the experiments, and their chemical composition is presented in

Table 1. Chemical composition of Ti6Al4V alloy (wt.%).

Al	V	Fe	C	N	H	O	Ti
5.5–6.75	3.5–4.5	0.30	0.08	0.05	0.015	0.20	Bal.

. The welding system comprised a 6000 W fiber laser (Wuhan Raycus Fiber Laser Technologies Co., Ltd., Wuhan, China), a KUKA robotic arm (KUKA Industries Automation (China) Co., Ltd., Kunshan, China), a wire feeding system (Corporate name: Panasonic Welding Systems (Tangshan) Co., Ltd., Tangshan, China). This configuration allowed for precise control of process parameters. Fifty-six groups of laser welding experiments were designed based on Latin Hypercube Sampling (LHS) to ensure comprehensive coverage of the multi-parameter space, which is critical for capturing nonlinear relationships between process parameters and outcomes, with detailed parameters listed in

Table 2. Parameter table for laser welding of Ti6Al4V alloy.

Sample No.	Gap (mm)	Defocus Amount (mm)	Laser Power (W)	Welding Speed (m/min)	Scanning Amplitude (mm)	Wire Feeding Speed (m/min)
1–20	0.0	0–2	1800–2200	2.0–2.2	0.0	0.00
21–34	0.1–0.5	0–3	1540–1900	1.5–1.8	0–1.2	0.00
35–46	0.2–0.5	1–3	1600–1900	1.6–1.8	0.0	1.44–1.68
47–56	0.3–0.5	3–4	1600–1900	1.5–1.8	1.2	1.56–1.68

. Laser power (ranging from 1540 to 2200 W, with values including 1540, 1600, 1700, …, 2200 W), welding speed (1.5 to 2.2 m/min, with specific levels of 1.5, 1.6, 1.8, 2.0, 2.1, and 2.2 m/min), defocus amount (0 to 4 mm, with increments of 1 mm: 0, 1, 2, 3, and 4 mm), scanning amplitude (0 to 1.2 mm, with values of 0 and 1.2 mm), wire feeding speed (1.44 to 1.68 m/min, including 1.44, 1.56, 1.6, and 1.68 m/min), and welding gaps (0 to 0.5 mm, with variations of 0, 0.1, 0.2, 0.3, 0.4, and 0.5 mm). The laser welding equipment is shown in Figure 1.

Before conducting laser welding experiments, it is necessary to mechanically polish the plate to remove oxides and impurities on the surface and then wipe the surface with anhydrous ethanol and dry it to prevent moisture on the surface of the plate from affecting the laser welding process.

Additionally, before designing the formal experiment, preliminary experiments on different configurations of the laser wire-filling welding process (including laser autogenous welding, oscillating laser welding, laser wire-filling welding, and oscillating laser wire-filling welding as specific setups) were conducted on Ti6Al4V alloy under different gaps. The results showed that different laser welding processes have different filling capabilities for welding gaps. It is shown that the filling ability of laser welding processes on welding gaps gradually increases in the order of laser autogenous welding, oscillating laser welding, laser wire-filling welding, and oscillating laser wire-filling welding.

The schematic diagrams of the four laser welding processes are shown in Figure 2. According to the preliminary experiment results, when the vibration diameter of the oscillating laser beam was 1.2 mm, and the vibration frequency was 90 Hz, this oscillating laser welding process achieved good weld quality on 2 mm thick Ti6Al4V alloy. Moreover, when conducting wire-filling welding for gaps between 0.2 and 0.5 mm, welded joints with good mechanical properties were obtained by setting wire feeding speeds of 1.44, 1.56, 1.6, and 1.68 m/min.

2.2. Data Collection and Preprocessing

After welding, the specimens were etched with Kroll’s reagent to expose the microstructure, which was subsequently observed via an optical microscope (Nanjing Jiangnan Yongxin Optics Co., Ltd., Nanjing, China). As illustrated in Figure 3, the martensite length and depth-to-width ratio of the weld fusion zone were measured at three characteristic positions (front, middle, and rear) of each sample, and the average values were determined. Martensite length and depth-to-width ratio were measured manually. Mechanical property tests were executed in accordance with GB/T standards [21]: tensile tests were conducted on an electronic universal testing machine at a loading rate of 1.0 mm/min; microhardness was measured via a Vickers hardness tester with a load of 10 kgf and a dwell time of 15 s; and fatigue tests were performed on a PLG - 200C fatigue testing machine (MTS Systems Corporation, Eden Prairie, MN, USA) at a stress amplitude of 300 MPa with a 70 Hz sinusoidal waveform. Figure 4 shows the microhardness, tensile strength, and fatigue life of the welded joint at different positions, which were measured according to standard protocols. Additionally, the average value of the above characteristics was calculated.

The acquired dataset underwent preprocessing via the min–max normalization technique, scaling all variables to the interval [0, 1]. An 8-fold cross-validation approach was employed to partition the dataset into training (80%) and testing (20%) subsets, serving to mitigate overfitting and ensure model generalizability. The details are as follows:

$$Z_i={\displaystyle \frac{x_i-x_{min}}{x_{max}-x_{min}}}$$

(1)

3. Experimental Results

There remains an irreplaceable need to investigate the synergistic effect of process parameters. In actual welding processes, process parameters (laser power, welding speed, and gap) do not act in isolation; instead, they collectively regulate microstructural characteristics (e.g., martensite size, depth-to-width ratio) via intermediary mechanisms such as heat input and molten pool behavior. Suppose this synergistic effect is ignored, and conclusions are drawn solely based on the influence of individual parameters. In that case, the resulting model will struggle to reflect the physical essence of real welding processes. Consequently, subsequent construction of machine learning models may be biased due to the lack of correlation among input variables.

This study on synergistic effects thus lays a key foundation for the development of the BP-WC model. On one hand, it clarifies the basis for selecting input variables of the model: process parameters (e.g., laser power, welding speed) and microstructural features (e.g., martensite length, depth-to-width ratio) must be jointly included in the input layer, rather than relying on a single dimension. On the other hand, it reveals the chain of “parameter interaction–microstructural evolution–performance variation,” which provides physical meaning for designing “weighting coefficients” in the BP-WC model.

Figure 5 shows the cross-sectional morphology of Ti6Al4V alloy laser-welded joints, along with the correlations between melting depth, melt width, and heat input density. In laser autogenous welding (Figure 5a,b), “light leakage” results in insufficient energy absorption, leading to defects such as incomplete penetration and surface collapse. The depth-to-width ratio increases under low heat input density (with melting depth growing faster than melt width) but decreases with excessive heat input (as melt width grows faster).

For gaps ≥ 0.2 mm, oscillating laser welding (Figure 5c,d, 0.3 mm gap) was employed to mitigate intensified “light leakage,” forming a symmetric “Y”-shaped joint without obvious pores. Melting depth, melt width, and depth-to-width ratio first increase and then decrease with heat input density: low input induces heat conduction welding, high input triggers deep melting, and gap-induced energy leakage eventually reduces the ratio.

For gaps ≥ 0.3 mm, laser wire-filling welding (Figure 5e,f, 0.4 mm gap) prevents defects, with joints exhibiting a “Y-H” transition. Melt width first decreases and then increases with heat input density, while melting depth and depth-to-width ratio show the opposite trend—low input prioritizes wire melting (with less energy absorbed by the base material), and high input saturates wire energy absorption (enabling more energy to be absorbed by the base material).

For gaps ≥ 0.4 mm, oscillating laser wire-filling welding (Figure 5g,h, 0.5 mm gap) suppresses “light leakage,” forming an “H”-shaped joint. Melt width decreases, while melting depth and depth-to-width ratio increase with heat input density, due to dominant energy absorption by the filler wire and molten metal filling.

Welding processes regulate joint morphology through the synergistic effect of energy distribution and material supplementation (filler wire). Heat input density drives transitions in welding modes, while gap size and filling strategy affect energy utilization efficiency, collectively governing the evolution of melting depth and melt width.

Figure 6 illustrates the microstructures of Ti6Al4V alloy laser-welded joints’ fusion zones and the correlation between martensite length and heat input density under different welding processes.

Figure 6a,b show laser autogenous welding: The fusion zone is composed of longitudinal/transverse acicular α′ martensite and a small amount of parallel acicular α phase. Martensite length first decreases and then increases with heat input density—low input enhances laser stirring, which inhibits growth, while high input weakens stirring and slows cooling, leading to coarsening.

Figure 6c,d present oscillating laser welding: The fusion zone has a similar structure (acicular α′ martensite and α phase), but the martensite length is significantly smaller than that in autogenous welding, attributed to enhanced molten pool stirring induced by the oscillating laser beam. It continues to decrease with increasing heat input, as the stirring effect is strengthened.

Figure 6e,f display laser wire-filling welding: The fusion zone contains acicular α′ martensite and Widmannstätten structure. The filler wire promotes martensite formation, resulting in a larger martensite length compared to autogenous welding; higher heat input, enhanced laser stirring, and homogenized solute diffusion gradually reduce the martensite length.

Figure 6g,h show oscillating laser wire-filling welding: The fusion zone consists of directional acicular α′ martensite and α phase, with a smaller martensite size than that in wire-filling welding alone. Martensite length decreases initially (strengthened oscillation stirring inhibits growth) but increases when heat input exceeds a threshold (stirring saturates, and cooling slows, allowing growth).

For Ti6Al4V alloy laser-welded joints, tensile properties, microhardness, and fatigue strength are core indices for evaluating comprehensive mechanical properties, with weld microstructure characteristics exerting a significant influence on them. Using the control variable method, the effects of individual microstructural features and their dominant mechanisms were analyzed to guide process parameter optimization.

Microhardness (Figure 7a,b): With similar depth-to-width ratios, microhardness decreases as martensite length increases; with similar martensite lengths, it first rises and then falls with increasing depth-to-width ratio (peaking at ~0.65). Martensite length has a more pronounced effect, as acicular α′ martensite forms a tightly interlocked “basket weave structure” that enhances hardness.

Tensile strength (Figure 7c,d): Acicular α′ martensite in a basket weave structure improves both strength and toughness. With similar depth-to-width ratios, tensile strength increases with martensite length; with similar martensite sizes, it first rises and then falls with the depth-to-width ratio (peaking at ~0.73). A reasonable depth-to-width ratio is critical—excessively large or small values deteriorate performance, while martensite length plays a more significant role within this range.

Fatigue life (Figure 7e,f): Trends mirror those of tensile strength. With similar depth-to-width ratios, fatigue life increases with martensite length; with similar martensite sizes, it improves with the depth-to-width ratio. This is attributed to the fact that a higher depth-to-width ratio reduces stress concentration (resulting in fewer crack sources), while longer martensite enhances toughness (slowing crack propagation).

4. Model

4.1. BP-WC

In laser welding of Ti6Al4V alloys, the “process parameters–microstructure–mechanical properties” correlation involves complex multi-parameter coupling and dynamic weight variations: Laser power and welding speed synergistically regulate heat input, affecting microstructural features like martensite length and depth-to-width ratio, which in turn determine mechanical properties via mechanisms such as grain boundary strengthening. Traditional orthogonal experiments and simple formulas struggle to quantify these nonlinear mappings. Neural networks offer potential for “performance-oriented” parameter inverse derivation, but pure BP inverse models have flaws: they ignore the physical significance of neural layer weights, leading to conflicts between results and actual laws, significant error fluctuations, and sensitivity to data noise, failing aerospace accuracy requirements.

As shown in Figure 8, the BP-WC model evolved from traditional BP flaws: Early BP forward models only unidirectionally predicted properties, with large error fluctuations; BP reverse models, swapping inputs/outputs and using forward weight pseudo-inverses, suffered from network asymmetry (e.g., mismatched hidden layer neurons), accumulating errors, and limiting accuracy. The BPWC model addresses these issues.

As shown in Figure 9, centered on bidirectional mapping and weight correction, the BP-WC model relies on a four-layer logic: forward network foundation, backward network inverse derivation, intermediate layer decoupling, and weight mechanism optimization.

The forward network takes process parameters as input, extracts features via two ReLU-activated hidden layers, and outputs mechanical properties, learning how parameters affect properties through microstructure to provide weight/bias benchmarks for backward derivation, resolving traditional BP’s lack of physical correlation.

The backward network inputs target mechanical properties, processes them via two ReLU-activated hidden layers, and outputs process parameters. It uses the forward network’s weight matrix pseudo-inverse for inverse operations, but hidden layer neuron asymmetry introduces errors, addressed by a weight correction mechanism fusing forward pseudo-inverse and backward weights, with gap-adaptive coefficients.

A microstructure intermediate layer decomposes high-dimensional mapping into “process parameters–microstructure” and “microstructure–mechanical properties” sub-problems. As a mechanistic link, it reflects thermal-induced phase transformation logic and reduces fitting difficulty via dimension reduction, stabilizing backward derivation.

4.2. Model Building

In laser welding of Ti6Al4V alloys, the mapping between process parameters (six items: laser power, welding speed, defocus distance, wire feeding speed, etc.), microstructure (two items: martensite length and depth-to-width ratio), and mechanical properties (three items: microhardness, tensile strength, and fatigue life) is strongly nonlinear, eluding accurate capture by linear equations or simple monomial functions. Traditional BP neural networks construct forward prediction models for such relationships, but these have limitations.

The BP forward model builds a “process parameters–microstructure–mechanical properties” mapping, with a three-layer structure: the “process parameters–microstructure” sub-model has six input neurons (for six parameters) and two output neurons (for two microstructural features); the “microstructure–mechanical properties” sub-model has two input neurons and three output neurons (for three properties). Trained on 56 samples (70% training, 30% validation) with ReLU activation and MAE loss, it achieves ~4% average error for microstructure (max 3.57% for martensite length, 5.33% for depth-to-width ratio) and ~5% for mechanical properties (Figure 10 and Figure 11). However, it shows significant error fluctuations and cannot directly inverse-derive process parameters.

The BP inverse model constructs an inverse mapping of “mechanical properties–microstructure–process parameters” to derive process parameters from target mechanical properties, with a structure symmetrical to the forward model: the “mechanical properties–microstructure” sub-model has three input neurons and two output neurons; the “microstructure–process parameters” sub-model has two input neurons and six output neurons. Trained on the same dataset with the MSE loss function, it shows low inverse accuracy (Figure 12): average microstructure inverse error is 5~6.7% and 6.73~7.14% for laser power and welding speed, with severe error fluctuations.

Core reasons: It is trained independently without leveraging the forward model’s physical mapping logic, causing weight matrix deviations; unequal hidden layer neurons (e.g., mismatched counts between forward “process parameters–microstructure” and inverse “microstructure–process parameters” layers) make the weight matrix non-square, requiring pseudo-inverse calculations that introduce extra errors.

As shown in Figure 13, to address these issues, the BP-WC (backpropagation weight correction) inverse derivation model is proposed. It is built on the inverse operation of the forward model, invoking its weight and bias matrices, and it incorporates weight coefficients from the inverse model for correction and uses a differential evolution algorithm to optimize the network structure, achieving synergistic optimization of forward and inverse models.

A key design of the BP-WC model is using microstructure (martensite length and depth-to-width ratio) as intermediate feature vectors to form a two-stage mapping chain of “process parameters–microstructure–mechanical properties”, rooted in welding mechanisms. This design reduces the dimensionality of complex mappings to enhance learning efficiency, endows physical interpretability to avoid the “black box” limitation, and buffers error propagation to reduce cumulative deviation.

4.3. Optimization Design

The basic architecture of the BP-WC model adopts a three-layer BP neural network, a choice driven by the advantage of three-layer networks in balancing nonlinear mapping capability and model complexity. The hierarchical design of the input layer, hidden layer, and output layer can not only fully capture the complex correlations among “process parameters–microstructure–mechanical properties” but also avoid overfitting caused by excessive layers, which is well-suited to the multi-parameter coupled nonlinear characteristics in laser welding of Ti6Al4V alloys.

The model realizes bidirectional mapping through the synergistic operation of forward and backward networks: Forward propagation undertakes the basic mapping task of “process parameters–microstructure–mechanical properties”, with its core being the calculation via the formula, where z^h represents the computation result of neurons in the h layer, a^h−1 is the input matrix, and W^h and b^h are the weights and biases of this layer, respectively:

$$z^h=a^{h-1}{W}^h+b^h$$

(2)

The introduction of the ReLU function can effectively address the gradient vanishing problem in deep network training. Therefore, after completing the linear transformation, nonlinear processing is performed via the ReLU activation function, ultimately providing reliable benchmark weights and bias matrices for backward inverse derivation. The backward inverse derivation, on the other hand, conducts inverse operations based on the weights and bias matrices of the forward model, starting with the following formula:

$$z^h=f^{-1}\left(a^h\right)$$

(3)

To address this, the model innovatively introduces a weight correction mechanism, which uses a dynamic correction coefficient to fuse the forward network’s pseudo-inverse matrix and the backward inverse network’s weights. Adaptively learned from training data, this coefficient adjusts the two matrices’ contribution ratios based on specific welding conditions (e.g., material properties, gaps). Ultimately, iterative optimization via an algorithm minimizes deviations between inverse results and actual process parameters, enhancing inverse derivation accuracy and stability.

$$w^{3-h}=a\times {\left(W^h\right)}^{-1}+b\times W^{3-h}$$

(4)

The forward pseudo-inverse matrix is fused with the backward network’s weight coefficients W^3−h, where correction coefficients a and b are dynamically adjusted by the welding gap (with distinct values for 0.3 mm and 0.5 mm gaps to minimize inverse errors under respective conditions). This corrects systematic errors from pseudo-inverse operations, enabling high-precision inverse derivation from target properties to process parameters.

To optimize model performance, hyperparameters are tuned via a differential evolution algorithm: Insufficient neurons cause underfitting, while excess leads to overfitting. As shown in Figure 14, for the BP forward model, MSA errors are minimized (0.06 and 0.08) when hidden layers of “process parameters–microstructure” and “microstructure–mechanical properties” sub-models have 24 and 12 neurons, respectively. For the inverse model, MSE errors are smallest (0.08) with 18 and 24 neurons in the hidden layers of “mechanical properties–microstructure” and “microstructure–process parameters” models (notably, MSE differs slightly—0.08 and 0.09—for 12and18 neurons in “mechanical properties–microstructure”). Thus, forward prediction uses MAE (robust to outliers), and backward inverse derivation uses MSE (sensitive to large errors) to balance task requirements.

4.4. Comparative Experiments

To verify the BP-WC model’s superiority, 16 test samples (30% of the total) were used for comparison with the traditional BP model (results in Figure 15).

Compared with the BP forward model, the BP-WC model has similar forward prediction errors (microstructure average ~2%, mechanical properties ~3%) but provides a more reliable weight benchmark for backward inverse derivation.

Compared with the BP inverse model, the BP-WC model shows significantly improved inverse accuracy: The average inverse error of microstructure (martensite length and depth-to-width ratio) is reduced to 2% (max 4.4%), outperforming the BP inverse model’s 5~6.7%. The average inverse error of process parameters (laser power and welding speed) is 3.95~4.49% (max 6.46%), better than the BP inverse model’s 6.73~7.14%.

Key reasons for error reduction are as follows: microstructure’s intermediate buffering reduces error accumulation; the weight correction mechanism corrects pseudo-inverse matrix deviation; and hyperparameters optimized by the differential evolution algorithm make the model more consistent with actual mappings.

Notably, comparing the BP-WC model before and after optimization (with/without the differential evolution algorithm) is crucial to verify the improvement effect of hyperparameter optimization (e.g., adjusting hidden layer neurons to 24 and 12) on accuracy.

Based on the differential evolution algorithm’s optimization results for BP forward and inverse models, their neural network structures were redesigned: 24 neurons for the “process parameters–microstructure” hidden layer and 12 for “microstructure–mechanical properties”. The redesigned models were trained and validated to provide parameters for BP-WC optimization. As shown in Figure 16, the optimized BP forward model’s “process parameters–microstructure” part has average prediction errors of 1.54% and 2.23% for microstructure characteristics, with significantly reduced fluctuations.

In Figure 17, the prediction results of the “microstructure–mechanical properties” part of the optimized BP forward model are shown. Comparing the above results with the prediction results before optimization, it can be found that the prediction accuracy of the BP forward model for mechanical properties improves, with the average error being less than 2% and the error fluctuations having decreased. The above results indicate that the addition of the differential evolution algorithm improves the accuracy and reliability of material property prediction.

In Figure 18, the inverse derivation results of the optimized BP inverse derivation model are presented. Comparing the above inverse derivation results with those in Figure 12, it can be found that the optimized model significantly improves the inverse derivation accuracy of microstructure characteristics, with only some samples having inverse derivation errors of microstructure characteristics exceeding 5%. Moreover, the inverse derivation accuracy of laser power and welding speed is also improved, with their average inverse derivation errors reduced to 4.17% and 4.84%, respectively. In addition, by comparing the inverse derivation results of different samples, it can be seen that the error fluctuations of the model significantly improve.

Based on the optimized BP forward and inverse derivation models for Ti6Al4V alloy laser welding, their weight coefficients and bias coefficients were obtained to establish the optimized BP-WC inverse derivation model. The inverse derivation model was then trained and validated, and the inverse derivation results of the optimized BP-WC inverse derivation model are shown in Figure 19. By comparing the inverse derivation results before and after optimization, it can be found that the optimized BP-WC inverse derivation model significantly improves the inverse derivation accuracy of microstructure characteristics and process parameters. Moreover, the error fluctuations obviously decrease: the average inverse derivation error for microstructure characteristics decreases by about 2%, while the average inverse derivation error for process parameters drops to 3%.

5. Results and Discussion

Taking the mechanical property indicators of aviation structural parts as the target (tensile strength ≥ 950 MPa, microhardness ≥ 400 HV10, and fatigue life ≥ 4 × 10⁵ cycles), the BP-WC inverse derivation model was used to perform inverse derivation of process parameters for Ti6Al4V alloy butt structures with gaps of 0.3 mm and 0.5 mm, obtaining two sets of key parameters. A galvo laser wire-filling welding process was adopted to carry out verification experiments. The inverse derivation process is shown in Figure 20: first, the process parameters were output through the BP-WC model, and then the BP forward model was used to predict whether the performance meets the standards, and finally, two sets of parameters were selected for actual welding (Specimen 1: gap 0.3 mm, laser power 1870 W, welding speed 1.67 m/min; Specimen 2: gap 0.5 mm, laser power 2060 W, welding speed 1.5 m/min). Experiments show that the weld formation of the two groups of specimens is good, with no obvious pores or incomplete penetration defects (Figure 21).

The prediction results of the BP forward model for microstructure are shown in Figure 22, with error data listed in

Table 3. Prediction errors of microstructure characteristics by the forward calculation model.

Specimen No.	Predicted Value of Martensite Length (μm)	Actual Value of Martensite Length (μm)	Prediction Error of Martensite Length (%)	Predicted Value of Depth-to-Width Ratio	Actual Value of Depth-to-Width Ratio	Prediction Error of Depth-to-Width Ratio (%)
1	67.6	67	0.90	0.69	0.64	7.81
2	74.5	73.6	1.09	0.74	0.71	4.23

. Martensite length: the prediction error ranges from 0.90% to 1.09% (Specimen 1: predicted 67.6 μm andactual 67 μm; Specimen 2: predicted 74.5 μm and actual 73.6 μm). Depth-to-width ratio: the prediction error ranges from 4.23% to 7.81% (Specimen 1: predicted 0.69 and actual 0.64; Specimen 2: predicted 0.74 and actual 0.71).

The prediction errors of the BP forward model for mechanical properties are shown in

Table 4. Prediction errors of mechanical properties by the forward calculation model.

Specimen No.	Predicted Microhardness (HV10)	Actual Microhardness (HV10)	Microhardness Error (%)	Predicted Tensile Strength (MPa)	Actual Tensile Strength (MPa)	Tensile Strength Error (%)	Predicted Fatigue Life (105 Cycles)	Actual Fatigue Life (105 Cycles)	Fatigue Life Error (%)
1	401.8	405.7	0.96	994.8	963.9	3.21	4.15	4.05	2.47
2	401.2	411.2	2.43	976	995.1	1.92	4.1	4.17	1.68

and Figure 23.

Microhardness: error of 0.96~2.43% (Specimen 1: predicted 401.8 HV and actual 405.7 HV; Specimen 2: predicted 401.2 HV and actual 411.2 HV); tensile strength: error of 1.92~3.21% (Specimen 1: predicted 994.8 MPa and actual 963.9 MPa; Specimen 2: predicted 976 MPa and actual 995.1 MPa); fatigue life: error of 1.68~2.47% (Specimen 1: predicted 4.15 × 10⁵ cycles and actual 4.05 × 10⁵ cycles; Specimen 2: predicted 4.1 × 10⁵ cycles and actual 4.17 × 10⁵ cycles). The overall error is ≤3.21%, which verifies the reliability of the inverse-derived parameters by the BP-WC model.

Experiments verified the process parameters inverse-derived by the BP-WC model: the prediction errors of microstructure and mechanical properties are both below 4%, significantly outperforming the BP inverse model before optimization (with errors of 6~7%), which proves that the weight correction mechanism and the optimization of the differential evolution algorithm are effective. The welded joints corresponding to the inverse-derived parameters meet the performance indicators of aviation structural parts, and the galvo laser wire-filling process has an excellent filling effect on gaps of 0.3~0.5 mm, providing a feasible solution for the welding process design of complex gap structures.

6. Conclusions

This article is based on a BP artificial neural network and builds forward and backward models for Ti6Al4V alloy laser welding. Furthermore, based on the weight coefficients of the BP forward and backward models, a more effective BP-WC reverse model was constructed. By dividing and training the collected data, the predictive performance of the BP model was verified, and a differential evolution algorithm was introduced to optimize the network model, ultimately obtaining a BP-WC reverse engineering model with better performance. This mainly includes the following features:

(1) The BP forward and inverse model for “process parameters–microstructure–mechanical properties” was established, where there was a hidden layer between the input layer, intermediate layer, and output layer. At the number of neurons in the hidden layer between process parameters and microstructure is 24, and the number of neurons in the hidden layer between microstructure and mechanical properties is 12, the model has the best prediction performance. Through the iterative training, the average error is less than 2%.

(2) The weight coefficients of the BP forward and reverse models were obtained. After weighting and correction processing, the “mechanical properties–microstructure–process parameters” BP-WC reverse model was established. The results confirmed that the reverse effect of this model is better than the BP reverse model, with an average error of about 3%, and the error fluctuation situation is significantly improved.

(3) The differential evolution algorithm was introduced to optimize the BP-WC inverse model by changing the number of neurons in the BP forward and inverse models. It facilitated the obtainance of weight coefficients that are closer to the actual mapping relationship, and improving the BP-WC inverse model. Additionally, the optimized BP forward model will be used as a validation tool for the reverse model to improve the accuracy of the reverse calculation of process parameters.