A Physics-Guided Two-Stage Learning Framework for Constitutive Modeling of TC4 Titanium Alloy: Validation Through Temperature and Strain-Rate Extrapolation

Abstract

Accurate constitutive modeling of TC4 titanium alloy at elevated temperatures is critical for process design and numerical simulation in aerospace manufacturing. However, purely data-driven deep neural networks (DNNs) often suffer from severe overfitting and may yield physically unreasonable predictions in data-sparse or strictly out-of-distribution (OOD) regions. To address this issue, this study proposes a physics-guided two-stage neural network framework, termed NN-PhysicsInit, for the constitutive modeling of TC4 alloy. In Stage I, a large synthetic dataset generated from a strain-compensated Arrhenius-type constitutive equation is used to pre-train the network, thereby introducing analytical prior knowledge into the initial topological space. In Stage II, the pre-trained model is fine-tuned using rigorously corrected experimental data obtained from isothermal compression tests conducted over 800–980 °C and 0.001–1 s⁻¹ to improve material-specific predictive accuracy. To evaluate generalization capability, a rigorous dual-perspective extrapolation validation scheme is designed separately in the temperature (1010 °C) and strain-rate (10 s⁻¹) dimensions. The results demonstrate that, compared with direct black-box training, the proposed framework successfully prevents non-physical divergence and better preserves macroscopic thermodynamic smoothness in unseen domains. Specifically, the extrapolation average absolute relative error (AARE) is significantly reduced from 34.21% to 14.34% in the temperature extrapolation task, and from 27.91% to 8.92% in the strain-rate extrapolation task. These findings confirm that physics-based initialization acts as a powerful implicit regularizer, effectively mitigating the extrapolation catastrophe while maintaining high fitting accuracy. The proposed framework provides a robust and practical strategy for the constitutive modeling of complex alloys under limited-data conditions.

1. Introduction

TC4 (Ti-6Al-4V) titanium alloy is widely utilized in aerospace, marine engineering, and other high-end manufacturing fields due to its high specific strength, excellent corrosion resistance, and favorable properties at elevated temperatures [1]. During hot working processes such as forging and superplastic forming, its flow behavior is strongly dependent on temperature, strain rate, and strain, governed by the synergistic effects of work hardening, dynamic recovery (DRV), and dynamic recrystallization (DRX) [2,3]. Consequently, accurate constitutive modeling of TC4 alloy is of paramount importance for process optimization, finite element method (FEM) simulation, and performance control of formed components [4].

Among conventional constitutive approaches, the strain-compensated Arrhenius-type equation remains a benchmark for describing the hot deformation of titanium alloys [5,6]. These analytical models are compact and physically interpretable, effectively capturing the coupled effects of thermomechanical parameters. However, because the functional form is predefined, their predictive accuracy is often limited when the material response becomes highly nonlinear, particularly in regimes involving pronounced flow softening or phase-related complexities [7,8]. As noted by Buchmayr [9], while traditional models provide a reliable baseline, they struggle to meet the precision requirements of modern digital twins in complex manufacturing environments.

With the development of machine learning, deep neural networks (DNNs) have attracted growing attention in constitutive modeling due to their strong nonlinear mapping capability [10,11]. Recently, comprehensive reviews, such as the latest exhaustive work published in Progress in Materials Science [12], have systematically highlighted the transformative impact of data-driven approaches in predicting the complex thermomechanical behaviors of metallic materials. Furthermore, advanced applications of artificial neural networks have successfully demonstrated superior predictive capabilities in capturing the highly non-linear flow characteristics of various alloys under extreme processing conditions [13]. In many cases, DNN-based models achieve high fitting accuracy within the training domain. Nevertheless, under limited and imbalanced datasets, purely data-driven models are prone to overfitting and often exhibit poor extrapolation behavior [14]. This issue is particularly critical in hot compression experiments, where high-strain-rate conditions typically yield much fewer data points because the deformation is completed rapidly [15]. Furthermore, high-strain-rate experimental data inherently contain localized noise and measurement artifacts (e.g., interfacial friction and adiabatic heating). Purely data-driven models lack explicit guidance from underlying deformation physics [16,17] and inevitably overfit these imperfections. As emphasized by Bock et al. [14], balancing model flexibility with physical robustness remains a core obstacle. Consequently, when applied to sparsely sampled or unseen regions, these “black-box” models produce unstable or trend-inconsistent predictions, which significantly reduces model reliability for digital manufacturing [18].

Therefore, it is highly desirable to develop a constitutive modeling strategy that combines the flexibility of neural networks with the robustness of physically motivated analytical models [19,20]. Recent advances in physics-guided and transfer-learning-based modeling have shown that prior physical knowledge can improve data efficiency and generalization performance [21,22,23,24,25,26]. Advanced frameworks have recently moved toward integrating physical metallurgy laws, such as the work by Li et al. [25] utilizing metallurgy-guided learning to predict microscopic deformation mechanisms. While such approaches are immensely valuable, providing a highly reliable, non-divergent macroscopic constitutive equation for complex aerospace FEM simulations requires a different perspective—one strictly governed by continuum mechanics and thermodynamic consistency.

While the phenomenological rheological behaviors of the TC4 titanium alloy have been extensively documented in previous decades, providing a rich foundation of experimental data, the direct application of pure data-driven deep learning models for FEM digital twins remains highly perilous due to the aforementioned extrapolation catastrophe involving extreme local gradients. Therefore, in this study, the well-documented TC4 alloy is strategically utilized as a “benchmark material” to propose and validate a novel methodological framework: a macroscopic physics-guided two-stage learning framework termed NN-PhysicsInit. Instead of embedding explicit physical residuals into the loss function, the framework introduces analytical prior knowledge through pre-training on a smooth synthetic dataset generated by a calibrated Arrhenius model, followed by fine-tuning with actual experimental data. This strategy acts as an implicit physical regularizer, filtering out experimental noise and constraining the initial parameter hypothesis space to align with thermodynamic boundaries.

In addition to the model design, this work emphasizes a highly stringent out-of-distribution (OOD) validation protocol. Unlike conventional data-driven studies that predominantly rely on random interpolation testing, a rigorous dual-perspective extrapolation strategy is designed to challenge the physical boundaries of the model. Specifically, the framework is blindly extrapolated across two critical dimensions: the thermodynamic boundary (temperature extrapolation to 1010 °C, crossing the macroscopic β-transus) and the kinetic boundary (strain-rate extrapolation to 10 s⁻¹, featuring severe data imbalance and dynamic softening). The scientific novelty of this work does not lie in discovering new deformation mechanisms of TC4, but in establishing a robust algorithmic bridge between classical thermodynamic bounds and high-capacity non-linear machine learning. This dual-perspective validation strictly proves that the embedded physical priors can effectively prevent the catastrophic extrapolation failure typical of pure black-box models, guaranteeing physically bounded predictions under unseen extreme conditions and providing a trustworthy digital foundation for aerospace component manufacturing.

The main contributions of this work are as follows:

• A novel macroscopic physics-guided learning framework (NN-PhysicsInit) is proposed. Unlike conventional mixed-data models that suffer from gradient clash, this decoupled two-stage strategy utilizes an analytical Arrhenius prior for topological pre-training (acting as an implicit regularizer), followed by empirical fine-tuning to accurately capture material-specific nonlinearities.
• A rigorous dual-perspective out-of-distribution (OOD) extrapolation scheme is designed. The generalization capability of the framework is systematically challenged across both thermodynamic (extrapolating across the macroscopic β-transus boundary at 1010 °C) and kinetic dimensions (extrapolating to the highly imbalanced 10 s⁻¹ domain).
• The “extrapolation catastrophe” inherent in purely data-driven models is successfully mitigated. The proposed framework effectively filters out inherent high-frequency experimental noise and measurement artifacts. It guarantees physically bounded and trend-consistent predictions under unseen extreme conditions, providing a trustworthy constitutive engine for advanced aerospace digital twins.

2. Materials and Methods

2.1. Experimental Materials and Procedures

The investigated material in this study was a commercial TC4 (Ti-6Al-4V) titanium alloy (provided by Baoti Group Ltd., Baoji, China), and its detailed chemical composition is listed in

Table 1. Chemical composition of the investigated TC4 titanium alloy (wt.%).

C	Fe	N	H	Al	O	V	Ti
0.04	0.24	0.013	0.015	6.36	0.20	4.37	Bal.

.

To obtain the fundamental flow stress data under various deformation conditions, isothermal hot compression tests were conducted on a Gleeble-3800 thermal simulator (Dynamic Systems Inc., Poestenkill, NY, USA). The complete experimental procedure and the designed thermomechanical pathways are schematically illustrated in Figure 1. Cylindrical specimens with a diameter of 8 mm and a height of 12 mm were machined along the axial direction of the as-received ingot. To minimize interfacial friction during compression and prevent heterogeneous deformation (barreling), both ends of the specimens were carefully polished, and tantalum foils combined with graphite lubricant were applied. All tests were performed in a vacuum environment to prevent oxidation at elevated temperatures.

Based on the typical hot working windows for the TC4 alloy, the deformation temperatures were set at 800, 850, 900, 950, 980, and 1010 °C. Concurrently, the strain rates were selected as 0.001, 0.01, 0.1, 1, and 10 s⁻¹, resulting in a comprehensive experimental matrix of 30 distinct deformation conditions.

The comprehensive rheological behaviors of the TC4 alloy under various temperatures and strain rates are illustrated in Figure 2. To rigorously eliminate measurement artifacts, the raw experimental data (faint lines) were recalculated using standard friction and adiabatic heating correction models, yielding the genuine isothermal flow stress curves (solid lines) [27]. Furthermore, the figure superimposes the analytical baseline calculated by the strain-compensated Arrhenius model. To explicitly demonstrate our data isolation strategy, the plot distinguishes between the interpolation points used for model calibration (marked with solid dots) and the out-of-distribution (OOD) extrapolation predictions (marked with crosses) for the unseen extreme conditions (i.e., at 10 s⁻¹ and 1010 °C).

2.2. Establishment of the Analytical Prior Model

To provide a physically motivated prior for network pre-training, a strain-compensated Arrhenius-type constitutive model was first established. The relationship among strain rate ($\dot{\epsilon }$), flow stress (σ), and deformation temperature (T) can be expressed as:

$$\dot{\epsilon }=A{\left[\sinh \left(\alpha \sigma \right)\right]}^n\exp \left(-Q/RT\right)$$

(1)

where A, n, and α are material constants, Q is the activation energy for hot deformation, and R is the universal gas constant.

Under low-stress and high-stress conditions, Equation (1) can be expressed as follows:

$$\dot{\epsilon }=A_1{\sigma }^{n_1}\exp (-Q/RT)$$

(2)

$$\dot{\epsilon }=A_2\exp \left(\beta \sigma \right)\exp \left(-Q/RT\right)$$

(3)

where A₁, A₂, n₁, and β are material constants, and α = β/n₁. The Zener–Hollomon parameter (Z) is defined as:

$$Z=\dot{\epsilon }\exp \left(Q/RT\right)=A{\left[\sinh \left(\alpha \sigma \right)\right]}^n$$

(4)

Taking logarithms of Equations (2) and (3) yields:

$$\ln \dot{\epsilon }=\ln A_1-Q/RT+n_1\ln \sigma$$

(5)

$$\ln \dot{\epsilon }=\ln A_2-Q/RT+\beta \sigma$$

(6)

From the fitted linear relationships of $\ln \dot{\epsilon }$ versus ln σ and $\ln \dot{\epsilon }$ versus σ, the parameters n₁ and β can be obtained.

Taking the logarithm of Equation (1) gives:

$$\ln \dot{\epsilon }=\ln \left(A\right)+n\ln \left[\sinh \left(\alpha \sigma \right)\right]-\left(Q/RT\right)$$

(7)

For a given temperature, the material constant n can be expressed as:

$$n={\left[{\displaystyle \frac{\partial \ln \dot{\epsilon }}{\partial \ln \left[\sinh \left(\alpha \sigma \right)\right]}}\right]}_T$$

(8)

For a given strain rate, the activation energy Q can be obtained from:

$$Q=Rn{\left[{\displaystyle \frac{\partial \ln \left[\sinh \left(\alpha \sigma \right)\right]}{\partial \left(1/T\right)}}\right]}_{\stackrel{.}{\epsilon }}$$

(9)

Substituting Equation (8) into Equation (9) yields:

$$Q=R{\left[{\displaystyle \frac{\partial \ln \dot{\epsilon }}{\partial \ln \left[\sinh \left(\alpha \sigma \right)\right]}}\right]}_T{\left[{\displaystyle \frac{\partial \ln \left[\sinh \left(\alpha \sigma \right)\right]}{\partial \left(1/T\right)}}\right]}_{\stackrel{.}{\epsilon }}$$

(10)

Taking the logarithm of Equation (4) yields:

$$\ln Z=\ln A+n\ln \left[\sinh \left(\alpha \sigma \right)\right]$$

(11)

Thus, the values of n and ln A can be obtained from the linear relation between ln [sinh (ασ)] and ln Z.

To enhance the predictive accuracy of the constitutive model, the influence of true strain on the flow stress was incorporated into the analysis. Derived from Equation (4), the constitutive equation correlating the flow stress with the Zener–Hollomon (Z) parameter can be expressed as follows:

$$\sigma ={\displaystyle \frac{1}{\alpha }}\mathrm{In}\left\{\right({\displaystyle \frac{Z}{A}}{)}^{{\displaystyle \frac{1}{n}}}+{[{({\displaystyle \frac{Z}{A}})}^{{\displaystyle \frac{2}{n}}}+1]}^{{\displaystyle \frac{1}{2}}}\}$$

(12)

To account for the strain dependence of the material behavior, the fundamental material constants (α, n, Q, and ln A) were treated as functions of the true strain. These parameters were sequentially calculated at strain intervals from 0.05 to 0.60 with a step of 0.05, and then smoothly fitted using sixth-order polynomial functions.

$$\begin{array}{l}{\displaystyle \ln A}={{\displaystyle 57.39+37.79\epsilon -674.52\epsilon }}^2{{\displaystyle +3081.96\epsilon }}^3{{\displaystyle -7275.80\epsilon }}^4{{\displaystyle +8687.73\epsilon }}^5{{\displaystyle -4105.77\epsilon }}^6\\ {{\displaystyle \alpha =0.02-0.02\epsilon +0.12\epsilon }}^2{{\displaystyle -0.23\epsilon }}^3{{\displaystyle -2.32\epsilon }}^4{{\displaystyle -0.01\epsilon }}^5{{\displaystyle -0.50\epsilon }}^6\\ {{\displaystyle n=2.76+0.64\epsilon -17.50\epsilon }}^2{{\displaystyle +92.15\epsilon }}^3{{\displaystyle -207.04\epsilon }}^4{{\displaystyle +211.07\epsilon }}^5{{\displaystyle -77.22\epsilon }}^6\\ {{\displaystyle Q=611.82+281.52\epsilon -5370.73\epsilon }}^2{{\displaystyle +25909.79\epsilon }}^3{{\displaystyle -60748.35\epsilon }}^4{{\displaystyle +72391.56\epsilon }}^5{{\displaystyle -34214.52\epsilon }}^6\end{array}$$

(13)

To further validate the reliability of the constructed strain-compensated Arrhenius model, the calculated material constants were compared with similar models reported in published literature for TC4 (Ti-6Al-4V) titanium alloys. As shown in Figure 3c, the calculated deformation activation energy (Q) in this study varies from approximately 480 to 620 kJ/mol depending on the true strain. This range is highly consistent with the typical activation energies required for the hot deformation of Ti-6Al-4V alloys, which are generally widely reported to be in a similar magnitude by recent comprehensive constitutive studies (e.g., Hu et al. [4] and Cai et al. [5]). Furthermore, while many conventional studies predominantly employ 4th- or 5th-order polynomials to describe the strain dependence of these parameters, this study utilizes a higher-order (6th-order) polynomial fitting, as detailed in Equation (13). Utilizing a 6th-order polynomial effectively minimizes the inherent regression errors of the analytical model, thereby providing a more precise and thermodynamically rigorous mathematical baseline for the subsequent Stage I pre-training of the neural network.

The resulting strain-compensated Arrhenius model was subsequently utilized to generate the large-scale synthetic dataset for the Stage I pre-training. It should be particularly emphasized that, to prevent data leakage during subsequent extrapolation validation, the calibration of this baseline Arrhenius model strictly utilized only the data designated for the training set (e.g., ≤980 °C or ≤1.0 s⁻¹), keeping the extrapolation targets entirely unseen. Following this rigorous data splitting, it should be noted that the polynomial coefficients listed in Equation (13) specifically correspond to the analytical baseline calibrated for the temperature extrapolation task. For the strain-rate extrapolation ablation study, the Arrhenius parameters (α, n, Q, and ln A) were independently re-calibrated using their respective training subsets. To avoid redundancy, those specific alternative coefficients are omitted here, as the mathematical structure and derivation procedure remain perfectly identical.

2.3. Physics-Guided Two-Stage Neural Network Framework (NN-PhysicsInit)

To improve constitutive modeling under limited and imbalanced data conditions, a two-stage neural network framework, termed NN-PhysicsInit, was developed. The key idea is to introduce analytical prior knowledge during initialization and then refine the model using experimental data.

Stage I: Pre-training with synthetic data: In the first stage, a large-scale synthetic dataset was generated using the calibrated strain-compensated Arrhenius model. Compared with the actual experimental dataset, which is sparse and unevenly distributed in some extreme deformation regions, the synthetic dataset was meticulously designed to cover the thermomechanical parameter space more uniformly. The input variables included deformation temperature (T), strain rate ($\dot{\epsilon }$), and true strain (ε), and the corresponding true stress (σ) values were calculated using the analytical model. This stage provides the neural network with a physically guided initialization. From a machine learning perspective, pre-training on smooth synthetic data constrains the initial parameter space and reduces the tendency of the model to overfit local fluctuations or noise in the experimental data. As illustrated in Figure 4, the synthetic dataset uniformly samples the temperature and logarithmic strain rate, projecting them into a smooth baseline stress mapping.

Stage II: Fine-tuning with experimental data. While the analytical Arrhenius prior provides a robust macroscopic boundary, it inherently lacks the non-linear capacity to capture the specific, localized dynamic softening features of the TC4 alloy at high strain rates. Therefore, the pre-trained network was further fine-tuned using the actual experimental stress–strain dataset. However, the critical challenge during this phase is to prevent the risk of prior degradation—a phenomenon where extensive unconstrained fine-tuning completely overwrites the physical priors, degrading the network back into a purely data-driven black box. To resolve this, the Stage II optimization was strictly regulated as a local, rather than an unconstrained global, search. By employing a significantly reduced learning rate (e.g., 1.5 × 10⁻⁴) combined with a strict early-stopping criterion monitored via validation loss, the network is restricted to moderate parameter adjustments. This optimized strategy ensures that the model successfully compensates for TC4-specific empirical non-linearities without compromising the global thermodynamic topology established in Stage I. The overall workflow of this proposed NN-PhysicsInit approach, integrating the physically guided topological initialization (Stage I) and the subsequent material-specific accuracy compensation (Stage II), is comprehensively illustrated in Figure 5.

It is imperative to clearly distinguish the proposed NN-PhysicsInit framework from existing hybrid constitutive models reported in the literature. Firstly, unlike standard Physics-Informed Neural Networks (PINNs) or conventional mixed-data training architectures that typically embed physical constraints as soft penalty residuals within the loss function—which often suffer from gradient clash as the network struggles to simultaneously minimize the loss for smooth physical priors and noisy local experimental variations—the proposed decoupled method injects physical priors through initial weight space topological mapping. This serves as an implicit regularizer rather than a loss penalty, allowing the network to first establish a globally stable thermodynamic manifold before performing localized optimization. Secondly, while conventional transfer learning transfers knowledge between different empirical datasets to accelerate convergence, the Stage I pre-training transfers knowledge from a deterministic analytical manifold to the network, deliberately restricting the hypothesis space to prevent catastrophic extrapolation. Finally, compared to traditional Arrhenius-ANN coupled frameworks that calculate analytical baselines and ANN residuals in parallel, NN-PhysicsInit absorbs the thermodynamic laws purely during the training phase. Consequently, it maintains the computational efficiency of a singular, end-to-end deep neural network during inference, making it highly robust for extreme out-of-distribution (OOD) scenarios in digital manufacturing.

Furthermore, it is critical to discuss the phenomenological consequences of excessive fine-tuning in Stage II. If the fine-tuning duration is extended indefinitely without strict early-stopping regularization, the neural network becomes overly biased towards minimizing the empirical loss on the limited experimental dataset. Consequently, the model begins to overfit the inherent high-frequency noise and local measurement artifacts. From a topological perspective, this excessive optimization leads to the severe degradation of the physical prior, where the globally smooth physical manifold constructed during Stage I is gradually overridden and distorted. As a result, the implicit thermodynamic bounds are lost, and the out-of-distribution (OOD) extrapolation performance degrades sharply, eventually regressing to the unstable behavior of a purely data-driven model. Therefore, determining the optimal fine-tuning stopping point—where material-specific non-linearities are sufficiently captured while the Stage I physical topology remains intact—is fundamentally essential for the success of this hybrid framework.

2.4. Extrapolation Validation Design and Implementation Details

2.4.1. Dual-Perspective Extrapolation Ablation Design

Before executing the dataset splitting, it is necessary to analyze the distribution of the collected experimental data. As illustrated in the coverage count heatmap (Figure 6), the rigorously corrected isothermal hot compression dataset exhibits an unbalanced distribution. For instance, the low strain rate domains (e.g., 0.001 s⁻¹) contain abundant sample points, whereas the high strain rate conditions (e.g., 10 s⁻¹) are relatively sparse due to the rapid completion of the dynamic deformation process. This inherent data imbalance is primarily caused by the intrinsic variations in test durations and the hardware limits of dynamic sampling rates during Gleeble high-speed compression. Such disparity inevitably leads to severe bias in standard data-driven models, which further highlights the necessity of injecting uniformly sampled physical priors (Stage I) to regularize the network.

This data sparsity poses a significant challenge for purely data-driven neural networks, increasing the risk of overfitting and distribution shift when transitioning to unseen conditions [27,28]. Therefore, to evaluate the out-of-distribution (OOD) generalization of the proposed NN-PhysicsInit architecture trained on this corrected thermodynamic baseline, this study adopted a dual-perspective extrapolation validation scheme instead of traditional random data division. The corrected isothermal experimental data corresponding to the highest temperature (1010 °C) and the highest strain rate (10 s⁻¹) were deliberately excluded from the training set to serve as testing domains:

• Temperature extrapolation ablation: Corrected experimental data obtained at 800, 850, 900, 950, and 980 °C were utilized for the training set, while the data at 1010 °C were reserved as a strictly unseen testing set. This strategy is designed to evaluate the model’s thermodynamic stability and extrapolation capability across the critical β-transus phase boundary.
• Strain rate extrapolation ablation: Corrected experimental data at strain rates of 0.001, 0.01, 0.1, and 1.0 s⁻¹ were allocated to the training set, while the data at the extreme strain rate of 10 s⁻¹ were reserved as an unseen test set. This approach evaluates the model’s kinetic robustness in the presence of inherent data imbalance and severe non-linear dynamic softening behaviors.

To further verify this OOD design, the statistical distribution of the true stress labels was analyzed. As depicted in the distribution comparison histogram (Figure 7), there is a noticeable data shift between the training set and the extrapolation test set. The high-stress regions (particularly exceeding 350 MPa) are mostly located within the extrapolation set. This distribution discrepancy indicates that the validation scheme functions as an extrapolation task rather than a standard interpolation, requiring appropriate physical constraints to stabilize predictions.

To systematically evaluate the effectiveness of the physics-guided training strategy under these conditions, an ablation framework was established. As summarized in

Table 2. Design of the comparative ablation study.

No.	Models Compared	Training Strategy
1	Arrhenius	Traditional analytical model (strain-compensated)
2	NN-Direct	Neural network trained from scratch using experimental data
3	NN-Arrhenius-Only	Neural network pre-trained using synthetic Arrhenius data only
4	NN-PhysicsInit (Ours)	Neural network pre-trained using synthetic data and fine-tuned using experimental data

, four distinct modeling approaches with different training strategies were deployed for comparison. This matrix aims to decouple and verify the contributions of the analytical physical prior (Stage I) and the experimental data fine-tuning (Stage II) to the final generalization capability.

2.4.2. Neural Network Architecture and Hyperparameters

The neural network had a topology of 3→128→128→64→1, consisting of one input layer, three hidden layers, and one output layer. The input variables were temperature, logarithmic strain rate, and true strain. The Sigmoid Linear Unit (SiLU) was used as the activation function, and the mean squared error (MSE) was used as the loss function. Although this architecture contains approximately 25,000 trainable parameters—which typically introduces a high risk of overfitting in standard data-driven models—this sufficient network capacity is intentionally retained to capture the severe non-linear dynamic softening at extreme strain rates. The inherent overfitting risk is fundamentally suppressed by the implicit regularization provided by the Stage I physics-guided pre-training, as detailed in Section 3.4.

During data preprocessing, the input variables were normalized using the Z-score method, and the flow stress target was transformed logarithmically. Model optimization was performed using the Adam optimizer together with a cosine annealing learning rate scheduler. Because the two extrapolation tasks differ in data distribution and difficulty, task-specific training hyperparameters were used. The detailed settings are summarized in

Table 3. Neural network architecture and task-specific training hyperparameter configurations.

Category	Parameter	Temperature Extrapolation	Strain Rate Extrapolation
Shared Architecture	Topology	3→128→128→64→1	3→128→128→64→1
	Input variables	(T/1000, ln($\dot{\epsilon }$), ε)	(T/1000, ln($\dot{\epsilon }$), ε)
	Activation/Loss	SiLU/MSE	SiLU/MSE
	Optimizer	Adam	Adam
	Weight decay	1 × 10−5	1 × 10−5
Dataset Splitting	Interpolation Temp.	800, 850, 900, 950, 980 °C	800, 850, 900, 950, 980, 1010 °C
	Extrapolation Temp.	1010 °C	—
	Interpolation Strain Rates	0.001, 0.01, 0.1, 1.0, 10.0 s−1	0.001, 0.01, 0.1, 1.0 s−1
	Extrapolation Strain Rate	—	10.0 s−1
	Synthetic dataset size	15,000	10,000
NN-Direct	Epochs/Batch size	2000/64	1500/64
	Initial learning rate	6 × 10−4	1 × 10−3
NN-Arrhenius-Only	Epochs/Batch size	1200/64	800/64
	Initial learning rate	8 × 10−4	1 × 10−3
NN-PhysicsInit: Stage I	Epochs/Batch size	1200/512	800/512
	Initial learning rate	8 × 10−4	1 × 10−3
NN-PhysicsInit: Stage II	Epochs/Batch size	1000/64	600/64
	Initial learning rate	1.5 × 10−4	2 × 10−4

. The neural network architecture and data processing were implemented using the Python programming language (version 3.9.12) and the PyTorch framework (version 1.12.1, Meta Platforms, Inc., Menlo Park, CA, USA).

2.4.3. Statistical Evaluation Metrics

To rigorously evaluate the model performance, the correlation coefficient (R), the coefficient of determination (R²), the average absolute relative error (AARE), and the root mean square error (RMSE) were introduced. These metrics are mathematically defined as follows:

$$R={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n\left({{\displaystyle y}}_i-\overline{{\displaystyle y}}\right)}\left({{\displaystyle \widehat{y}}}_i-\overline{\widehat{y}}\right)}{\sqrt{{\displaystyle {\sum }_{i=1}^n{{\displaystyle \left({{\displaystyle y}}_i-\overline{y}\right)}}^2}·{\displaystyle {\sum }_{i=1}^n{{\displaystyle \left({{\displaystyle \widehat{y}}}_i-\overline{\widehat{y}}\right)}}^2}}}}$$

(14)

$${{\displaystyle R}}^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{{\displaystyle \left({{\displaystyle y}}_i-{{\displaystyle \widehat{y}}}_i\right)}}^2}}{{\displaystyle {\sum }_{i=1}^n{{\displaystyle \left({{\displaystyle y}}_i-\overline{y}\right)}}^2}}}$$

(15)

$$AARE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|\frac{y_i-{\widehat{y}}_i}{y_i}\right|}\times 100\%$$

(16)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{{\displaystyle \left(y_i-{\widehat{y}}_i\right)}}^2}}$$

(17)

3. Results

3.1. Comparison of Predictive Performance in Training and Extrapolation Domains

The predictive performance of NN-PhysicsInit was compared with that of the Arrhenius model, NN-Direct, and NN-Arrhenius-Only.

To ensure statistical robustness, all reported neural network metrics represent the average results from 5 random restarts of the training process. As shown in

Table 4. Statistical metrics of different models in the temperature extrapolation task.

No.	Models Compared	Data Set	R	R2	AARE (%)	RMSE (MPa)
1	Arrhenius	Training set (800–980 °C)	0.9803	0.9609	16.30	22.22
		Testing set (1010 °C)	0.9625	0.9263	25.72	13.49
2	NN-Direct	Training set (800–980 °C)	0.9978	0.9957	3.51	6.34
		Testing set (1010 °C)	0.9807	0.9617	34.21	14.49
3	NN-Arrhenius-Only	Training set (800–980 °C)	0.9803	0.9610	16.37	22.22
		Testing set (1010 °C)	0.9627	0.9267	25.74	13.48
4	NN-PhysicsInit	Training set (800–980 °C)	0.9949	0.9899	7.92	9.73
		Testing set (1010 °C)	0.9287	0.8625	14.34	11.12

, under this rigorous evaluation, the NN-Direct model achieves the lowest training error, with an average AARE of 3.51% and an RMSE of 6.34 MPa, indicating a strong fitting ability within the training domain. However, its performance deteriorates substantially in the unseen 1010 °C domain, where the extrapolation AARE surges to 34.21%. It should be noted that the determination coefficient (R²) of NN-Direct remains deceptively high in this testing set despite its poor AARE. This is mainly because the overall stress magnitude in the high-temperature domain is inherently low; thus, moderate absolute deviations can produce massive relative errors. In this extreme scenario, AARE serves as a much more sensitive and rigorous metric for evaluating extrapolation inaccuracies. Meanwhile, the Arrhenius model and NN-Arrhenius-Only exhibit nearly identical performance, indicating that the pre-training stage perfectly enables the network to reproduce the analytical baseline. However, both inherit the intrinsic approximation errors of the empirical Arrhenius formulation, thereby demonstrating limited predictive accuracy. The detailed statistical results of independent repeated runs are summarized in Tables S1–S4 of the Supplementary Materials.

By comparison, the proposed NN-PhysicsInit framework provides an exceptional balance between fitting performance and extrapolation stability. Its AARE in the 1010 °C testing set is significantly reduced to 14.34%, and its RMSE is reduced to 11.12 MPa. To understand the significance of this improvement, it is crucial to consider the underlying thermodynamic phase transition involved in this specific extrapolation task. At 1010 °C, the TC4 alloy crosses its macroscopic β-transus boundary. Although the single Arrhenius prior (calibrated primarily below this temperature) inherently lacks the domain-specific parameters for the single β-phase region, the NN-PhysicsInit framework seamlessly overcomes this physical gap. It effectively utilizes its Stage II fine-tuning to adaptively compensate for the macroscopic phase-shift and the associated stress reduction. Ultimately, this robust performance at 1010 °C, coupled with the high-strain-rate extrapolation at 10 s⁻¹, constitutes a rigorous dual-perspective validation. The consistency observed across these two orthogonal extrapolation perspectives (thermodynamic and kinetic) confirms that the proposed framework reliably preserves macroscopic physical trends and accurately self-corrects, even when the baseline analytical model reaches its theoretical limit.

A similar trend is observed in the kinetic strain-rate extrapolation task (

Table 5. Statistical evaluation metrics of different models on the strain rate training and testing sets.

No.	Models Compared	Data Set	R	R2	AARE (%)	RMSE (MPa)
1	Arrhenius	Training set $\dot{\epsilon }$ ≤ 1 S−1	0.9607	0.9229	23.14	23.84
		Testing set $\dot{\epsilon }$ = 10 S−1	0.9778	0.9561	16.83	51.54
2	NN-Direct	Training set $\dot{\epsilon }$ ≤ 1 S−1	0.9978	0.9956	2.69	5.00
		Testing set $\dot{\epsilon }$ = 10 S−1	0.9534	0.9090	27.91	59.85
3	NN-Arrhenius-Only	Training set $\dot{\epsilon }$ ≤ 1 S−1	0.9608	0.9231	23.11	23.81
		Testing set $\dot{\epsilon }$ = 10 S−1	0.9777	0.9559	16.87	51.48
4	NN-PhysicsInit	Training set $\dot{\epsilon }$ ≤ 1 S−1	0.9900	0.9802	10.92	10.66
		Testing set $\dot{\epsilon }$ = 10 S−1	0.9787	0.9578	8.92	25.34

). The purely data-driven NN-Direct model exhibits excellent fitting performance within the training domain (with an AARE of 2.69% and an RMSE of 5.00 MPa); however, it suffers a severe loss of accuracy in the strictly unseen 10 S⁻¹ testing domain, where the extrapolation RMSE predictably surges to 59.85 MPa and the AARE reaches 27.91%. In striking contrast, the proposed NN-PhysicsInit framework significantly reduces the extrapolation AARE to 8.92% and the RMSE to 25.34 MPa, demonstrating superior predictive robustness against the highly non-linear dynamic softening behaviors characteristic of extreme strain rates.

Taken together, the quantitative evaluations across Table 4 and Table 5 confirm that the direct application of unconstrained black-box models is highly perilous for reliable constitutive extrapolation. The proposed dual-stage framework successfully mitigates this “extrapolation catastrophe” by seamlessly combining the thermodynamic reliability of analytical priors with the flexibility of data-driven empirical corrections.

To rigorously justify the necessity of the proposed framework, five conventional machine-learning models, including Random Forest, ExtraTrees, XGBoost, SVR-RBF and KNN, were evaluated as benchmarks. As detailed in Supplementary Tables S5 and S6, while these models show acceptable interpolation accuracy, they suffer from catastrophic divergence in out-of-distribution (OOD) tasks. For instance, in the 1010 °C temperature extrapolation task, the AARE of the SVR-RBF model surges to 80.20%, whereas NN-PhysicsInit maintains a robust 14.34%. Similarly, for the 10 s⁻¹ strain-rate extrapolation, the AARE of SVR-RBF reaches 39.93%, while our framework achieves a high-precision prediction of 8.92%. This stark contrast compellingly proves that without inherent thermodynamic constraints, unconstrained data-driven algorithms fail to capture the physical boundaries of metal deformation.

The correlation scatter plots in Figure 8 and Figure 9 are consistent with the quantitative statistical results presented above. The purely data-driven NN-Direct model exhibits high accuracy within the interpolation training domain; however, it shows significant scattering and deviation from the ideal diagonal line when evaluated in the out-of-distribution (OOD) extrapolation regions. By comparison, the proposed NN-PhysicsInit framework demonstrates a much more stable and tightly bounded error distribution under these unseen extreme conditions. Benefiting from the implicit regularization of the pre-trained physical baseline, the dual-stage framework effectively mitigates the overfitting variance associated with unconstrained black-box models while reducing the systematic deviation inherent to the empirical Arrhenius equation.

3.2. Comparison of Stress–Strain Curve Fitting in the Training Domain

To intuitively evaluate the constitutive responses predicted by different models, four highly indicative cases were extracted from the full dataset to demonstrate the fitting performance under various thermomechanical conditions within the training domain. These representative curve comparisons are shown in Figure 10.

As clearly observed in the selected representative cases (Figure 10), the classical analytical Arrhenius model (blue dashed line) outlines the macroscopic thermodynamic contour but suffers from significant systematic deviations—such as massive overestimation at low strain rates (e.g., Figure 10a) and noticeable underestimation at high strain rates (e.g., Figure 10b)—due to the rigid constraints of its predefined mathematical structure. The proposed NN-PhysicsInit framework successfully utilizes this analytical prior as a topological baseline and precisely compensates for these empirical errors through the second-stage data fine-tuning, pulling the predictions effectively back into the accurate experimental domain.

More importantly, Figure 10 explicitly reveals the fundamental behavioral differences between the models when encountering experimental artifacts. As prominently exhibited in Figure 10c (900 °C, 0.001 s⁻¹), the raw experimental data (grey solid line) inevitably contains an anomalous, upward-drifting tail at large strains. This is likely a measurement artifact caused by dynamic instrumental interferences (such as excessive interfacial friction or high-temperature oxidation) rather than true material strain-hardening. The purely data-driven NN-Direct model aggressively minimizes the empirical training loss, essentially “memorizing” this non-physical localized trend. Although this phenomenon explains its deceptively low training error, it exposes the catastrophic sensitivity of unconstrained networks to corrupted local data.

In striking contrast, the proposed NN-PhysicsInit framework acts as a regularized continuum, exhibiting exceptional robustness against such artifacts. It deliberately resists overfitting this localized pseudo-hardening, strictly maintaining the physically correct downward trajectory (dynamic softening) dictated by the Arrhenius prior. This compellingly proves that, even within the interpolation training domain, the proposed dual-stage framework willingly sacrifices marginal local fitting accuracy on corrupted points to achieve a perfect equilibrium between macroscopic physical reliability and data-driven nonlinear capacity.

To comprehensively illustrate the fitting accuracy within the training domain, a detailed visual comparison of the flow behavior across all training temperatures (800–980 °C) and strain rates (0.001–1.0 s⁻¹) among the four modeling approaches is provided in Figure S1 of the Supplementary Materials.

3.3. Extrapolation Behavior in Unseen Deformation Domains

The differences among the four models become more apparent in the extrapolation tasks. Figure 11 and Figure 12 compare the predicted stress–strain curves in the unseen temperature domain (1010 °C) and the unseen strain-rate domain (10 s⁻¹), respectively.

The limitations of the purely data-driven approach (NN-Direct) are strikingly apparent in these out-of-distribution (OOD) regions. As highlighted by the red cross marks, these regions indicate significant non-physical oscillations and inaccurate trend predictions. For instance, at 1010 °C and 1.0 s⁻¹, while the actual material exhibits continuous work hardening, the NN-Direct model erroneously predicts a dynamic softening trend. Similarly, under the 10 s⁻¹ extrapolation condition, it yields concave shapes and substantial stress drops that severely deviate from the typical thermodynamic principles of metal deformation [29].

The fundamental reason why the purely data-driven model and the proposed NN-PhysicsInit model exhibit such divergent extrapolation behaviors lies in their inherent topological constraints. Devoid of any prior physical knowledge, the NN-Direct model merely performs unconstrained mathematical curve fitting, aggressively overfitting the localized high-frequency noise or data imbalance present in the training set. When forced into OOD domains, these overfitted mathematical artifacts are non-physically amplified, resulting in the aforementioned catastrophic stress oscillations and unnatural softening phenomena.

In striking contrast, the NN-PhysicsInit behaves robustly because macroscopic physical laws—specifically, the monotonic influence of thermal activation energy and the Zener–Hollomon parameter—are implicitly encoded into its neural weights during the Stage I pre-training [30,31]. This physical skeleton strictly governs the boundaries of the neural network’s hypothesis space. Therefore, even when navigating blindly through unseen extreme conditions, the model resists chaotic local fluctuations and naturally preserves the thermodynamic smoothness inherited from Arrhenius continuum mechanics. Consequently, at 10 s⁻¹, NN-PhysicsInit effectively mitigates the non-physical fluctuations of the data-driven model and partially corrects the systematic underestimation of the pure Arrhenius model (which struggles to capture intense dislocation proliferation at extreme speeds). Furthermore, at 1010 °C, this implicit regularization restricts the prediction boundaries, capturing thermodynamically reasonable evolutionary trends. Ultimately, this dual-stage physical injection strategy transforms the neural network into a highly reliable extrapolator, guaranteeing physically bounded predictions under unprecedented processing conditions [32].

3.4. Analysis of Parameter Distributions and Their Relation to Model Behavior

To further examine the effect of the two-stage training strategy, the layer-wise weight distributions of the neural network models were compared in both extrapolation tasks, as shown in Figure 13 and Figure 14.

Analysis of the layer-wise probability density distributions of the network weights provides insights into the structural robustness of the proposed framework. From the perspective of machine learning theory, a model with a widely dispersed weight distribution—such as the purely data-driven NN-Direct model, where the standard deviation in the shallow mapping layer (Layer 0) is approximately σ ≈ 0.35–0.37—tends to possess a highly complex and unconstrained hypothesis space. This characteristic inherently increases the risk of fitting high-frequency experimental noise, leading to potential overfitting.

However, by conducting Stage I pre-training on the analytical synthetic dataset, the physical prior effectively acts as an implicit regularizer. As shown by the NN-Arrhenius-Only distributions, this physical injection confines the parameter hypothesis space from a widely dispersed state to a more concentrated, zero-centered distribution. During the subsequent Stage II fine-tuning on real experimental data, the NN-PhysicsInit framework maintains this thermodynamically guided weight initialization. Rather than reverting to a scattered state, its standard deviation in Layer 0 remains bounded at σ ≈ 0.14–0.16. This structural restriction helps prevent weight explosion and promotes the macroscopic topological smoothness of the predicted constitutive surfaces, thereby improving the model’s generalization capability under extreme unseen conditions.

4. Conclusions

In this study, a physics-guided two-stage deep learning framework (NN-PhysicsInit) was developed to describe the hot deformation behavior of TC4 titanium alloy. Unlike conventional neural networks that merely pursue interpolation accuracy, this work rigorously evaluated the model’s out-of-distribution (OOD) extrapolation robustness. The main explicit findings are drawn as follows:

(1) Topological constraint via physical initialization: By decoupling the optimization process into analytical pre-training and experimental fine-tuning, the Arrhenius prior acts as an effective implicit regularizer. It restricts the neural network weights to a concentrated, thermodynamically consistent distribution (with a bounded standard deviation of σ ≈ 0.14–0.16), which mitigates the risk of overfitting inherent in unconstrained black-box models.

(2) Robustness against data imbalance at extreme strain rates: For the blind extrapolation at the highest strain rate (10 s⁻¹), the purely data-driven model overfitted local noise, producing non-physical stress predictions with an average absolute relative error (AARE) of 27.91%. In contrast, the NN-PhysicsInit model filtered out spurious high-frequency fluctuations and successfully captured the macroscopic dynamic softening trend, significantly reducing the extrapolation AARE to 8.92%.

(3) Generalization across the phase transus: During the temperature extrapolation task at 1010 °C (above the β-transus), the physical injection constrained the network from generating diverging predictions. While the purely analytical baseline inevitably exhibits systematic deviations when traversing this phase shift, the dual-stage framework adaptively compensates for the phase-induced stress reduction. Consequently, the extrapolation AARE was reduced to 14.34% (compared to 34.21% for the direct-training model), confirming that the physical prior provides a highly reliable structural baseline even across complex thermodynamic boundaries.

Limitations and Future Directions: It should be noted that the current framework is constructed primarily from a macroscopic phenomenological perspective. As highlighted by Li et al. [25], integrating explicit microstructural features into the machine learning framework is essential for a more fundamental understanding of physical metallurgy mechanisms. Future work will incorporate EBSD and TEM characterizations to establish a comprehensive cross-scale constitutive model.