## 2 Acquisition of Experimental Stress-strain Data
### The Basic Principles of SVR  
Support vector machine (SVM) is a machine learning method according to the structural risk minimization principle and statistical learning theory [21]. SVM is primarily used for regression analysis and classification, so it is classified into support vector regression (SVR) and support vector classification (SVC). In SVR, linearly-inseparable low-dimensional data are mapped into linearly-separable multidimensional data by using the kernel function.  
Figure 2: True stress-strain curves of the Ti–10V–2 Fe–3 Al alloy at different strain rates and temperatures: (**a**) 948 K; (**b**) 973 K; (**c**) 998 K; (**d**) 1023 K; (**e**) 1048 K; (**f**) 1073 K; (**g**) 1098 K; (**h**) 1123 K.  
SVR has several advantages as follows: firstly, in SVR, linearly-inseparable low-dimensional data are mapped into linearly-separable high-dimensional data by using the kernel function, and this mapping can be briefly expressed by Equation (1). With the help of the kernel function, SVR can avoid the curse of dimensionality and construct the linear discriminant function in high dimensional space to implement the nonlinear discrimination in the original space. Secondly, comparing SVR to artificial neural network (ANN), the computational process of SVR is robust, and SVR will obtain the globally-optimal solution; while ANN easily falls into local extreme values. SVR has a complete theoretical basis and strong generalization ability, and it does not need to try many network topologies to achieve a high accuracy level.  
\[x\rightarrow\Phi(x)=(g_{1}\Phi_{1}(x),g_{2}\Phi_{2}(x),\...\,g_{n}\Phi_{n}(x)) \quad g_{n}\in R,\Phi_{n}\in R \tag{1}\]  
where \(x\) is the input variable; \(\Phi(x)\) is the mapping function; \(g_{1},g_{2},\...\,g_{n}\) are constants.  
An SVR equipped with the radial basis function (RBF) expressed by Equation (2) can achieve a higher regression accuracy. Therefore, the RBF was adopted in this work.  
\[k(x_{i},x)=\exp(-\gamma\ |\ |\ x_{i}-x\ |\ |^{2}),\gamma=\frac{1}{2\tau^{2}} \tag{2}\]  
where \(\gamma\) and \(\tau^{2}\) are variable parameters of the RBF. A suitable parameter \(\tau^{2}\) will avoid over-fitting and under-fitting of data in SVR.  
It is assumed that the original data are \((x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3}),\...\,(x_{i},y_{i}),\...\,(x_{l},y_{l}),x_{i},y_{i}\in R\), and the function \(f(x)\) is able to estimate all data. In SVR, the \(y=f(x)\) can be expressed by Equation (3):  
\[f(x)=\mathbf{\omega}\cdot\mathbf{x}+b \tag{3}\]  
where \(\mathbf{\omega}\) is a multidimensional column vector; \(b\) is a bias term. Additionally, the optimal function can be expressed as:  
\[\min\frac{1}{2}\ |\ |\ \mathbf{\omega}\ |\ |^{2}+C\sum_{i=1}^{l}\ (\xi_{\text{t}}+\xi_{i} \text{*}) \tag{4}\]  
s.t.  
\[\left\{\begin{array}{l}y_{i}-\mathbf{\omega}\cdot x_{i}-b\leq\zeta+\xi_{i}\\ \mathbf{\omega}\cdot x_{i}+b-y_{i}\leq\zeta+\xi_{i}\text{*}\\ \xi_{\text{t}},\xi_{\text{t}}\text{*}\geq 0\end{array}\right. \tag{5}\]  
where \(\mathbf{\omega}\) is a multidimensional column vector; \(C\) is a penalty factor; \(\xi_{\text{t}}\) and \(\xi_{\text{t}}\text{*}\) are slack variables, which can improve regression accuracy; \(\zeta\) is an insensitive loss function, which greatly impacts the regression accuracy of SVR. In this work, the input variables \(x\) of SVR include strain (\(\varepsilon\)), strain rate (\(\dot{\varepsilon}\)) and temperature (\(T\)), and the target output \(f(x)\) is the flow stress (\(\sigma\)) of as-forged Ti-10V-2 Fe-3 Al alloy.  
The regression function in SVR can be expressed by Equation (6):  
\[f(x)=\sum_{i=1}^{l}\ (\alpha_{i}-\alpha_{i}\text{*})k(x_{i},x)+b \tag{6}\]  
where \(\alpha_{i}\) is the Lagrange multiplier; \(k(x_{i},x)\) is a kernel function; \(b\) is a bias term.