## 2 Method
### The Stress-Strain Adaptive Predictive Model (SSAPM)
#### 2.3.1 Hybrid Stress-Strain Modeling Framework  
The SSAPM introduces a hybrid representation where the stress tensor \(\sigma\) is expressed as a superposition of a mechanistic component \(\mathcal{M}\), grounded in classical constitutive laws, and a data-driven correction term \(\mathcal{D}\), designed to capture material complexities and heterogeneities beyond traditional models:  
\[\sigma=\mathcal{M}(\epsilon,\mathbf{p})+\mathcal{D}(\epsilon,\mathbf{x}), \tag{10}\]  
where \(\epsilon\) represents the strain tensor, \(\mathbf{p}\) denotes state-dependent variables (e.g., plastic strain, internal damage metrics), and \(\mathbf{x}\) includes auxiliary factors such as temperature, strain rate, or prior loading history. The mechanistic component \(\mathcal{M}\) encapsulates elastic and inelastic material behavior. For linear elastic responses, it follows [PERSON]'s law:  
\[\mathcal{M}(\epsilon)=\mathbf{C}:\epsilon, \tag{11}\]  
where \(\mathbf{C}\) is the elasticity tensor, a fourth-order tensor that governs stress-strain relationships in isotropic or anisotropic materials. For nonlinear and inelastic responses, \(\mathcal{M}\) accounts for additional terms:  
\[\mathcal{M}(\epsilon,\mathbf{p})=\mathcal{F}(\epsilon)+\mathcal{H}(\mathbf{p}), \tag{12}\]  
Figure 1: The Stress–Strain Adaptive Predictive Model (SSAPM). This model integrates multi-scale feature extraction, hybrid stress–strain modeling, and adaptive learning mechanisms to enhance stress–strain predictions. The framework consists of shallow and deep feature extraction modules, hybrid attention blocks (HAB), computationally efficient dimensionality reduction techniques, and a physics-consistent learning paradigm. Key components include the Channel Attention Block (CAB), Layer Normalization (LayerNorm), and Multi-Layer Perceptron (MLP), enabling the SSAPM to balance data-driven learning with fundamental physical constraints. The integration of physics-informed and machine learning techniques ensures accurate, scalable, and interpretable predictions in structural mechanics applications. where \(\mathcal{F}(\varepsilon)\) incorporates nonlinear elastic responses, often through polynomial or exponential functions of \(\varepsilon\), and \(\mathcal{H}(\mathbf{p})\) models plastic deformation or viscoelastic effects. We considered incorporating viscoelastic effects to account for time-dependent stress-strain behavior. However, since our primary formulation follows a static framework, as defined in Equation (1), the inclusion of viscoelasticity would require a more comprehensive reformulation to explicitly address dynamic effects. Given that viscoelasticity inherently involves time-dependent stress evolution, its presence within a purely static context lacks justification. The data-driven term \(\mathcal{D}\), in contrast, employs a neural network to adaptively learn material behavior from experimental data, expressed as  
\[\mathcal{D}(\varepsilon,\mathbf{x})=\boldsymbol{\
abla}(\varepsilon, \mathbf{x};\Theta)=\sum_{i=1}^{k}\alpha_{i}\phi_{i}(\varepsilon,\mathbf{x}), \tag{13}\]  
where \(\phi_{i}\) are basis functions (e.g., nonlinear activation functions in the neural network), \(\alpha_{i}\) are learnable coefficients, and \(\Theta\) encapsulates the network parameters. The network is trained to minimize the residual stress:  
\[\mathcal{R}=\|\boldsymbol{\sigma}_{\text{exp}}-(\mathcal{M}(\varepsilon, \mathbf{p})+\mathcal{D}(\varepsilon,\mathbf{x}))\|^{2}, \tag{14}\]  
where \(\boldsymbol{\sigma}_{\text{exp}}\) denotes experimentally observed stress. This hybridization allows SSAPM to balance theoretical rigor with empirical adaptability, enabling precise predictions of stress-strain behavior under diverse loading conditions. Furthermore, the inclusion of physical constraints, such as enforcing equilibrium:  
\[\
abla\cdot\boldsymbol{\sigma}+\mathbf{f}=0, \tag{15}\]  
ensures that the model adheres to fundamental physical principles while flexibly addressing complex material responses.