## 2 Method
### The Stress-Strain Adaptive Predictive Model (SSAPM)
#### 2.3.2 Adaptive Learning with Physics-Consistency  
The SSAPM leverages a robust multi-objective loss function to ensure that its predictions are not only data-driven but also grounded in the fundamental laws of physics (as shown in Figure 2). This hybrid objective simultaneously minimizes the error in predicted stresses while enforcing physical consistency through governing equations. The loss function is defined as  
\[\mathcal{L}=\lambda_{1}\|\boldsymbol{\sigma}_{\text{pred}}-\boldsymbol{\sigma }_{\text{true}}\|^{2}+\lambda_{2}\|\
abla\cdot\boldsymbol{\sigma}_{\text{pred }}-\mathbf{f}\|^{2}+\lambda_{3}R(\Theta), \tag{16}\]  
where \(\boldsymbol{\sigma}_{\text{pred}}\) and \(\boldsymbol{\sigma}_{\text{true}}\) represent the predicted and true stress tensors, respectively, \(\mathbf{f}\) denotes external forces, and \(R(\Theta)\) is a regularization term designed to constrain the complexity of the neural network parameters \(\Theta\). The terms \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) are weighting factors that balance the contributions of data fidelity, equilibrium satisfaction, and regularization. To ensure physical adherence, the second term in the loss function incorporates the equilibrium constraint:  
\[\
abla\cdot\boldsymbol{\sigma}_{\text{pred}}+\mathbf{f}=0, \tag{17}\]where \(\
abla\cdot\mathbf{\sigma}_{\text{pred}}\) represents the divergence of the predicted stress tensor. This enforces that the internal stresses within the material equilibrate with the external forces. Furthermore, the regularization term \(R(\Theta)\) penalizes overly complex solutions, typically defined as  
\[R(\Theta)=\|\Theta\|_{2}^{2}+\alpha\|\
abla_{\epsilon}\mathbf{\
abla}\|^{2}, \tag{18}\]  
where \(\|\Theta\|_{2}^{2}\) controls the size of the model parameters, and the gradient regularization \(\|\
abla_{\epsilon}\mathbf{\
abla}\|^{2}\) smoothens the data-driven component \(\mathbf{\
abla}\) with respect to strain \(\epsilon\), ensuring physical plausibility. The optimization process dynamically adjusts \(\Theta\), the parameters of the data-driven correction term \(\mathbf{\
abla}\), using a gradient-based method such as Adam or L-BFGS. The gradients of \(\mathcal{L}\) with respect to \(\Theta\) are computed as  
\[\frac{\partial\mathcal{L}}{\partial\Theta}=2\lambda_{1}\Big{(}\sigma_{\text{ pred}}-\sigma_{\text{true}}\Big{)}\frac{\partial\sigma_{\text{pred}}}{ \partial\Theta}+2\lambda_{2}\Big{(}\
abla\cdot\mathbf{\sigma}_{\text{pred}}-\mathbf{ \mathsf{f}}\Big{)}\frac{\partial(\
abla\cdot\mathbf{\sigma}_{\text{pred}})}{ \partial\Theta}+\lambda_{3}\frac{\partial R(\Theta)}{\partial\Theta}. \tag{19}\]  
The equilibrium constraint term can also incorporate boundary conditions, such as stress-free surfaces or prescribed traction, encoded as  
\[\mathbf{\sigma}_{\text{pred}}\cdot\mathbf{\mathsf{n}}=\mathbf{\mathsf{t}}, \tag{20}\]  
where \(\mathbf{\mathsf{n}}\) is the outward normal vector on the boundary, and \(\mathbf{\mathsf{t}}\) is the applied traction. This ensures that the predicted stress distributions are consistent with both internal equilibrium and external boundary conditions. By iteratively minimizing \(\mathcal{L}\), the SSAPM achieves a harmonious integration of data-driven adaptability and physical realism. The adaptive learning framework enables the model to generalize across varying material heterogeneities and non-linear behaviors, while the regularization and physics-consistency terms prevent overfitting, ensuring robustness and interpretability. This holistic optimization process empowers SSAPM to tackle complex material modeling challenges with precision and reliability. Figure 2: Comparison of adaptive learning mechanisms with physics-consistency for stress–strain prediction tasks.