## 2 Method
### Adaptive Stress-Strain Optimization Strategy (ASSOS)
#### 2.4.3 Surrogate Models and Physics-Informed Constraints  
To significantly improve computational efficiency while preserving model accuracy, ASSOS integrates surrogate models such as Gaussian processes, neural networks, or polynomial chaos expansions. These surrogate models are trained to approximate complex constitutive relations, enabling rapid evaluation of stress responses. The surrogate function is expressed as  
\[\sigma=\hat{\mathcal{F}}(\varepsilon;\theta), \tag{40}\]  
where \(\hat{\mathcal{F}}\) represents the surrogate model, parameterized by \(\theta\), and \(\varepsilon\) is the strain tensor. The surrogate model is trained using high-fidelity simulation or experimental data, minimizing the discrepancy:  
\[\mathcal{L}_{\text{train}}=\frac{1}{N}\sum_{i=1}^{N}\|\sigma_{\text{true}}^{( i)}-\hat{\mathcal{F}}(\varepsilon^{(i)};\theta)\|^{2}, \tag{41}\]  
where \(\sigma_{\text{true}}^{(i)}\) and \(\varepsilon^{(i)}\) are the stress and strain samples, respectively, from the training dataset of size \(N\).  
By embedding surrogate models into the optimization workflow, ASSOS reduces the computational overhead associated with solving nonlinear constitutive relations or finite element simulations. For example, in scenarios where material behavior is highly nonlinear, the surrogate model can approximate the stress-strain response within an acceptable error margin, enabling faster iterations.  
In addition to surrogate modeling, ASSOS incorporates physics-informed constraints directly into the optimization process to maintain physical plausibility and stability. For incompressible materials, the trace of the strain tensor is constrained to zero:  
\[\text{Tr}(\varepsilon)=0, \tag{42}\]  
where \(\text{Tr}(\varepsilon)\) is the volumetric strain. This constraint prevents spurious volumetric changes during deformation, ensuring compliance with material incompressibility.  
To guarantee energy stability, ASSOS minimizes the total potential energy \(\mathcal{E}\), defined as  
\[\mathcal{E}=\int_{\Omega}\biggl{(}\frac{1}{2}\sigma:\varepsilon-\mathbf{f} \cdot\mathbf{u}\biggr{)}d\Omega, \tag{43}\]  
where \(\mathbf{u}\) represents the displacement vector, \(\mathbf{f}\) denotes external forces, and \(\Omega\) is the domain of the material. The energy gradient is used to drive the optimization process:  
\[\frac{\partial\mathcal{E}}{\partial\sigma}=\varepsilon-\
abla\cdot\mathbf{u}, \tag{44}\]  
ensuring stability and convergence.  
Termination criteria are established to stop the optimization process once the stress and strain fields satisfy predefined tolerances. These criteria are defined as \[\|\mathbf{R}\|<\epsilon,\quad\|\Delta\sigma\|<\delta, \tag{45}\]  
where \(\mathbf{R}\) is the residual stress imbalance, and \(\Delta\sigma\) is the stress update between iterations. \(\epsilon\) and \(\delta\) are user-defined thresholds for residual error and stress change, respectively.  
To further enhance the robustness of the surrogate models, ASSOS incorporates physics-informed neural networks (PINNs). These models enforce physical laws such as equilibrium \(\mathbf{\
abla}\cdot\mathbf{\sigma}=\mathbf{f}\) and boundary conditions during training, effectively embedding domain knowledge into the surrogate:  
\[\mathcal{L}_{\text{PINN}}=\mathcal{L}_{\text{train}}+\lambda_{\text{phys}}\| \mathbf{\
abla}\cdot\mathbf{\sigma}-\mathbf{f}\|^{2}, \tag{46}\]  
where \(\lambda_{\text{phys}}\) is a weighting factor balancing data-driven and physics-based loss terms.