## 2 Methodological Evolution  
### Foundational Framework Development  
**Mathematical Formulation and Verification. ** The foundational architecture of Physics-Informed Neural Networks (PINNs) establishes a unified framework for integrating physical constraints through composite loss functions. The mathematical formulation can be rigorously expressed as:  
\[\mathcal{L}=\lambda_{\text{data}}\mathcal{L}_{\text{data}}+\lambda_{\text{PDE }}\mathcal{L}_{\text{PDE}}+\lambda_{\text{BC/IC}}\mathcal{L}_{\text{BC/IC}} \tag{1}\]  
where \(\mathcal{L}_{\text{data}}\) represents the data-driven term, which supervises known measurement points, and \(\lambda_{\text{data}}\) is its corresponding weight coefficient; \(\mathcal{L}_{\text{PDE}}\) denotes the PDE residual term,enforcing the physical laws, and \(\lambda_{\text{PDE}}\) is its corresponding weight coefficient; \(\mathcal{L}_{\text{BC/IC}}\) is the boundary/initial condition constraint term, and \(\lambda_{\text{BC/IC}}\) is its corresponding weight coefficient [7]. Figure 1 complexity scores account for asymptotic scaling: FEM's \(O(n^{d})\) vs. PINNs' \(O(kn\log n)\) (\(k\sim d\)). Though \(k\) introduces dimension dependence, PINNs avoid exponential growth. Figure 1 systematically compares this paradigm with traditional numerical methods and pure data-driven approaches across five critical dimensions: grid dependency, data efficiency, physical consistency, computational complexity, and scalability. Notably, PINNs achieve optimal balance between physical consistency and data efficiency, overcoming limitations of conventional approaches. While PINNs generalize across physics domains, quantitative comparisons in Figure 1 use PDE-specific benchmarks (e.g., Burgers/Navier-Stokes systems) to ensure consistent evaluation. **Experimental Verification. ** The framework has been rigorously validated through canonical PDE benchmarks:  
* 1D Burgers Equation: Training time: 2.1 h; Adaptive activation functions reduce relative \(L^{2}\) error to 3.21 \(\times\) 10\({}^{-4}\), demonstrating 40% faster convergence than baseline models [21]. * 2D Navier-Stokes Equations: Training time: 8.5 h; Parallel MLP architecture achieves 8.76 \(\times\) 10\({}^{-3}\) error in vortex shedding prediction, comparable to finite volume methods with 60% less computational cost [22]. * High-Dimensional Poisson Equation: Training time: 14.3 h; Domain decomposition strategies enable solutions in 10D parameter spaces with 6.78 \(\times\) 10\({}^{-2}\) relative error [23]. Collocation points for Burgers (10\({}^{4}\) samples) required 12 s (Latin Hypercube Sampling), while LES data for Navier-Stokes consumed 72 h (ANSYS Fluent). Figure 1: Paradigm Comparison: Traditional vs. Data-Driven vs. PINNs (evaluated on PDE-specific benchmarks. **Taxonomy of PINN Architectures. ** While establishing the foundational framework, diverse PINN variants have emerged with distinct architectural properties, classified in Table 1 across four critical dimensions. Common properties unifying all PINNs include embedded PDE residuals in loss functions (Equation (1)), automatic differentiation for gradient computation, and mesh-free collocation point sampling. Key differences driving specialization manifest through variations in network depth and complexity (typically 4-12 layers), activation function selection (ranging from physics-informed to standard options), and training data generation principles (adaptive versus precomputed approaches). These architectural distinctions directly impact performance metrics shown in Table 2, where XPINN's domain decomposition enables 5x faster 3D simulations through Swish activations that maintain interface continuity, while HFD-PINN's hybrid finite-difference modules require ReLU activations to ensure discrete stability in coupled numerical-neural systems. **Core Architectural Components. ** Table 2 catalogs critical modules that constitute modern PINN architectures. **Theoretical Guarantees.