Physics-Informed Neural Networks: A Review of Methodological Evolution, Theoretical Foundations, and Interdisciplinary Frontiers Toward Next-Generation Scientific Computing

Physics-informed neural networks (PINNs) have emerged as a transformative methodology integrating deep learning with scientific computing. This review establishes a three-dimensional analytical framework to systematically decode PINNs’ development through methodological innovation, theoretical breakthroughs, and cross-disciplinary convergence. The contributions include threefold: First, identifying the co-evolutionary path of algorithmic architectures from adaptive optimization (neural tangent kernel-guided weighting achieving 230% convergence acceleration in Navier-Stokes solutions) to hybrid numerical-deep learning integration (5× speedup via domain decomposition) and second, constructing bidirectional theory-application mappings where convergence analysis (operator approximation theory) and generalization guarantees (Bayesian-physical hybrid frameworks) directly inform engineering implementations, as validated by 72% cost reduction compared to FEM in high-dimensional spaces (p<0.01,n=15 benchmarks). Third, pioneering cross-domain knowledge transfer through application-specific architectures: TFE-PINN for turbulent flows (5.12±0.87% error in NASA hypersonic tests), ReconPINN for medical imaging (▵SSIM=+0.18±0.04 on multi-institutional MRI), and SeisPINN for seismic systems (0.52±0.18 km localization accuracy). We further present a technological roadmap highlighting three critical directions for PINN 2.0: neuro-symbolic, federated physics learning, and quantum-accelerated optimization. This work provides methodological guidelines and theoretical foundations for next-generation scientific machine learning systems.