## 1 Introduction
Existing reviews (e.g., [19]) lack strategic foresight regarding the core features of PINN 2.0 (e.g., neural-symbolic reasoning [20] and uncertainty quantification frameworks), making it challenging to bridge the gap from technological prototypes to industrial-grade tools in the field. This paper proposes a three-dimensional analytical framework--comprising methodology-theory-application--to systematically deconstruct the evolution of PINNs' methodology, theoretical deepening, and interdisciplinary integration mechanisms:  
* Methodological Perspective. The co-evolutionary path of adaptive optimization \(\rightarrow\) domain decomposition \(\rightarrow\) hybrid numerical-deep learning is revealed. This progressive approach is based on the idea of the _multi-modal_ problem [17]. The idea of the _multi-modal_ problem is to find a _multi-modal_ problem [17]. sion spans from the foundational residual weighting strategy [11] to neural tangent kernel-guided dynamic optimization [12], ultimately achieving multi-scale coupling with traditional finite element methods (Section 2.2.3). * Theoretical Perspective. A dual-pillar framework is established, combining convergence proofs and generalization guarantees. Operator approximation theory [13] is employed to rigorously analyze approximation error bounds, while a Bayesian-physics hybrid framework [18] is used for uncertainty quantification (Section 3). * Application Perspective. An interdisciplinary knowledge transfer paradigm is constructed, encompassing physical modeling--life sciences--earth systems. This includes the development of the TFE-PINN architecture for turbulence simulations (reducing error by 62% in NASA benchmark tests), the introduction of ReconPINN for medical image reconstruction (improving SSIM by \(0.18\pm 0.04\) across multi-center MRI data), and the implementation of a seismic early warning system with 0.5 km localization accuracy (Section 4). * The contributions of this paper can be summarized in three key aspects:
* Systematic Deconstruction of Methodology. This paper introduces, for the first time, the algorithmic evolution pathway of adaptive optimization \(\rightarrow\) domain decomposition \(\rightarrow\) numerical-deep learning hybrid (Section 2), revealing the common design principles underlying technological breakthroughs. * Bidirectional Mapping Between Theory and Application. We establish quantitative links between convergence analysis (Section 3.1) and generalization guarantees (Section 3.2) with practical performance improvements, addressing the issue of theoretical elegance but limited practicality. * Interdisciplinary Roadmap Design. Common challenges are distilled from computational physics (Section 4.1), biomedical sciences (Section 4.2), and earth sciences (Section 4.3), and integration paradigms are planned for neural-symbolic reasoning (Section 5.1), federated learning (Section 5.2), and quantum enhancement (Section 5.3). The structure of the paper is as follows: Section 2 analyzes the evolution of the PINNs methodology, with a focus on adaptive optimization strategies (Section 2.2.1), domain decomposition architectures (Section 2.2.2), and the deep coupling mechanism with traditional numerical methods (Section 2.2.3). Section 3 constructs the theoretical framework from the perspectives of convergence (Section 3.1) and generalization (Section 3.2). Section 4 demonstrates the unique advantages of PINNs in solving multi-scale, multi-physics problems through interdisciplinary case studies. Section 5 explores frontier topics such as enhancing interpretability through neural-symbolic systems, enabling privacy-preserving modeling with federated learning, and accelerating high-dimensional optimization via quantum computing. Section 6 summarizes the technical challenges and proposes a research roadmap for the development of PINN 2.0.