## 1 Introduction  
The solution of Partial Differential Equations (PDEs) forms the cornerstone of scientific computing and engineering modeling, with widespread applications across critical fields, including fluid mechanics, quantum physics, biomedical engineering, and climate simulation. Traditional numerical methods, such as the Finite Element Method (FEM) [1], Finite Difference Method (FDM) [2], and Finite Volume Method (FVM) [3], discretize continuous problems into algebraic systems through spatial approximation. While these methods demonstrate robustness in regular geometrical domains and low-dimensional settings, their fundamental shortcomings become increasingly evident as problem complexity escalates. Specifically, the computational cost of high-dimensional problems increases exponentially with dimensionality, a phenomenon known as the curse of dimensionality [4]. Additionally, the generation of grids for complex geometrical boundaries is resource-intensive and susceptible to numerical errors [5]. Furthermore, when addressing multi-physics coupling, sparse observational data, or dynamic boundary conditions, traditional methods are often reliant on empirical assumptions and manual parameter tuning, undermining their robustness. In parallel, purely data-driven deep learning approaches, such as Convolutional Neural Networks (CNNs) [6], have made significant strides in fields like image recognition. However, these methods exhibit a black-box nature, which leads to a loss of physical consistency and a marked decline in generalization performance under conditions of sparse data. The introduction of Physics-Informed Neural Networks (PINNs) marks a paradigm shift in the relationship between machine learning and computational physics, transitioning from a loose collaboration to a deep coupling. By embedding conservation laws, such as the residuals of governing equations, boundary conditions, and initial conditions, as soft constraints within the neural network's loss function, and utilizing automatic differentiation techniques for precise encoding of physical laws [7], PINNs overcome the dimensional limitations of traditional numerical methods and the physical ignorance inherent in data-driven models, thereby achieving physics-guided generalization [8]. The core advantages of PINNs are reflected in three aspects: mesh-free solutions (avoiding discretization errors), multi-modal integration (compatibility with experimental data and physical priors), and high-dimensional scalability (polynomial complexity growth). However, the research landscape of PINNs exhibits characteristics of rapid technological iteration, delayed theoretical validation, and dispersed application exploration, and existing review literature has failed to address the following key questions systematically:  
* Lack of Historical Continuity in Methodological Evolution. Most reviews (e.g., [9; 10]) categorize studies based on application domains or algorithmic modules, thereby disconnecting the intrinsic logical progression from foundational framework development (e.g., residual weighting strategies [11]), algorithmic innovations (e.g., adaptive optimization [12]), to theoretical advancements (e.g., convergence proofs [13]). Disconnection Between Theoretical Analysis and Engineering Practice: Some studies emphasize mathematical rigor (e.g., generalization error bounds [14]) but fail to elucidate its practical value in guiding training stability. Other works extensively present engineering cases (e.g., [15]), yet lack theoretical explanations for algorithmic failure modes. * Insufficient Exploration of Interdisciplinary Collaborative Innovation. Emerging directions such as quantum computing to accelerate PINN training [16] and federated learning for distributed physical modeling [17] have yet to develop a systematic framework. Additionally, the influence of biological neuron dynamics on the architecture of PINNs remains largely at the metaphorical level [18]. * Insufficient Forward-Looking Technological Roadmap.