Physics-Informed Neural Networks as an Integrated Computational Methodology: A Critical Review of Formulation, Training, and Evaluation

Physics-informed neural networks (PINNs) have developed from residual constrained neural approximations into a heterogeneous family of scientific computing formulations, combining governing equations, observations, numerical operators, and optimization. This review examines how formulation, training, and evaluation interact within an integrated computational methodology. A structured and iterative review assembled 853 records from Scopus, IEEE Xplore, ScienceDirect, MDPI, reference tracing, and topic-specific searches; 514 references are cited in the revised manuscript. The review analyzes representation, physics enforcement, differentiation, collocation, loss construction, optimization, architectural and theoretical developments, and multidimensional evaluation. The analysis shows that reliability depends on these coupled choices and that no universal architecture, optimizer, enforcement strategy, or sampling strategy exists. An evidence-linked co-design framework, qualitative design-performance dependency matrix, and failure-mode diagnostic map are therefore developed to connect design choices with performance and guide redesign. Low prediction or residual error alone does not establish conservation, stability, robustness, uncertainty calibration, or computational efficiency. The open PINN Review Atlas links the synthesis to evidence, metrics, mathematical formulations, and related data as an interactive resource to help researchers explore the field and support future studies. PINNs are likely to serve as problem-dependent trainable components of hybrid scientific computing workflows rather than universal replacements for traditional numerical solvers.