Physics-informed neural networks and neural operators for PDEs: A review of architectures, training pathologies, and method selection.

Physics-informed neural networks (PINNs) and neural operators are increasingly used to learn solutions and solution operators for partial differential equations, but the literature remains fragmented across architectures, training objectives, and benchmark protocols. This review organizes these developments for a neural-network audience. We separate two design axes that are often conflated: representation level (solution-level vs. operator-level) and training paradigm (data-driven vs. physics-informed). The resulting taxonomy positions classical PINNs, domain-decomposition and weak-form variants, Fourier and transformer-based neural operators, and hybrid physics-informed operators within a common design space. We further analyze recurring training pathologies, including gradient imbalance, spectral bias, optimization stiffness, and constraint enforcement, and relate them to PDE properties through a PDE Complexity Matrix with explicit scoring criteria. The review combines a 60-study PRISMA-screened methodological core set with a 155-paper contextual corpus covering benchmarks, applications, and operator-learning studies. Across the surveyed evidence, PDE complexity emerges as a useful organizing lens: classical PINNs remain attractive for sparse-data and inverse settings, whereas operator-level and hybrid models are often better suited to parametric, multiscale, or repeated-query regimes. We conclude with standardized evaluation recommendations, a method-selection flowchart, and open problems in generalization, uncertainty quantification, and foundation-model-based scientific learning.