Physics-informed neural networks for differential equation solutions: A comprehensive review

Physics-Informed Neural Networks (PINNs) embed governing differential equations into training, enabling solutions of ODEs and PDEs. This review consolidates theoretical foundations (expressive capacity, automatic differentiation) and core methods (loss design, constraint enforcement, sampling, optimization), while surveying applications from baseline PINNs to advanced variants such as multi-physics coupling, domain decomposition, frequency-enhanced representations, and operator-learning hybrids. Comparative synthesis links architectural and training choices to equation type, data conditions, and computational budgets. A unified benchmarking framework is proposed with standard PDE tasks, accuracy metrics, collocation budgets, and transparent reporting for fair comparison with classical solvers. Evidence positions PINNs as complementary to traditional methods-effective for inverse problems, data assimilation, irregular domains, and parametric inference-while challenges remain in scalability, spectral bias, constraint enforcement, and reproducibility. The review offers a coherent synthesis with actionable guidance for scientific and engineering applications.