Physics-guided machine learning is unlocking new capabilities in modeling complex systems

Physics-Informed Neural Networks (PINNs) have emerged as a powerful paradigm at the intersection of deep learning and scientific computing, offering a novel approach to solving forward and inverse problems governed by partial differential equations (PDEs). By embedding physical laws into the training of neural networks through constrained loss formulations, PINNs unify data-driven learning with mechanistic modeling. This survey provides an in-depth review of the theoretical foundations, algorithmic advances, and practical applications of PINNs across diverse scientific and engineering disciplines. We begin by exploring the motivation and historical context for physics-informed learning, contrasting it with traditional numerical methods and conventional machine learning models. The core methodologies underlying PINNs—including loss construction, network architecture design, optimization techniques, and extensions to stochastic and multi-physics systems—are discussed in detail. A taxonomy of recent variants, such as variational PINNs, probabilistic PINNs, domain-decomposed PINNs, and operator-learning PINNs, is presented to highlight the rapid diversification of the field. Representative applications in fluid dynamics, materials science, geophysics, biomedical engineering, and climate modeling are examined to demonstrate the real-world impact of PINNs. Despite their promise, PINNs face several challenges, including optimization stiffness, high computational cost, generalization limitations, sensitivity to noise, and lack of theoretical guarantees. We provide a critical analysis of these issues and survey emerging solutions. Looking ahead, we identify future research directions such as foundation models for physics, self-supervised learning, symbolic and probabilistic hybrid frameworks, scalable training strategies, and autonomous scientific discovery. By systematically bridging the gap between data and physical laws, PINNs represent a foundational step toward interpretable, generalizable, and trustworthy scientific machine learning. This review aims to serve as a comprehensive reference for researchers and practitioners seeking to understand and advance the frontiers of physics-informed neural computation.