Embedding Physics into Machine Learning: A Review of Physics Informed Neural Networks as Partial Differential Equation Forward Solvers

Abstract Partial Differential Equations (PDEs) are a fundamental class of mathematical models widely used for modeling continuous systems across scientific and engineering disciplines. Physics Informed Machine Learning (PIML), which utilises both data and scientific knowledge, provides a powerful framework that bridges Artificial Intelligence (AI) and PDEs. Among PIML methods, Physics Informed Neural Networks (PINNs) have emerged as a representative and widely adopted approach. This paper offers a structured, problem-oriented review of recent developments in the use of PINNs as PDE forward solvers. We aim to help readers grasp the key trends and interrelations across methodological advances and practical applications. From a methodological perspective, we review existing approaches with a focus on Machine Learning (ML) models and representations, optimization objectives and strategies, as well as datasets and training procedures. From an application perspective, we examine the task characteristics and the applicability of PINNs in domains such as fluid dynamics, heat transfer, solid mechanics, magnetism, and highlight several practical software toolkits and benchmarks. Despite the remarkable progress made in PINN research, significant challenges remain in addressing complex real-world problems. Accordingly, we discuss current limitations in generalization capability, training efficiency, and optimization difficulty, and outline promising directions for future improvements.