A Physics-Informed Neural Network Approach to Numerical Solution of Partial Differential Equations

Abstract Physics-informed neural networks (PINNs) have become a revolutionary computational paradigm that combines the power of data-driven learning with the laws of physics to solve forward and inverse problems of partial differential equations (PDEs). Formally introduced by Raissi et al., PINNs substitute the residue of a PDE, along with the initial and boundary conditions, into the loss function of a neural network, which computes the necessary derivatives through automatic differentiation. Unlike classical numerical solvers, like the finite element method or finite difference method, PINNs are mesh-less, easy to adapt to irregular geometries, can treat high dimensional problems naturally, and provide a natural framework to integrate sparse observational information with the underlying physical model. This review summarizes the mathematical background of PINNs, architectures and training approaches to address their limitations, and the key variants, such as extended PINNs (XPINNs), conservative PINNs (cPINNs), and operator-learning frameworks like DeepONet (Deep Learning Operator Network), and Fourier neural operators. A survey of applications in fluid mechanics, heat transfer, solid mechanics, biomedical modeling and geophysics is conducted. The continued difficulties with training stability, spectral bias, multi-scale problem and theoretical convergence guarantees are discussed and future prospects for adaptive sampling, causal training and hybrid solver–network schemes are highlighted.