PINNs

Physics-informed neural networks (PINNs) are a class of machine learning models that embed physical laws, typically described by partial differential equations (PDEs), into the training objective to approximate solutions of physical systems. Introduced in a 2017 framework by Raissi, Perdikaris, and Karniadakis, PINNs seek to learn a map that satisfies both observed data and the governing equations.

In a PINN, a neural network represents the unknown solution field over space and time. The loss function combines a data term, which enforces agreement with measurements, and a physics term, which enforces the PDE residuals via automatic differentiation. Boundary and initial conditions are included as additional terms. By evaluating the network and its derivatives, PINNs enforce the strong form of the equations at a set of collocation points, avoiding explicit meshing in some domains.

PINNs are used for forward problems, where the PDE and conditions are known and data may be sparse, and inverse problems, where unknown parameters, sources, or even governing terms are inferred from data. They can handle noisy data, irregular geometries, and multi-fidelity information, and they are often mesh-free. Training can be computationally intensive and sensitive to the weighting of loss terms, especially when PDE residuals dominate or data are scarce.

Variants and extensions include domain-decomposition approaches (XPINN), time-marching formulations that treat time as an input, and physics-informed surrogate models for parameter estimation, uncertainty quantification, and equation discovery. PINNs have been applied across physics and engineering fields, including fluid dynamics, solid mechanics, heat conduction, and geophysics.