pinn

PINN, short for Physics-Informed Neural Network, is a machine learning framework for solving forward and inverse problems involving partial differential equations (PDEs). It trains a neural network to approximate the solution of a PDE while enforcing the governing physical laws through the training objective.

In a PINN, the neural network takes spatial and temporal coordinates as input and outputs the solution value. The loss function combines several terms: a data misfit term for observed values, a PDE residual term computed via automatic differentiation, and boundary and initial condition terms. The PDE residual enforces that the network output satisfies the differential equation at a set of collocation points, while boundary and initial terms enforce conditions. Training uses gradient-based optimizers, often a combination of Adam and L-BFGS.

PINNs can handle noisy measurements and incomplete data, and can solve inverse problems to estimate unknown parameters by treating them as trainable variables. Variants and extensions include domain decomposition approaches (XPINN), adaptive weighting of loss terms, and improved encoding schemes to capture sharp features.

Applications span fluid dynamics, elasticity, electromagnetics, geophysics, reaction-diffusion systems, and material science. The framework is particularly attractive when data are scarce or expensive to obtain and when incorporating known physics can improve generalization. Limitations include sensitivity to hyperparameters and loss term balancing, difficulty with stiff or highly nonlinear PDEs, challenges in scaling to very high dimensions, and potential extrapolation issues outside the training domain.

The approach is associated with the work of Raissi, Perdikaris, and Karniadakis, who popularized the method in the late 2010s as a means to unify data and physics in neural networks.