“Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” by Maziar Raissi, Paris Perdikaris, and George Em Karniadakis, appeared in the Journal of Computational Physics in 2019 (volume 378, pages 686-707). It gave a name and a clean recipe to an idea that has since spread across scientific computing: train a neural network so that it not only fits any measured data but also satisfies the governing equations of physics.
The trick is in the loss function. A standard network is penalized only for missing data points. A physics-informed network adds a second penalty: the network’s own derivatives, computed by automatic differentiation, are plugged into the partial differential equation that the system is supposed to obey, and any residual from that equation is also punished. The result is a network that respects conservation of mass, momentum, or energy even in regions where there is little or no data, because the physics fills the gaps.
The paper showed the approach working both ways. In the forward direction it solved equations such as Burgers’, Schrodinger’s, and Navier-Stokes from sparse information. In the inverse direction it discovered unknown parameters of an equation directly from noisy observations, which is exactly the situation a scientist or engineer faces when a model has free constants to calibrate.
For a general reader, PINNs matter because they bridge two worlds that used to be separate: the data-driven world of machine learning and the equation-driven world of physics and engineering. That makes them a practical tool for problems where data is expensive but the underlying laws are known, from fluid dynamics to medical imaging to materials.