a physics-informed neural network
An ordinary neural network learns only from data: show it many examples and it fits the pattern, but it knows nothing about physics and will happily violate a conservation law to match noise. A physics-informed neural network, or PINN, fixes this by telling the network the rules of nature directly. It learns a solution to a differential equation by being penalized whenever it disobeys that equation.
Here is the mechanism in plain steps. You want a function u(x, t) that solves some differential equation, say a heat or wave equation. You represent u by a neural network with adjustable weights. Because automatic differentiation can compute exact derivatives of the network's output with respect to its inputs, you can plug the network straight into the differential equation and measure the residual — how far it is from satisfying the equation — at a scatter of sample points inside the domain. The training loss is then the sum of two parts: how badly the equation residual is violated at interior points, plus how badly the known boundary and initial conditions are missed. Minimizing this combined loss by gradient descent nudges the weights until the network both fits the data you have and respects the physics everywhere. No mesh is required, which is appealing in high dimensions or awkward geometries where building a grid is painful.
PINNs are attractive for inverse problems (inferring an unknown coefficient from sparse measurements), for high-dimensional PDEs, and for blending scattered sensor data with governing equations. But honesty is essential: they are not a free lunch over classical solvers. For standard well-posed PDEs a finite-element or finite-difference method is usually faster, more accurate, and comes with convergence guarantees that PINNs lack. PINN training can be slow and finicky, the loss landscape is hard, and there is generally no error bound on the result. They are a promising tool for the problems classical methods handle poorly, not a wholesale replacement.
Suppose you have a few temperature readings inside a metal plate and you want the full temperature field plus the plate's unknown thermal conductivity. A PINN represents the temperature as a network, adds the steady heat equation as a loss term, and trains so the network matches the readings AND satisfies the equation — recovering both the field and the missing conductivity at once, something a plain data fit could not do.
A PINN recovers a hidden coefficient by forcing the network to obey the governing equation.
PINNs are not generally faster or more accurate than classical solvers for standard PDEs — finite elements usually win and come with convergence guarantees PINNs lack. Their niche is inverse problems, high dimensions, and data-physics fusion, not replacing FEM wholesale.