A physics-informed neural network is trained to fit observed data while also obeying physical equations, such as flow or mass balance laws.
A physics-informed neural network (PINN) is a neural network whose training includes the governing equations of the system, not just observed data. The approach was set out in a widely cited 2019 paper by Raissi, Perdikaris and Karniadakis. The network takes inputs such as position and time and outputs a quantity such as water depth, velocity or hydraulic head. During training, the loss has two main parts. The data term measures how far the outputs are from observations. The physics term measures how badly the outputs break the governing equation, such as the shallow water equations or the groundwater flow equation, checked at many points across the domain. The derivatives the equation needs are computed with automatic differentiation. A weight, λ, sets the balance between the two terms. Boundary and initial conditions are usually added as further terms. In water, PINNs have been tested for groundwater flow and contaminant transport, open-channel flow under the Saint-Venant equations, and estimating unknown parameters such as hydraulic conductivity from sparse head measurements. The physics term acts as a guide where data are scarce, which is common in hydrology. They need a known governing equation, some observations to fit, and careful setup of the domain and conditions. They do not replace a well-tested numerical model for routine design work. Limits: training can be slow and hard to get right, especially for sharp fronts, shocks, or wetting and drying in floods, and results depend on how the loss terms are weighted. A PINN is usually trained for one domain and one set of conditions, so a new case often needs new training. A low training loss does not prove the solution is accurate; check results against a standard solver or field data.