Physics-Informed Neural Networks (PINNs) are neural networks trained to solve partial differential equations (PDEs) — by embedding the physical laws (like Navier-Stokes or Maxwell's equations) directly into the loss function, ensuring the output respects physics.

What Is a PINN?

• Goal: Approx solution $u(x,t)$ to a PDE.
• Loss Function: $L = L_{data} + L_{physics}$.
• $L_{data}$: Standard MSE on observed data points.
• $L_{physics}$: Residual of the PDE. (e.g., if $f = ma$, penalize outputs where $f

eq ma$).

• No Data?: Can be trained with zero data, just boundary conditions + physics equation.

Why PINNs Matter

• Data Efficiency: Drastically reduces data needs because physics provides strong regularization.
• Extrapolation: Standard NN fails outside training range; PINNs follow physics even where no data exists.
• Inverse Problems: Can infer hidden parameters (e.g., viscosity) from observation data.

Physics-Informed Neural Networks are scientific theory meets deep learning — using AI to accelerate simulations while keeping them grounded in reality.