Physics-Informed Neural Networks (PINNs)

Physics-Informed Neural Networks (PINNs) represent one of the most exciting convergences of artificial intelligence and fundamental physical laws.

The Limitation of Pure Data-Driven AI

Standard deep learning architectures treat problems as pure black-box regression. If you train a neural network to predict fluid flow based only on sensor readings, it will frequently generate predictions that violate fundamental laws—such as the conservation of mass or energy.

The PINN Formulation

In a PINN, physical equations are encoded directly into the neural network's loss function:

\mathcal{L}_{\text{total}} = \mathcal{L}_{\text{data}} + \lambda \mathcal{L}_{\text{physics}}

Where the physics loss \mathcal{L}_{\text{physics}} measures the residual of the governing differential equation. For example, in the 1D Burgers' equation:

f = \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2}

The loss enforces:

\mathcal{L}_{\text{physics}} = \frac{1}{N} \sum_{i=1}^N |f(x_i, t_i)|^2

Using automatic differentiation (AD), the neural network's gradients are computed with respect to the input coordinates (x, t), ensuring that the network's predictions strictly adhere to Navier-Stokes or wave equations.

Future Directions

As computational power grows, combining PINNs with browser-based WebGPU will allow us to run physics-accurate surrogates directly on consumer laptops, a paradigm we are actively exploring at MetaMotion.