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Deep LearningMathematics & Statistics

physics-informed neural network

nounAlso: PINN, PINNs, physics-informed machine learning

In plain wordsA neural network that is penalized during training whenever its output breaks a known physical law, such as an equation of fluid flow, so it can model a system well even from few measurements.

  1. A neural networkneural network, n.: A kind of computer program that learns a task from examples by tuning thousands to billions of connections between simple number-crunching units, an arrangement loosely inspired by nerve cells in the brain., usually a multilayer perceptronmultilayer perceptron, n.: A neural network made of several layers of artificial neurons, each connected to every neuron in the next layer. It is the simplest kind of deep network and a component of almost every modern model. taking space and time coordinates as input, trained to approximate the solution of a differential equation by adding to its loss the residual of the governing equations, computed exactly at sampled points by automatic differentiationautomatic differentiation, n.: A technique that lets a computer work out exactly how a program's output would change if its inputs changed slightly, by tracking every arithmetic step. It lets AI software compute training updates automatically., together with terms for boundary and initial conditions and any observed data. Because the equations enter only as a penalty, a soft constraint that also regularizesregularization, n.: Any technique that stops a model from memorizing its training examples too closely — often by penalizing complexity — so that it works better on new data. the fit, the same setup can solve an equation without a mesh (a forward problem) or infer unknown coefficients from sparse measurements (an inverse problem). PINNs are used in fluid dynamics, heat transfer, biomechanics and geophysics. Known difficulties include slow and unstable training, imbalanced loss terms, a bias towards smooth solutions and poor performance on stiff, chaotic or multiscale problems, where classical solvers or neural operatorsneural operator, n.: A neural network that learns how a whole physical system responds, such as how a fluid flows given its starting state, so it can stand in for slow simulations and answer new cases quickly. often do better.

    The PINN recovered the material's diffusion coefficient from only twelve temperature sensors.

L(θ)=Ldata(θ)+λN∑i=1N∥N[uθ](xi,ti)∥2\mathcal{L}(\theta) = \mathcal{L}_{\text{data}}(\theta) + \frac{\lambda}{N}\sum_{i=1}^{N} \big\lVert \mathcal{N}[u_\theta](x_i, t_i) \big\rVert^2

Origin

Introduced by Maziar Raissi, Paris Perdikaris and George Em Karniadakis in two 2017 arXiv preprints, published together in the Journal of Computational Physics (2019). Isaac Lagaris, Aristidis Likas and Dimitrios Fotiadis had trained networks to satisfy differential equations in 1998.

Related

See also
  • neural operatorneural operator, n.: A neural network that learns how a whole physical system responds, such as how a fluid flows given its starting state, so it can stand in for slow simulations and answer new cases quickly.
  • neural ordinary differential equationneural ordinary differential equation, n.: A neural network that, instead of passing data through a fixed stack of layers, describes how the data changes continuously over 'time' and uses an equation solver to follow that change.
  • automatic differentiationautomatic differentiation, n.: A technique that lets a computer work out exactly how a program's output would change if its inputs changed slightly, by tracking every arithmetic step. It lets AI software compute training updates automatically.
  • regularizationregularization, n.: Any technique that stops a model from memorizing its training examples too closely — often by penalizing complexity — so that it works better on new data.
  • multilayer perceptronmultilayer perceptron, n.: A neural network made of several layers of artificial neurons, each connected to every neuron in the next layer. It is the simplest kind of deep network and a component of almost every modern model.
  • implicit neural representationimplicit neural representation, n.: A way of storing an image, shape or scene as a small neural network that, given any coordinate, returns what is there, such as a color or density, instead of as a grid of pixels.

Sources

  1. Raissi, M., Perdikaris, P. & Karniadakis, G. E. (2019). Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations. Journal of Computational Physics, 378, 686–707.

Mentioned in

implicit neural representationimplicit neural representation, n.: A way of storing an image, shape or scene as a small neural network that, given any coordinate, returns what is there, such as a color or density, instead of as a grid of pixels., neural operatorneural operator, n.: A neural network that learns how a whole physical system responds, such as how a fluid flows given its starting state, so it can stand in for slow simulations and answer new cases quickly.