Notation and Abbreviations

Math notation:

Symbol

Meaning

\(A\)

matrix

\(\eta\)

learning rate or step size

\(\Gamma\)

boundary of computational domain \(\Omega\)

\(f^{*}\)

generic function to be approximated, typically unknown

\(f\)

approximate version of \(f^{*}\)

\(\Omega\)

computational domain

\(\mathcal P^*\)

continuous/ideal physical model

\(\mathcal P\)

discretized physical model, PDE

\(\theta\)

neural network params

\(t\)

time dimension

\(\mathbf{u}\)

vector-valued velocity

\(x\)

neural network input or spatial coordinate

\(y\)

neural network output

Summary of the most important abbreviations:

ABbreviation

Meaning

BNN

Bayesian neural network

CNN

Convolutional neural network

DL

Deep Learning

GD

(steepest) Gradient Descent

MLP

Multi-Layer Perceptron, a neural network with fully connected layers

NN

Neural network (a generic one, in contrast to, e.g., a CNN or MLP)

PDE

Partial Differential Equation

PBDL

Physics-Based Deep Learning

SGD

Stochastic Gradient Descent