A glossary of the mathematical notation used throughout the course, in roughly the order it first appears. Use this as a lookup for any symbol whose meaning isn’t obvious from context.
Variables and indexing conventions
| t |
time (independent variable) |
| x, y, z |
spatial coordinates |
| \mathbf{x} = (x, y, z) |
spatial position vector |
| z = 0 |
sea surface (in the column model, Unit 8) |
| z = -H |
column bottom (depth H below the surface) |
| \eta(x, y, t) |
sea-surface elevation anomaly (Unit 1, Unit 6) |
| T(z, t) |
temperature, column model (Unit 9) |
| u, v |
horizontal velocity components |
| w |
vertical velocity component |
| \mathbf{u} = (u, v, w) |
full velocity vector |
| \mathbf{u}_h = (u, v) |
horizontal velocity vector |
| \psi(t) |
unknown boundary forcing (Unit 1’s river source) |
| \theta, \phi |
network parameter vectors (PINN unknowns) |
Indices: i for collocation / sample points, j for components, k for time steps. Discrete grid indices use i, j over (x, y) and n over t (e.g. u_i^n for the value at grid cell i, time step n).
Derivatives
| \dot{x}, \ddot{x} |
first and second time derivatives, \frac{dx}{dt} and \frac{d^2 x}{dt^2} |
| \dfrac{\partial f}{\partial x} |
partial derivative of f with respect to x |
| \partial_x f, f_x |
shorthand for \partial f / \partial x |
| \partial_t f, \partial_z f |
likewise for t and z |
| \partial^2_x f, f_{xx} |
second partial derivative \partial^2 f / \partial x^2 |
| \nabla f |
gradient: (\partial_x f, \partial_y f, \partial_z f) |
| \nabla\!\cdot\!\mathbf{u} |
divergence: \partial_x u + \partial_y v + \partial_z w |
| \nabla \times \mathbf{u} |
curl |
| \nabla^2 f |
Laplacian: \partial^2_x f + \partial^2_y f + \partial^2_z f |
| \nabla_h = (\partial_x, \partial_y) |
horizontal gradient |
| \nabla_h\!\cdot\!\mathbf{u}_h |
horizontal divergence |
Vector and tensor operations
| \mathbf{a}\!\cdot\!\mathbf{b} |
dot product (scalar): \sum_i a_i b_i |
| \mathbf{a} \otimes \mathbf{b} |
outer / tensor product |
| \|\mathbf{a}\|, \|\mathbf{a}\|_2 |
Euclidean norm: \sqrt{\mathbf{a}\!\cdot\!\mathbf{a}} |
| \|f\|_{L^2} |
L^2 norm of a function: \sqrt{\int f^2} |
| \hat{\mathbf{n}} |
outward unit normal to a boundary |
| \mathbf{u}\!\cdot\!\hat{\mathbf{n}} = 0 |
no-flux (impermeable) boundary condition |
| A^\top |
matrix transpose |
| A^{-1} |
matrix inverse |
| \det(A), \mathrm{tr}(A) |
determinant, trace |
Probability, statistics, loss functions
| \mathbb{E}[X] |
expectation of random variable X |
| \mathrm{Var}(X), \sigma^2 |
variance |
| \mathcal{N}(\mu, \Sigma) |
Gaussian with mean \mu, covariance \Sigma |
| \sim |
“distributed as” (e.g. X \sim \mathcal{N}(0, 1)) |
| \mathcal{L}(\theta) |
loss function evaluated at parameters \theta |
| \hat R(f), R(f) |
empirical risk, true risk (Unit 2 §2.1) |
| \ell(\hat y, y) |
per-sample loss |
| \arg\min_\theta f(\theta) |
the \theta that minimises f |
Network architectures and PINN-specific symbols
| u_\theta, T_\theta, \eta_\theta |
neural-network parameterisations of a field |
| \theta |
parameter vector of an MLP / KAN |
| \sigma |
activation function (sigmoid, \tanh, ReLU, Swish, …) |
| W_\ell, b_\ell |
weights and biases of layer \ell |
| \mathcal{L}_{\text{PDE}} |
PDE-residual loss term |
| \mathcal{L}_{\text{BC}} |
boundary-condition loss term |
| \mathcal{L}_{\text{IC}} |
initial-condition loss term |
| \mathcal{L}_{\text{data}} |
data-misfit loss term |
| \lambda_r, \lambda_b, \lambda_i, \lambda_d |
weights on the above |
| \{t_i\}_{i=1}^{N_r}, \{(\mathbf{x}_i, t_i)\} |
collocation points |
| r_\theta(\mathbf{x}, t) |
residual of the PDE evaluated at (\mathbf{x}, t) |
| \gamma(\mathbf{x}) = [\sin(B\mathbf{x}), \cos(B\mathbf{x})] |
Fourier-feature embedding (Unit 7 §7.3) |
Optimisation
| \eta (learning rate) |
step size in gradient descent (not the surface elevation!) |
| \nabla_\theta \mathcal{L}, g_t |
gradient of the loss with respect to parameters |
| m_t, v_t |
first- and second-moment estimates in Adam |
| \beta_1, \beta_2 |
exponential-decay rates in Adam |
| H_t, B_t |
true Hessian / quasi-Newton approximation in L-BFGS |
| \mu |
momentum coefficient |
| Adam, L-BFGS, SGD |
named optimisers (Unit 2 §2.5) |
A handful of letters carry different meanings in different contexts; check the local section:
- \eta is the learning rate in optimisation and the surface-elevation anomaly in shallow-water physics.
- \kappa is eddy diffusivity in oceanography and a matrix condition number in numerical analysis.
- \sigma is the sigmoid activation, the standard deviation of noise, and a Rayleigh-damping rate depending on context.
- H is column depth (m) in the column model and the Hessian matrix in optimisation.
- N is number of training samples, the grid size, and the buoyancy frequency depending on context.
PDE classification quick-reference
For a second-order linear PDE a\, u_{xx} + 2b\, u_{xy} + c\, u_{yy} + \ldots = 0:
| Elliptic |
< 0 |
Laplace: u_{xx} + u_{yy} = 0 |
| Parabolic |
= 0 |
Heat: u_t = \alpha\, u_{xx} |
| Hyperbolic |
> 0 |
Wave: u_{tt} = c^2\, u_{xx} |
(Full discussion in Unit 6 §6.1.)
Common dimensionless groups
| Reynolds Re |
UL/\nu |
inertia vs. viscosity |
| Péclet Pe |
UL/\kappa |
advection vs. diffusion |
| Richardson Ri |
N^2/(\partial_z U)^2 |
buoyancy vs. shear |
| Courant–Friedrichs–Lewy CFL |
c\,\Delta t/\Delta x |
numerical wave speed |
| Strouhal St |
fL/U |
unsteadiness vs. advection |