Skip to content

Cheat Sheet¤

Data Types¤

  • \([B,C,H,W,\cdots]\): [Batch, Channel, x, y, \(\cdots\)]
  • \([B,T,C,H,W,\cdots]\): [Batch, Trajectory, Channel, x, y, \(\cdots\)]
  • SpatialTensor/SpatialArray: A tensor/array in physical space.
  • FourierTensor/FourierArray: A tensor/array in Fourier space, i.e., the tensor is a complex tensor.

Operators¤

Basic Operators¤

  • \(\mathbf{u}\): \(n\) d vector field.
  • \(\phi\): scalar field (0d vector field.).
  • \(i\): index of the coordinate.
  • \(I\): maximum index of the coordinate.
  • \(u_i\): \(i\) th component of the vector field.
Operator Equation Is linear operator
SpatialDerivative \(\frac{\partial ^n}{\partial i} \phi\) True
Gradient \(\nabla \phi = \left[\begin{matrix}\frac{\partial \phi}{\partial x} \\\frac{\partial \phi}{\partial y} \\\cdots \\\frac{\partial \phi}{\partial I} \\\end{matrix}\right]\) True
Divergence \(\nabla \cdot \mathbf{u} = \sum_{i=0}^I \frac{\partial u_i}{\partial i}\) True
Laplacian \(\nabla^2\mathbf{u}=\left[\begin{matrix}\sum_{i=0}^I \frac{\partial^2 u_x}{\partial i^2 } \\ \sum_{i=0}^I \frac{\partial^2 u_y}{\partial i^2 } \\ \cdots \\ \sum_{i=0}^I \frac{\partial^2 u_I}{\partial i^2 } \\ \end{matrix} \right]\) True
Biharmonic \(\nabla^4\mathbf{u}=\left[\begin{matrix}(\sum_{i=0}^I\frac{\partial^2}{\partial i^2 })(\sum_{j=0}^I\frac{\partial^2}{\partial j^2 })u_x \\ (\sum_{i=0}^I\frac{\partial^2}{\partial i^2 })(\sum_{j=0}^I\frac{\partial^2}{\partial j^2 })u_y \\ \cdots \\ (\sum_{i=0}^I\frac{\partial^2}{\partial i^2 })(\sum_{j=0}^I\frac{\partial^2}{\partial j^2 })u_i \\ \end{matrix} \right]\) True
Curl (2D input) \(\nabla \times \mathbf{u} = \frac{\partial u_y}{\partial x}-\frac{\partial u_x}{\partial y}\) False
Curl (3D input) \(\nabla \times \mathbf{u} = \left[\begin{matrix} \frac{\partial u_z}{\partial y}-\frac{\partial u_y}{\partial z} \\ \frac{\partial u_x}{\partial z}-\frac{\partial u_z}{\partial x} \\ \frac{\partial u_y}{\partial x}-\frac{\partial u_x}{\partial y} \end{matrix} \right]\) False
ConservativeConvection \(\nabla \cdot \mathbf{u}\mathbf{u}=\left[\begin{matrix}\sum_{i=0}^I \frac{\partial u_i u_x }{\partial i} \\\sum_{i=0}^I \frac{\partial u_i u_y }{\partial i} \\ \cdots\\ \sum_{i=0}^I \frac{\partial u_i u_I }{\partial i} \\ \end{matrix} \right]\) False
Convection \(\mathbf{u} \cdot \nabla \mathbf{u}=\left[\begin{matrix}\sum_{i=0}^I u_i\frac{\partial u_x }{\partial i} \\\sum_{i=0}^I u_i\frac{\partial u_y }{\partial i} \\\cdots\\\sum_{i=0}^I u_i\frac{\partial u_I }{\partial i} \\\end{matrix} \right]\) False
KS Convection \(\frac{1}{2}\|\nabla \phi\|^2=\frac{1}{2}\sum_{i=0}^{I}(\frac{\partial \phi}{\partial i})^2\) False

All the above operators can be imported from torchfsm.operator module. Corresponding functions that directly apply the operator to the input tensor are also available in torchfsm.functional module.

  • \(\mathbf{u}\): vector field.
  • \(u,v\): components of 2d velocity field.
  • \(\omega\): vorticity scalar of a 2d velocity.
  • \(p\): pressure.
Operator Equation
VorticityConvection \((\mathbf{u}\cdot\nabla) \omega\)
NSPressureConvection \(-\nabla (\nabla^{-2} \nabla \cdot (\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}-f))-\left(\mathbf{u}\cdot\nabla\right)\mathbf{u} + \mathbf{f}\)
Vorticity2Velocity \([u,v]=[-\frac{\partial \nabla^{-2}\omega}{\partial y},\frac{\partial \nabla^{-2}\omega}{\partial x}]\)
Velocity2Pressure \(-\nabla^{-2} (\nabla \cdot (\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}-f))\)
Vorticity2Pressure \(\begin{matrix}\mathbf{u}=[u,v]=[-\frac{\partial \nabla^{-2}\omega}{\partial y},\frac{\partial \nabla^{-2}\omega}{\partial x}]\\ p= -\nabla^{-2} (\nabla \cdot (\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}-f))\end{matrix}\)

All the above operators can be imported from torchfsm.operator module. Corresponding functions for VorticityConvection, Vorticity2Velocity, Velocity2Pressure and Vorticity2Pressure are also available in torchfsm.functional module.

Equations¤

Operator Equation
Burgers \(\frac{\partial \mathbf{u}}{\partial t} =-\mathbf{u} \cdot \nabla \mathbf{u} + \nu \nabla^2 \mathbf{u}\)
KuramotoSivashinsky \(\frac{\partial \phi}{\partial t}=-\frac{\partial^2 \phi}{\partial x^2} -\frac{\partial^4 \phi}{\partial x^4} - \phi\frac{\partial\phi}{\partial x}\)
KuramotoSivashinskyHighDim \(\frac{\partial \mathbf{\phi}}{\partial t}=-\nabla^2 \phi- \nabla^4 \phi - \frac{1}{2} \| \nabla \phi \|^2\)
KortewegDeVries \(\frac{\partial \phi}{\partial t}=-c_1\frac{\partial^3 \phi}{\partial x^3} + c_2 \phi\frac{\partial\phi}{\partial x}\)
NavierStokesVorticity \(\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla) \omega = \frac{1}{Re} \nabla^2 \omega + \nabla \times \mathbf{f}\)
NavierStokes \(\frac{\partial\mathbf{u}}{\partial t}=-\nabla (\nabla^{-2} \nabla \cdot (\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}-f))-\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}+\nu \nabla^2 \mathbf{u} + \mathbf{f}\)