updates, airfoils test

physicalloss.md

@@ -5,3 +5,50 @@ Using the equations now, but no numerical methods!
 
 Still interesting, leverages analytic derivatives of NNs, but lots of problems
 
+---
+
+Some notation from SoL:
+
+The following PDEs typically work with a continuous
+velocity field $\mathbf{u}$ with $d$ dimensions and components, i.e.,
+$\mathbf{u}(\mathbf{x},t): \mathbb{R}^d \rightarrow \mathbb{R}^d $.
+For discretized versions below, $d_{i,j}$ will denote the dimensionality
+of a field such as the velocity,
+with domain size $d_{x},d_{y},d_{z}$ for source and reference in 3D.
+
+% with $i \in \{s,r\}$ denoting source/inference manifold and reference manifold, respectively.
+%This yields $\vc{} \in \mathbb{R}^{d \times d_{s,x} \times d_{s,y} \times d_{s,z} }$ and $\vr{} \in \mathbb{R}^{d \times d_{r,x} \times d_{r,y} \times d_{r,z} }$
+%Typically, $d_{r,i} > d_{s,i}$ and $d_{z}=1$ for $d=2$.
+
+For all PDEs, we use non-dimensional parametrizations as outlined below,
+and the components of the velocity vector are typically denoted by $x,y,z$ subscripts, i.e.,
+$\mathbf{u} = (u_x,u_y,u_z)^T$ for $d=3$.
+
+Burgers' equation in 2D. It represents a well-studied advection-diffusion PDE:
+
+$\frac{\partial u_x}{\partial{t}} + \mathbf{u} \cdot \nabla u_x =
+  \nu \nabla\cdot \nabla u_x + g_x(t), 
+  \\
+  \frac{\partial u_y}{\partial{t}} + \mathbf{u} \cdot \nabla u_y =
+  \nu \nabla\cdot \nabla u_y + g_y(t)
+$, 
+
+where $\nu$ and $\mathbf{g}$ denote diffusion constant and external forces, respectively.
+
+Burgers' equation in 1D without forces with $u_x = u$:
+%\begin{eqnarray}
+$\frac{\partial u}{\partial{t}} + u \nabla u = \nu \nabla \cdot \nabla u $ .
+
+---
+
+Later on, Navier-Stokes, in 2D:
+
+$
+    \frac{\partial u_x}{\partial{t}} + \mathbf{u} \cdot \nabla u_x =
+    - \frac{1}{\rho}\nabla{p} + \nu \nabla\cdot \nabla u_x  \\
+    \frac{\partial u_y}{\partial{t}} + \mathbf{u} \cdot \nabla u_y =
+    - \frac{1}{\rho}\nabla{p} + \nu \nabla\cdot \nabla u_y  \\
+    \text{subject to} \quad \nabla \cdot \mathbf{u} = 0,
+$
+
+