draft model equations

physicalloss.md

@@ -7,7 +7,48 @@ Still interesting, leverages analytic derivatives of NNs, but lots of problems
 
 ---
 
-Some notation from SoL:
+% \newcommand{\pde}{\mathcal{P}}         % PDE ops
+% \newcommand{\pdec}{\pde_{s}}
+% \newcommand{\manifsrc}{\mathscr{S}}    % coarse / "source"
+% \newcommand{\pder}{\pde_{R}}
+% \newcommand{\manifref}{\mathscr{R}}
+
+% vc - coarse solutions
+% \renewcommand{\vc}[1]{\vs_{#1}}            % plain coarse state at time t
+% \newcommand{\vcN}{\vs}                     % plain coarse state without time 
+% vc - coarse solutions, modified by correction
+% \newcommand{\vct}[1]{\tilde{\vs}_{#1}}     % modified / over time at time t
+% \newcommand{\vctN}{\tilde{\vs}}            % modified / over time without time
+% vr - fine/reference solutions
+% \renewcommand{\vr}[1]{\mathbf{r}_{#1}}            % fine / reference state at time t , never modified
+% \newcommand{\vrN}{\mathbf{r}}                     % plain coarse state without time 
+
+% \newcommand{\project}{\mathcal{T}}           % transfer operator fine <> coarse
+% \newcommand{\loss}{\mathcal{L}}              % generic loss function
+% \newcommand{\nn}{f_{\theta}}
+% \newcommand{\dt}{\Delta t}                   % timestep
+% \newcommand{\corrPre}{\mathcal{C}_{\text{pre}}}            % analytic correction , "pre computed"
+% \newcommand{\corr}{\mathcal{C}}                         % just C for now...
+% \newcommand{\nnfunc}{F} % {\text{NN}}
+
+
+Some notation from SoL, move with parts from overview into "appendix"?
+
+
+
+We typically solve a discretized PDE $\mathcal{P}$ by performing discrete time steps of size $\Delta t$. 
+Each subsequent step can depend on any number of previous steps,
+$\mathbf{u}(\mathbf{x},t+\Delta t) = \mathcal{P}(\mathbf{u}(\mathbf{x},t), \mathbf{u}(\mathbf{x},t-\Delta t),...)$, 
+where
+$\mathbf{x} \in \Omega \subseteq \mathbb{R}^d$ for the domain $\Omega$ in $d$
+dimensions, and $t \in \mathbb{R}^{+}$.
+
+Numerical methods yield approximations of a smooth function such as $\mathbf{u}$ in a discrete
+setting and invariably introduce errors. These errors can be measured in terms
+of the deviation from the exact analytical solution.
+For discrete simulations of
+PDEs, these errors are typically expressed as a function of the truncation, $O(\Delta t^k)$ 
+for a given step size $\Delta t$ and an exponent $k$ that is discretization dependent.
 
 The following PDEs typically work with a continuous
 velocity field $\mathbf{u}$ with $d$ dimensions and components, i.e.,
@@ -41,14 +82,49 @@ $\frac{\partial u}{\partial{t}} + u \nabla u = \nu \nabla \cdot \nabla u $ .
 
 ---
 
-Later on, Navier-Stokes, in 2D:
+Later on, additional equations...
+
+
+
+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{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  \\
+    - \frac{1}{\rho}\nabla{p} + \nu \nabla\cdot \nabla u_y  
-    \text{subject to} \quad \nabla \cdot \mathbf{u} = 0,
+    \\
+    \text{subject to} \quad \nabla \cdot \mathbf{u} = 0
 $
 
 
+
+Navier-Stokes, in 2D with Boussinesq:
+
+%$\frac{\partial u_x}{\partial{t}} + \mathbf{u} \cdot \nabla u_x$
+%$ -\frac{1}{\rho} \nabla p $
+
+$
+  \frac{\partial u_x}{\partial{t}} + \mathbf{u} \cdot \nabla u_x = - \frac{1}{\rho} \nabla p 
+  \\
+  \frac{\partial u_y}{\partial{t}} + \mathbf{u} \cdot \nabla u_y = - \frac{1}{\rho} \nabla p + \eta d
+  \\
+  \text{subject to} \quad \nabla \cdot \mathbf{u} = 0,
+  \\
+  \frac{\partial d}{\partial{t}} + \mathbf{u} \cdot \nabla d = 0 
+$
+
+
+
+Navier-Stokes, in 3D:
+
+$
+  \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 
+  \\
+  \frac{\partial u_z}{\partial{t}} + \mathbf{u} \cdot \nabla u_z = - \frac{1}{\rho} \nabla p + \nu \nabla\cdot \nabla u_z 
+  \\
+  \text{subject to} \quad \nabla \cdot \mathbf{u} = 0.
+$