Geo-Modeling

Overview¶

• Inspiration - Anatomy of a PDE

• Data Discretization

• I - State Space Models

• II - Spatial Operators

• III - Reduced Order Models

• IV - TimeSteppers

Anatomy of a PDE¶

• State

• Discretization

• Equation of Motion - Spatial Operators

• Initial Conditions

• Boundary Conditions

• TimeSteppers

Data Discretization¶

• Unstructured, e.g., Weather Stations

• Irregular, e.g., Continents, Countries...

• Curvilinear, e.g., 2D Lat-Lon Coordinates, SWATHs

• Rectilinear, e.g., Geographic Coordinates

• Regular, e.g., Images, GeoStationary Satellites

Parameterization¶

I - State Space Models¶

• Emission Distribution, $y_t \sim p(y_t|z_t,\theta)$

• Transition Distribution, $z_t \sim p(z_t|z_{t-1},x_t,\theta)$

• Initial Distribution, $z_0 \sim p(z_0|\theta)$

• Parameter Distribution, $\theta \sim p(\theta)$

II - Spatial Operators¶

• Differentiation

  • Exact - Symbolic, AutoDiff

  • Approximate - Finite Difference/Volume

• Neural Operators

  • Fully Connected

  • AutoDifferential + Symbolic

  • Convolutions + Finite Difference

  • Spectral Convolutions + PseudoSpectral

  • Graphical NNs + Finite Element

  • Free-Form

• Free-Form Complexity

  • Linear

  • Basis Function

  • Non-Linear

III - Reduced Order Models¶

• Encoder + Decoder, $z = f(u), u = g(z)$

• Bijections, Surjections, Stochastic

• Separable (POD), $u(s,t) = \sum \phi(s)\psi(t)$

• Structured

IV - TimeSteppers¶

• Integration

• AutoRegressive

• Taylor - Euler

• Sigma Points - Unscented

• Quadrature - Runge-Kutta

• Monte Carlo

• Hybrid - Leap-Frog, Adam-Bashforth