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