
# Flow Matching Implementation

## Overview
This section details our implementation of flow matching for weather prediction, which integrates:
- Probability paths on manifolds
- Spherical geometry for Earth's surface
- Physics-constrained ODE solvers

## Core Components

### Probability Paths
The `ProbPath` class provides the foundation for continuous-time flow matching:

### Spherical Manifold
The `Sphere` class handles geometric operations on Earth's surface:

### Weather ODE Solver
The `WeatherODESolver` manages time evolution with physics constraints:

## Integration Example

## Physics Constraints
The implementation maintains:
- Mass conservation through divergence-free velocity fields
- Energy conservation via soft constraints
- Proper spherical geometry for Earth's surface

## Advanced Usage
See the examples directory for detailed implementations.
