introduction to flow models

https://www.pi.website/blog/pi07 https://diffusion.csail.mit.edu/2026/index.html

Time dependent vector fields

• A time dependent vector field maps a point ‘x’ in space and time ‘t’ to a velocity vector
• mathematically we define such field as u(x,t) or just as u_t(x)
• ex- wind direction (the velocity of wind at every point in space and time would be different)
• interestingly the velocity field is responsible to change the trajectories of points as:

• the central question which we want to address is:
  • if a point starts at x₀ at t=0 and follows the vector field, where does the particle reach at a later time?
  • to answer this question we need to solve the above differential equation
  • the idea is to use the velocity vector to advance the point in small time increments

defining flow

• collection of trajectories which evolve according to the vector field

so in sequence we have Velocity Field → Trajectories → Flow

Let’s take an example:

• suppose we have a velocity vector field given by :- u_t(x) = -𝛉_x
• then the flow field is given as follows (how can we check if this is valid?)

differential equation:

if the flow field is replaced in the differential equation and it satisfies the equation it is a valid flow field:

Once we have an ODE, how do we simulate it an find the trajectories?

• We use Euler method in inference to simulate the trajectories and find the final state!
• We simply take small steps in the direction of vector field

Flow models:

• goal is to convert a simple dist p_init to a complex dist p_data
• simulation of an ODE is a natural choice for this transformation
• a flow model is described by the following ODE:
• our goal is to make the endpoint x₁ of the trajectory have dist p_data
• the neural network that predicts this ODE is aka neural ode

Links:

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