Adjoint Liouville Dynamics for Forcing Inference from Observations of Particle Density Distributions

The Liouville equation is a partial differential equation that describes the conservation of particle density in a dynamic system. The characteristic lines coincide with the trajectories of particles in this system. The density distribution along the trajectory is governed by an ordinary differential equation, derived using the material derivative. We aim to find the optimal parameters for the forcing function in the dynamic system so that the distribution density of the group of characteristic lines best matches a target distribution. This leads to an optimization problem, which can be solved using the adjoint method. When random effects are present in the system, we model the dynamics using stochastic differential equations (SDE). However, traditional Itô integration is not suitable for non-Gaussian Wiener increments. To address this, we reformulate the SDE from a deterministic perspective using random variables. These random variables can vary, and the resulting simulated process agrees reasonably well with the distribution obtained from the traditional Fokker–Planck equation. This allows us to reconstruct the true particle statistics without relying on Itô integration or Gaussian white noise.