# Aggregation-Diffusion Equations

Aggregation-diffusion equations are recast as a constrained optimal control problem whose control is the transport velocity, parameterized by a neural network and trained on a variational objective evaluated along particle trajectories, so mass conservation and density positivity hold by construction without a spatial mesh. Earlier structure-preserving schemes mostly relied on mesh-based discretization whose cost grows quickly with dimension; the framework splits total error into neural approximation, optimization, Monte Carlo sampling and time discretization, and is tested on Doi-Onsager, Patlak-Keller-Segel, porous-medium and Fokker-Planck models.

Aggregation-diffusion equations are recast as a constrained optimal control problem whose control is the transport velocity, parameterized by a neural network and trained on a variational objective evaluated along particle trajectories, so mass conservation and density positivity hold by construction without a spatial mesh. Earlier structure-preserving schemes mostly relied on mesh-based discretization whose cost grows quickly with dimension; the framework splits total error into neural approximation, optimization, Monte Carlo sampling and time discretization, and is tested on Doi-Onsager, Patlak-Keller-Segel, porous-medium and Fokker-Planck models.