Zero-penalty limit in the penalized mean-field normalizing flow formulation

Determine whether, in the mean-field variational formulation for normalizing flows that is regularized by adding a small quadratic control penalty (ε/2)∫||U_t||^2 to the action—yielding the control law U_t(X_t) = (1/ε)(P_t − ∇_x log ρ_t(X_t))—the limit ε → 0 recovers the classical optimal transport formulation.

Background

In the postscript section on normalizing flows, the paper derives an Eulerian action that, when translated into a McKean–Pontryagin framework (with Σ = 0), leads to a singular appearance of the control: variation with respect to the control does not determine it and instead imposes the constraint P_t = ∇_x log ρ_t(X_t). This suggests that many bridging families (ρ_t, U_t) satisfying the continuity equation would be critical points of the unregularized action, indicating degeneracy.

To address this degeneracy, the author proposes a small quadratic penalty on the control, introducing an ε-dependent action and deriving the control U_t(X_t) = (1/ε)(P_t − ∇_x log ρ_t(X_t)). The author then poses the explicit question of what happens in the limit ε → 0, asking whether this limit recovers optimal transport.