PDPO: Parametric Density Path Optimization

We introduce Parametric Density Path Optimization (PDPO), a novel method for computing action-minimizing paths between probability densities. The core idea is to represent the target probability path as the pushforward of a reference density through a parametric map, transforming the original infinite-dimensional optimization over densities to a finite-dimensional one over the parameters of the map. We derive a static formulation of the dynamic problem of action minimization and propose cubic spline interpolation of the path in parameter space to solve the static problem. Theoretically, we establish an error bound of the action under proper assumptions on the regularity of the parameter path. Empirically, we find that using 3-5 control points of the spline interpolation suffices to accurately resolve both multimodal and high-dimensional problems. We demonstrate that PDPO can flexibly accommodate a wide range of potential terms, including those modeling obstacles, mean-field interactions, stochastic control, and higher-order dynamics. Our method outperforms existing state-of-the-art approaches in benchmark tasks, demonstrating superior computational efficiency and solution quality. Source code https://github.com/SebasGutHdz/PDPO/tree/main . Here, K(ρ, v) denotes the transportation energy, and F (ρ) represents a potential term that captures interactions among particles or with the environment. When K(ρ, v) = 1 2 |v| 2 ρ and F (ρ) = 0, this reduces to the classical Wasserstein geodesic problem. Directly solving such action-minimizing problems poses substantial mathematical and computational challenges. In low dimensions (e.g., 2 or 3), the associated PDE system-derived from the first-order optimality conditions-can be tackled using classical numerical methods [1, 5, 6, 14] . However, these approaches do not scale well with dimension and become computationally infeasible in high-dimensional settings. Recent advances in machine learning have greatly expanded the range of high-dimensional problems that can be tackled effectively. A leading example is the Generalized Schrödinger Bridge Method (GSBM) [20] , originally developed for stochastic optimal control (SOC) problems [24] . GSBM learns forward and backward vector fields by modeling conditional densities and velocities, drawing inspiration from stochastic interpolants [2], and approximates Gaussian path statistics via spline-based optimization. In parallel, [29] introduced an algorithm for problems with linear energy potentials, leveraging Kantorovich duality and amortized inference to efficiently compute c-transforms and transportation costs. For Mean-Field Games, APAC-Net [17] casts the primal-dual formulation as a convex-concave saddle point problem, trained using a GAN-style adversarial framework. Our method builds upon prior work on parametric probability distributions [21, 33, 15] , extending these ideas to address boundary-valued action-minimizing density problems. Our formulation is an extension of the static OT formulation to the parameter space, whereas in [16], they extend the dynamic formulation. In the remainder of this section, we outline our methodology, with full technical details deferred to Section 3.