PDPO: Parametric Density Path Optimization

TL;DR

PDPO tackles the challenge of finding action-minimizing paths between probability densities in high dimensions by parameterizing pushforward maps with Neural ODEs and interpolating them in parameter space using cubic Hermite splines. This turns the original infinite-dimensional optimization into a finite-dimensional bilevel problem over spline control points and boundary parameters, with a tractable action functional combining kinetic and potential terms. The method supports a broad class of energies, including obstacles, mean-field interactions, and momentum terms, and demonstrates superior efficiency and accuracy versus state-of-the-art baselines across several challenging benchmarks, including high-dimensional opinion dynamics. Boundary initialization via Flow Matching and reuse of boundary parameters further enhance practicality, making PDPO scalable to complex density transport tasks.

Abstract

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. The source code will be publically available after the revision process.

Figures (16)

• Figure 1: Visualization of our framework. (a) Illustrates the three main components: the space of probability densities $\mathcal{P}(\mathbb{R}^d)$ (top), the admissible parameter space $\Theta \subseteq \mathbb{R}^d$ (left), and the sampling space $\mathbb{R}^d$ containing an obstacle (right). (b) and (c) depict the pushforwards of the optimized control points $(T_{\theta_i})_{\#}\,\lambda$ for $i = 0, \ldots, K$ and the continuous spline trajectory $(T_{\theta(t)})_{\#}\,\lambda$, respectively. Time is color-coded, with red indicating $t = 0$ and yellow indicating $t = 1$.
• Figure 2: S-curve-C: (a) Pushforward densities at control points, (b) Pushforward densities along the interpolated trajectory, (c) Density path produced by GSBM, (d) Density path produced by APAC-Net
• Figure 3: Solutions for generalized momentum-minimizing problems (a-c). Solutions for kinetic energy minimizing problems (d-f).
• Figure 4: Opinion depolarization in 1000 dimensions. Top: 2D PCA projections of the distributions. Bottom: directional similarity histograms. Left: target unimodal distribution. Middle: PDPO solution. Right: DeepGSB solution.
• Figure 5: No pertaining for $\theta_0$ and $\theta_1$.

Theorems & Definitions (12)

• Definition 1
• Theorem 1
• Theorem 2
• proof
• proof
• Theorem 3
• proof
• Corollary 1
• Corollary 2
• Theorem 4