Time-integrated Optimal Transport Overview

• Time-integrated Optimal Transport is a framework that extends classical optimal transport by incorporating time-dependent dynamics to analyze evolving distributions.
• It employs rigorous formulations such as the dynamical Benamou–Brenier approach, spatio-temporal minimax metrics, and neural PDE-based models to capture temporal transformations.
• TiOT supports practical applications including time-series analysis, network flow optimization, and spatio-temporal clustering through efficient algorithmic implementations.

Time-integrated Optimal Transport (TiOT) generalizes classical optimal transport to account for temporal evolution, yielding dynamic models and metrics that capture the movement, alignment, or transformation of distributions over time. Unlike static OT, which seeks an optimal instantaneous map or coupling between two distributions based on a fixed cost, TiOT integrates transport costs, constraints, or objectives across a temporal domain, enabling principled analysis and optimization of dynamic processes ranging from time-series analysis to flow networks and spatio-temporal pattern discovery.

1. Mathematical Formulations and Frameworks

TiOT encompasses several rigorous mathematical formulations, with central examples including dynamical, hybrid minimax, spatio-temporal, graph-based, and neural PDE-based models.

Dynamical Benamou–Brenier Formulation

The foundational dynamic OT formulation by Benamou and Brenier describes the Wasserstein-2 distance as the minimal kinetic energy required to advect one probability measure $\mu$ to another $\nu$ via a time-dependent continuity equation: $$\inf_{\rho, v} \int_0^1 \int_{\mathcal{X}} \frac{1}{2} \|v(x,t)\|^2 \rho(x,t)\, dxdt \quad\text{s.t.}\quad \partial_t \rho + \nabla \cdot (\rho v) = 0,\;\rho(\cdot,0) = \mu,\;\rho(\cdot,1) = \nu.$$ This continuous trajectory formulation is the archetype for many TiOT generalizations to discrete time, space, and structured domains (Lavenant et al., 2018, Dong et al., 1 Nov 2025, Buzun et al., 23 Jul 2025).

Spatio-Temporal Minimax Metric

Modern TiOT metrics for time series, such as the robust minimax framework, recast dissimilarity between probability measures on feature-time space using a joint cost: $$c_{ij}(w) = \bigl(w\,\|x_i - y_j\|_p^p + (1-w)|t_i - s_j|^p\bigr)^{1/p}$$ and define the TiOT distance as

$$\mathcal{D}_p(\alpha, \beta) = \max_{w\in[0,1]} \mathcal{W}_{p,w}(\alpha, \beta),$$

where the inner minimum is a standard discrete optimal transport problem with cost $c_{ij}(w)$, and the outer maximization finds the most discriminating time/feature weighting (Nguyen et al., 26 Dec 2025).

Spatio-Temporal Alignment with Regularized OT and Soft-DTW

Another important TiOT realization aligns two spatio-temporal objects $(X_t), (Y_{t'})$ by combining entropic/unbalanced OT in space and soft-DTW in time: $$\operatorname{STA}_\gamma(X,Y) = \operatorname{soft\text{-}DTW}_\gamma(C),\;\; C_{tt'} = \operatorname{OT}_\varepsilon^\tau(X_t, Y_{t'})$$ where each $C_{tt'}$ describes the spatial OT cost between time slices, and the global alignment cost is computed via a differentiable "soft-min" over all temporal matchings (Janati et al., 2019).

Time-Parameterized and Graph-Structured TiOT

Extensions to multi-period logistics and traffic flow incorporate explicit time indices in the transport plans: $$\min_{\gamma^1, \dots, \gamma^N} \sum_{t=1}^N \langle C^t, \gamma^t \rangle,$$ with per-step capacities or sparsity constraints, and only require the aggregate flow to meet total demand (Shi, 14 Feb 2025). For flow over graphs, TiOT is cast as a convex, time-discretized optimization that respects edge capacity via fundamental diagrams, yielding global uniqueness and efficient ADMM solvers (Dong et al., 1 Nov 2025).

Neural Hamilton–Jacobi Flows

Parameterized PDE-based TiOT constructs the optimal time-indexed Monge maps via viscosity solutions to the Hamilton–Jacobi equation: $$\partial_t \phi(t,x) + H(x, \nabla_x \phi(t,x)) = 0,\quad x(t) = x(0) + t \nabla \phi_0(x(0)),$$ where $\phi$ is realized by a neural network and transport interpolation $T_t(x) = x + t \nabla \phi_0(x)$ (Park et al., 30 Sep 2025, Buzun et al., 23 Jul 2025).

2. Theoretical Properties and Metric Structure

TiOT models are constructed to preserve key properties from classical optimal transport, such as metricity, geodesic interpolation, and robustness under perturbations.

• Metric Properties: The minimax TiOT distance $\mathcal{D}_p$ satisfies non-negativity, symmetry, identity of indiscernibles, and triangle inequality, and is bi-Lipschitz equivalent to the Wasserstein metric (Nguyen et al., 26 Dec 2025).
• Geodesic Structure: Time-integrated transport yields displacement interpolations (Wasserstein geodesics), preserving Riemannian metric structure even in discrete domains (e.g., triangle meshes) and under finite-difference schemes (Lavenant et al., 2018).
• Quadratic Growth: Soft-DTW-based TiOT models exhibit quadratic growth in cost under temporal shifts, providing sensitivity to misalignment in time that standard DTW lacks (Janati et al., 2019).

3. Algorithmic Implementations and Complexity

TiOT frameworks span classical convex optimization, specialized dynamic programming, neural PDE optimization, and hybrid block coordinate descent.

• Entropic-regularized Sinkhorn Algorithms: Dual approaches and block coordinate descent (BCD) schemes enable efficient approximation of large-scale TiOT problems, e.g., eTiOT has per-iteration cost $O(mn)$ and empirical convergence within 2--3$\times$ the Sinkhorn runtime for classical OT (Nguyen et al., 26 Dec 2025).
• Soft-DTW Dynamic Programming: Quadratic-time DP for temporal alignment is feasible for moderate sequence lengths, GPU parallelization and batching are standard (Janati et al., 2019).
• ADMM for Graph- and Surface-based TiOT: For dynamic flow on networks or meshes, splitting methods solve convex cone programs or variational discretizations with provable convergence and polynomial complexity (Dong et al., 1 Nov 2025, Lavenant et al., 2018).
• Heuristic Search for Sparse Plans: Combinatorial constraints (e.g., per-step route sparsity) are handled by importance-score-based heuristics, achieving $\sim$25\% suboptimality at $\sim$10\% of brute-force computational cost (Shi, 14 Feb 2025).
• Neural PDE Solvers: For time-indexed OT maps, neural networks parameterize potentials governed by Hamilton–Jacobi or Hamilton–Jacobi–Bellman equations, trained using residual losses and distributional matching (e.g., via MMD) (Park et al., 30 Sep 2025, Buzun et al., 23 Jul 2025).

4. Applications and Empirical Results

TiOT models have been applied across domains requiring alignment, interpolation, or regulation of structured temporal data.

• Time Series Analysis: Robust pairwise metrics for time-series, capturing both time-alignment and distributional similarity, outperform DTW and Wasserstein distances in classification and lag analysis tasks (Nguyen et al., 26 Dec 2025, Janati et al., 2019).
• Spatio-temporal Clustering and Visualization: STA-based TiOT cleanly separates brain imaging or handwritten letter classes, outperforming methods that use only temporal or spatial features (Janati et al., 2019).
• Graph and Network Flows: TiOT under fundamental diagram constraints models congestion-aware routing and achieves unique, convex-optimal solutions for city-scale traffic networks (Dong et al., 1 Nov 2025).
• Distribution Interpolation and Surface Flows: Discrete-surface TiOT produces geodesic interpolants and supports Dirichlet problems, gradient flows, and crowd motion modeling on meshes (Lavenant et al., 2018).
• Control and Regulation: In discrete-time LQR with distributional endpoints, TiOT yields closed-form steering laws and efficient numerical transport-based regulation of multi-agent systems (Badyn et al., 2021).
• Logistics and Resource Planning: Time-parameterized OT enables capacity- and sparsity-constrained multi-period plans for supply chains and logistics, with significant complexity reduction via problem reformulation (Shi, 14 Feb 2025).

5. Extensions, Limitations, and Outlook

Ongoing work in TiOT addresses generalizations, improved robustness, and faster algorithms.

• Generalized Costs and Conditionals: Neural Hamilton–Jacobi and HOTA frameworks accommodate general strictly convex cost functions, class-conditional transports, and non-smooth potential terms (Park et al., 30 Sep 2025, Buzun et al., 23 Jul 2025).
• Regularization and Scalability: Entropic smoothing, separable-kernel approximations, and low-rank OT reduce complexity for high-dimensional or long-horizon problems (Nguyen et al., 26 Dec 2025, Janati et al., 2019).
• Discrete and Structured Domains: TiOT generalizes to graphs, meshes, and other discrete structures via carefully constructed continuous/discrete analogues (Dong et al., 1 Nov 2025, Lavenant et al., 2018).
• Robustness to Balance Parameters: Minimax TiOT formulations eliminate manual tuning, providing automatic tradeoffs between time and feature similarity (Nguyen et al., 26 Dec 2025).
• Limitations: Current TiOT metrics can be limited to scalar time-indexing; multivariate or structured time extensions are open questions (Nguyen et al., 26 Dec 2025). Outer maximizations remain low-dimensional but could be extended to more complex cost families.
• Open Challenges: Faster convergence rates, generalization to stochastic process couplings with causality constraints, and integration with deep generative modeling remain active research directions.

6. Comparative Overview

TiOT Category	Principle	Key Algorithms/Properties
Dynamical/Benamou–Brenier	Minimum kinetic action	Convex optimization/ADMM; Geodesic transport (Lavenant et al., 2018, Dong et al., 1 Nov 2025)
Spatio-temporal Minimax	Worst-case over weights	Sinkhorn/BCD, automatic balancing (Nguyen et al., 26 Dec 2025)
STA (Soft-DTW+OT)	Joint space-time align.	Soft-DTW DP, Sinkhorn, quad penalty (Janati et al., 2019)
Neural PDE/Characteristic	Closed-form maps	Neural net PDE, MMD matching (Park et al., 30 Sep 2025, Buzun et al., 23 Jul 2025)
Time-parameterized Planning	Sequential constraints	LP reduction, heuristic search (Shi, 14 Feb 2025)

This collection of TiOT models demonstrates a comprehensive theoretical and computational toolkit, ranging from PDE-based neural interpolants to scalable, robust time-series metrics and dynamic network flows. Continuing advances are likely to further extend the reach and scalability of time-integrated optimal transport models across computational science, engineering, and data analysis.