Perturbed Wasserstein Gradient Flow (PWGF)

• PWGF is an extension of classical Wasserstein Gradient Flow that integrates explicit or implicit perturbations to address nonconvex objectives and ensure local well-posedness.
• It leverages modified JKO schemes, noise-injected dynamics, and parameterized approximations to model high-dimensional systems and maintain stability.
• The framework offers practical insights into escaping saddle points, preserving structure, and enabling applications in PDEs, generative models, and topological data analysis.

Perturbed Wasserstein Gradient Flow (PWGF) is a framework for evolution equations and optimization on the space of probability measures that extends classical Wasserstein gradient flow (WGF) through explicit or implicit perturbations in the underlying functional, gradient flow dynamics, or numerical algorithm. PWGF arises in diverse contexts: handling nonconvex objectives, enforcing robust well-posedness for energies lacking global convexity, incorporating noise or stochasticity, modeling interacting particle systems, ensuring structure preservation, and approximating high-dimensional dynamics by reduced order parameterizations. Theoretical and computational investigation of PWGF is fundamental to robustly addressing nonconvexity, sensitivity, and modeling errors in Wasserstein optimization and evolution settings.

1. Core Mechanism and Definitions

Classical Wasserstein gradient flow is the evolution of a probability measure μ(t) on ℝ^d that follows the steepest-descent dynamics of an energy functional E(μ) with respect to the Wasserstein (optimal transport) metric W₂. In formal terms, the evolution is governed by

$$\partial_t \mu_t + \nabla \cdot \left( \mathbf{v}_t \mu_t \right) = 0, \quad \mathbf{v}_t = -\nabla_{\mu} E(\mu_t)$$

where the tangent vector field $\mathbf{v}_t$ is the Wasserstein gradient. The JKO (Jordan–Kinderlehrer–Otto) scheme for time discretization constructs (for time step τ)

$$\mu_{k+1} = \operatorname{argmin}_\mu \left\{ \frac{1}{2\tau} W_2^2(\mu, \mu_k) + E(\mu) \right\}$$

PWGF generalizes this by introducing a perturbation, either in the energy functional, the dynamics, the optimization constraints, or the numerical update:

• Functional-level: $E(\mu) = E_0(\mu) + \varepsilon P(\mu)$.
• Dynamics-level: addition of noise or stochastic drift, $$d\mu_t = -\nabla_{\mu} E(\mu_t) dt + \sqrt{2D} dR_t$$
• Scheme-level: extra regularization/barrier terms in minimization, e.g., logarithmic barrier $-\mu \log(\rho)$ for enforcing positivity.

Fundamentally, the "perturbation" can be smooth, nonsmooth, stochastic, geometric, or structure-inducing.

2. Analytical Foundations: Restricted Convexity and Well-Posedness

A recurring analytical obstacle for higher-order or nonconvex energies is the failure of global displacement convexity, preventing global existence and uniqueness results in the standard theory. The notion of restricted λ-convexity addresses this challenge (Kamalinejad, 2011):

• Restricted λ-convexity: The functional E is λ-convex on sublevel sets or small Wasserstein balls around a reference measure; i.e., for $μ_0, μ_1$ in a set $B_\delta(\mu) \cap \{E \leq c\}$ and geodesic $μ_s$,

$$E(μ_s) \leq (1-s)E(μ_0) + sE(μ_1) - \frac{\lambda}{2}s(1-s)W_2^2(μ_0, μ_1)$$

In the presence of a perturbation, if the perturbing term is sufficiently regular and "small" (in the sense of preserving restricted λ-convexity locally), this local convexity supplies uniform compactness and contraction estimates for the minimizing movement (JKO) scheme, yielding well-posedness and uniqueness for "perturbed" WGF on sublevel sets—even where the total energy is nonconvex or only degenerate-convex. This machinery is used to handle thin-film equations and quantum drift-diffusion (Kamalinejad, 2011), as well as general perturbed flows (Kinderlehrer et al., 2015, Kell, 2014).

3. Numerical and Algorithmic Schemes for PWGF

The variational structure and energy-dissipation properties of Wasserstein gradient flows enable robust time- and space-discretizations that inherit stability and monotonicity, as well as flexibility in handling perturbations:

• JKO with perturbations: For time step τ and perturbation Ρ(μ),

$$\mu_{k+1} = \arg\min_{\mu} \left\{ \frac{1}{2\tau}W_2^2(\mu, \mu_k) + E_0(\mu) + \varepsilon Ρ(\mu) \right\}$$

This admits adaptation to PWGF by varying the choice of Ρ, e.g., entropy, interaction, or external fields (Kamalinejad, 2011, Kinderlehrer et al., 2015).

• Robust discretizations: Upstream-mobility two-point flux finite volumes, TPFA finite volume with interior-point methods, and regularized Lagrangian schemes can incorporate perturbations in both the energy and dissipation, while guaranteeing nonnegativity, mass conservation, and discrete energy dissipation (Cancès et al., 2019, Natale et al., 2020, Cheng et al., 21 Jun 2024).
• Stochastic and noise-injected dynamics: PWGF naturally accommodates additive or multiplicative noise, either by direct SDE formulations (Sebag et al., 2023, Wang et al., 5 Dec 2024, Ren et al., 15 Jun 2025) or as a byproduct of particle approximations and controlled Euler–Maruyama schemes (for instance, to ensure differential privacy or model stochastic interacting particle systems).

A table summarizing these approaches:

Discretization Strategy	Type of Perturbation	Analytical/Practical Benefit
Energy-perturbed JKO (min movement)	Smooth/convex term/entropy	Uniform well-posedness, stability
Logarithmic barrier in FV schemes	Positivity-enforcing	Numerical robustness
SDE-based particle flows	Noise/Differential privacy	Privacy, stochastic modeling
Hessian-guided perturbation	Nonconvex optimization	Saddle point escape, 2nd order

4. Advanced Constructions and Modern Applications

Escaping Nonconvexity: Hessian-Guided PWGF

Standard WGF provably finds only first-order stationary points for nonconvex functionals; in many cases (especially in high-dimensional machine learning) this may mean convergence to unstable or non-optimal local minima. PWGF (Yamamoto et al., 21 Sep 2025) improves on this by injecting Gaussian-process-based noise adapted to the local curvature (via the Wasserstein Hessian), leading to efficient escape from saddle points. The perturbation is: $$μ ← (Id + η_p ξ)_{\#} μ, \quad ξ \sim \mathcal{GP}(0, K_μ), \quad K_μ(x, y) = \int ∇_μ^2 F(μ, x, z)∇_μ^2 F(μ, z, y) μ(dz)$$ This noise is structured to concentrate along the eigenvectors associated with negative curvature, ensuring efficient descent toward second-order stationary points, whose achievement is further controlled by complexity bounds polynomial in $1/ε$ and $1/δ$ for desired first- and second-order tolerances.

Stochastic Intrinsic Gradient Flow

PWGF is also realized through stochastic quantization techniques: starting from a symmetric Dirichlet form corresponding to a Gaussian-based reference measure $\Lambda$ on $\mathcal{P}_2$, one perturbs with $e^{-W_F}$ to obtain a Gibbs-type measure $\Lambda_F$, and constructs a Markov process obeying the stochastic EVI (martingale) problem (Ren et al., 15 Jun 2025): $dμ_t = - D^{γ}W_F(μ_t) dt + dR_t$ where $R_t$ is a symmetric Markov noise process, and $D^γ W_F$ is the (possibly weighted) intrinsic Wasserstein gradient. This framework enables perturbations via stochastic processes and is especially relevant for modeling equilibrium fluctuations, stochastic PDE limits, and structure-preserving sampling.

Parameterized, Reduced-Order, and Geometry-Driven PWGF

There is now substantial interest in constructing parameterized or reduced-order Wasserstein gradient flows, both for scalability and high-dimensional applications (Li et al., 2019, Jin et al., 29 Apr 2024). One enforces the flow at the level of parametrized push-forward families (e.g., normalizing flows or neural networks mapping reference densities), turning the infinite-dimensional WGF into an ODE on the parameter space Θ equipped with a pullback Wasserstein metric: $$\dot{\theta} = -\hat{G}(\theta)^{\dagger} \nabla_{\theta} F(\theta)$$ with $\hat{G}(\theta)$ the empirical or analytic pullback metric. Perturbations at the parameter level are possible (e.g., noise in θ, extra regularizations in the energy or via injected random projections), enabling fast, scalable approximations for complex PDEs, mean-field models, and generative frameworks.

Perturbation for Structure Preservation and Privacy

In domains requiring preservation of global or geometric structure (e.g., images, point clouds), perturbed flows are used where the "mobility operator" maps the Wasserstein gradient to an "alignment-preserving" gradient (Zhang et al., 16 Jul 2024). For privacy-preserving data synthesis (especially in generative latent variable models), gradient flows for smoothed Sliced Wasserstein distances with Gaussian convolution naturally yield flows that are differentially private (Sebag et al., 2023). Here, the perturbation (Gaussian smoothing) not only improves regularity, but also implements a privacy mechanism.

5. Theoretical Guarantees: Robustness, Stability, and Convergence

PWGF analysis builds on convexity (even if only restricted/approximate), energy-dissipation estimates, and variational structure. The minimizing movement scheme, when coupled with uniform lower semicontinuity of the energy and slopes (typically enforced by convexity or controlled perturbations), yields compactness, convergence to continuum flows, and stability under perturbation. In particle methods, Γ-convergence of the discrete energy together with metric slope lower semicontinuity ensures that particle approximations of PWGF converge to the true continuum flow (Lei, 7 Jan 2025).

Complexity results for nonconvex functionals under Hessian-guided PWGF guarantee polynomial-time convergence to second-order stationary points or global optimizers in "strictly benign" functional landscapes (Yamamoto et al., 21 Sep 2025).

6. Applications and Extensions

• Nonconvex measure optimization: Accurate optimization of probability distributions under nonconvex loss functions with convergence to global optima or certified second-order stationary points.
• Generative modeling and privacy: Flow-based generative models for synthetic data that encode privacy or geometric constraints via perturbations.
• High-dimensional PDEs and sampling: Scalable numerical simulation for kinetic, porous medium, aggregation, and filtering equations, circumventing curse of dimensionality by parameterization or particle methods.
• Transfer learning and distributional domain shifts: Wasserstein-over-Wasserstein flows (Bonet et al., 9 Jun 2025) adapt entire labeled distributions across domains; perturbations enable tracking under stochasticity, data noise, or class mismatch.
• Topological data analysis: Evolving persistence diagrams under perturbed flows enables robust analysis of topological features under noisy dynamical conditions (Wang et al., 5 Dec 2024).

7. Limitations and Open Directions

The effectiveness of PWGF critically depends on:

• The magnitude and structure of the perturbation: Too large a perturbation may destroy well-posedness, cause instability, or move the flow outside controllable sublevel sets.
• Regularity and convexity: Preserving (restricted) λ-convexity is crucial; pathological functionals may break the variational framework.
• Numerical resolution: For high-dimensional or singular flows, the parameterized or particle-based flows are only as accurate as the capacity and calibration of the reduced model.
• Quantitative convergence: While qualitative robustness under perturbation is established, quantitative error rates, especially for stochastic/intrinsic flows and in the presence of structured noise, remain an active research area.

For further mathematical and algorithmic details, see:

• "Well-posedness of Wasserstein Gradient Flow Solutions of Higher Order Evolution Equations" (Kamalinejad, 2011) (restricted λ-convexity, JKO, stability under perturbation)
• "Hessian-guided Perturbed Wasserstein Gradient Flows for Escaping Saddle Points" (Yamamoto et al., 21 Sep 2025) (second-order optimality, nonconvex optimization)
• "Parameterized Wasserstein Gradient Flow" (Jin et al., 29 Apr 2024) (parameterized models, scalable simulation)
• "Stochastic intrinsic gradient flows on the Wasserstein space" (Ren et al., 15 Jun 2025) (intrinsic gradient, stochastic Dirichlet forms)
• "Differentially Private Gradient Flow based on the Sliced Wasserstein Distance" (Sebag et al., 2023) (privacy via flow perturbation)
• "A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations" (Kinderlehrer et al., 2015) (flow interchange method in perturbed evolution)
• "Flowing Datasets with Wasserstein over Wasserstein Gradient Flows" (Bonet et al., 9 Jun 2025) (WoW flows for hierarchical datasets)

These works collectively form the core landscape for both the theory and application of Perturbed Wasserstein Gradient Flows in contemporary analysis and data science.