Gradient Flow Algorithms for Density Propagation in Stochastic Systems

Abhishek Halder; Kenneth F. Caluya

REVIEW 3 major objections 4 minor 1 cited by

Gradient Flow Algorithms for Density Propagation in Stochastic Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A time-only discretization can compute transient joint probability densities of stochastic systems without gridding the state space.

desk verdict Promising point-cloud Sinkhorn/JKO density propagation, but the central claim is unproven: no consistency for the outer split, so treat it as a strong heuristic until that is fixed. read the letter →

arxiv 1908.00533 v2 pith:Q5FKVYKE submitted 2019-08-01 math.OC cs.LGcs.SYeess.SY

Kenneth F. Caluya , Abhishek Halder This is my paper · ORCID

classification math.OCcs.LGcs.SYeess.SY MSC 49Q2235Q8460H3565C30

keywords Fokker-Planck-KolmogorovequationWassersteingradientflowJKOschemeentropicregularizationSinkhorniterationuncertaintypropagationMcKean-VlasovdynamicsThompsonmetric

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that transient joint probability density functions for Itô stochastic differential equations can be computed by discretizing time alone, never the state space. The core idea is to view the Fokker-Planck-Kolmogorov PDE as a Wasserstein gradient flow and to replace it by the Jordan-Kinderlehrer-Otto proximal recursion, which minimizes Wasserstein distance plus a free-energy term at each small time step. The paper shows how to make that infinite-dimensional recursion numerically concrete: an entropic regularization turns each proximal step into a cone-preserving fixed-point iteration whose unique solution gives updated probability weights on a moving point cloud. For systems already in the JKO canonical form, and for several systems that can be transformed into it, the paper argues that this yields fast, nonparametric, grid-free propagation of joint densities. A sympathetic reader would care because this offers a route around the curse of dimensionality that limits conventional PDE solvers.

What carries the argument

The load-bearing object is the JKO proximal recursion, a variational replacement of the Fokker-Planck-Kolmogorov PDE in which each small time step solves $\arg\inf_{\varrho} \frac{1}{2} W^2(\varrho_{k-1},\varrho) + h F(\varrho)$ with $W$ the 2-Wasserstein metric and $F$ the free energy. Entropic regularization of the inner optimal-transport coupling makes the optimal matrix have the Sinkhorn form $m_{\mathrm{opt}}(i,j) = \exp(\lambda_0(i)h/\epsilon) \exp(-C_k(i,j)/(2\epsilon)) \exp(\lambda_1(j)h/\epsilon)$, algebraicizing the problem. The resulting fixed-point equations are solved by a block-coordinate iteration whose contraction is established in the Thompson metric, $d_T(z,\tilde{z}) = \log \max_i \max(z_i/\tilde{z}_i, \tilde{z}_i/z_i)$, on the positive orthant, using Perron-Frobenius positivity of the kernel $\Gamma_k = \exp(-C_k/2\epsilon)$. The point-cloud locations advance independently by the Euler-Maruyama scheme, so the whole algorithm is a temporal recursion on weighted scattered data rather than a spatial discretization.

What would settle it

Run the Ornstein-Uhlenbeck benchmark of Section IV-A1 with the same $N=400$ and $\epsilon=5\times 10^{-2}$ while shrinking $h$ from $10^{-2}$ to $10^{-6}$, and compare the proximal weighted mean and variance against the exact formulas $\mu_0 \exp(-at)$ and $(\sigma_0^2 - 1/(a\beta)) \exp(-2at) + 1/(a\beta)$; if the error does not systematically decrease as $h \to 0$, the claimed consistency of the split scheme is false. Separately, compute the Thompson contraction factor of the linear map $z \mapsto \Gamma_k z$ for random positive $z$ with a randomly generated cost matrix $C_k$; if any eigenvalue of the associated projective map has modulus at least one, the asserted Perron-Frobenius strict contractivity fails.

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Extended reading notes

Core claim

The central claim is that the proximal recursion $\varrho_k = \arg\min_{\varrho} \frac{1}{2} W^2(\varrho_{k-1}, \varrho) + h F(\varrho)$ is computationally tractable in higher dimensions once the transport term is entropy-regularized and dualized. The optimality conditions collapse to the coupled equations $y \odot (\Gamma_k z) = \varrho_{k-1}$ and $z \odot (\Gamma_k^\top y) = \xi_{k-1} \odot z - \beta\epsilon/h$, with the updated density given by $\varrho_k = z \odot (\Gamma_k^\top y)$. The authors prove that the block-coordinate iteration for these equations is strictly contractive in the Thompson metric on the positive orthant, so Algorithm 1 has a unique fixed point and converges; combining this weight update with Euler-Maruyama moves of the particle locations gives the full density-propagation algorithm. They demonstrate the scheme on linear Gaussian systems, a bimodal nonlinear system, a McKean-Vlasov mean-field problem, a multiplicative-noise process via the Lamperti transform, and a six-state mixed conservative-dissipative satellite dynamics example.

Load-bearing premise

The load-bearing premise is that combining Euler-Maruyama point-location updates with the Sinkhorn-style weight updates reproduces the JKO scheme, and hence the Fokker-Planck-Kolmogorov PDE, as the time step $h$ tends to zero; the paper proves contraction of the inner fixed-point iteration but does not prove this outer consistency, and it also asserts rather than proves the Perron-Frobenius strict contraction of $\Gamma_k$.

Editorial extensions

If this is right

Transient joint PDFs for nonlinear Itô SDEs with gradient drift can be computed as weighted point clouds at a cost that scales with the number of particles, not with the state-space volume, so the method sidesteps the curse of dimensionality.
Each proximal weight update provably converges to a unique fixed point, giving a deterministic inner loop with a global contraction guarantee for every physical time step.
Systems outside JKO canonical form, including multiplicative noise and mixed conservative-dissipative drift, can be brought into the framework by coordinate transforms or modified transport cost functions, as shown for the CIR process and a satellite problem.
The McKean-Vlasov case with interaction potentials is handled by a semi-implicit free energy, so mean-field density evolution is available at the same proximal-scheme cost.
Reported runtimes near $10^{-6}$ seconds per proximal update at $N=400$, against a physical time step of $10^{-3}$ seconds, indicate the scheme is fast enough for repeated density propagation in filtering and control loops.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the split scheme is only consistent in the limit $h\to 0$, practical accuracy will depend on the joint tuning of $h$, the regularization $\epsilon$, and particle number $N$; the paper proves contraction of the inner fixed-point iteration but not consistency of the outer split, so the observed accuracy in examples may not hold uniformly across parameters.
Because the inner iteration is a positive-cone contraction, an adaptive time-stepping rule driven by the measured Thompson contraction rate could reduce the number of physical steps while preserving the fixed-point guarantee.
The same dual Sinkhorn structure can be applied to nonlinear filtering by replacing the free-energy term with a data-likelihood term, yielding an optimal-transport-based Bayesian update for the prior density propagated by this scheme.
The fixed-point equations match the structure of entropic optimal transport, so the transient densities computed here may coincide with Schrödinger-bridge interpolations in the small-noise limit, tying uncertainty propagation to stochastic control.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a serious attempt to give JKO computational teeth, and the numerics are good enough that I believe the algorithm does something real. But the paper overclaims: the step from the inner Sinkhorn recursion to the PDE is not proved, and the missing piece is not cosmetic.

What's new is the specific point-cloud proximal recursion (Euler-Maruyama for support, entropic OT for weights) with a Thompson-metric contraction proof for the inner loop. That combination is not in the JKO/Sinkhorn literature. The extensions to CIR and to a conservative-dissipative satellite problem show real scope.

The numerics are a genuine plus. The OU, multivariate LTI, stationary bimodal, and CIR examples all match analytic solutions visually, and the 6D satellite marginals are plausible. That is not easy to get from a toy method.

Soft spots, in order of size. First, no consistency theorem for the split scheme: as h goes to zero, the support points move by Euler-Maruyama while the weights solve an entropic OT problem with fixed epsilon. Since epsilon is not sent to zero or scaled by h, the limit is not obviously the JKO gradient flow. The paper proves only that the inner fixed point exists and is unique for each step. The claim that the split approximates the FPK PDE rests on unverified heuristic. A numerical error-versus-h study would partially address this, but it is not in the paper.

Second, the alleged inconsistency in (33b). I checked the math. If the printed equation is read as z times (Gamma^T y) equals xi times z^{-beta*epsilon/h}, it is consistent with the proof and with Algorithm 1. The typesetting is ambiguous and should be fixed, but it is not a load-bearing flaw. The real flaw is the missing outer convergence.

Third, the Thompson contractivity of Gamma_k is asserted via Perron-Frobenius without proof. This is repairable, since the power map theta_1 is already a strict contraction, so the composite is contractive even if Gamma_k is merely non-expansive.

Fourth, the numerical validation would be stronger with quantitative error metrics, a description of how weights relate to the continuous PDF (weights live in the simplex but are compared directly to density values), and wall-clock comparisons against standard PDE solvers. The timing plots are too thin to support the "remarkably fast" claim.

Who this is for: researchers in uncertainty propagation, filtering, or computational optimal transport who want a practical alternative to grid-based FPK. The paper deserves a serious referee. I would send it out, with the expectation of major revision: add a consistency argument or a careful numerical convergence study, clarify the point-cloud representation, and clean up the typography. The inner-loop result is worth keeping.

Referee Report

3 major / 4 minor

Summary. The paper develops a computational framework for propagating joint probability density functions (PDFs) of stochastic systems described by Fokker-Planck-Kolmogorov (FPK) and McKean-Vlasov equations. The approach is based on the Jordan-Kinderlehrer-Otto (JKO) gradient-flow formulation, discretizing time while keeping the state space continuous. The proposed algorithm represents the density as a weighted point cloud, advances the support points via Euler-Maruyama, and updates the weights by solving an entropic-regularized optimal transport proximal recursion. The inner fixed-point iteration (Algorithm 1) is proved contractive in the Thompson metric (Theorem 3), and numerical examples for linear Gaussian, nonlinear non-Gaussian, McKean-Vlasov, CIR, and a satellite dynamics problem are compared against analytical solutions or stationary densities.

Significance. If the full algorithmic pipeline were proved convergent to the underlying PDEs as the time step tends to zero, the paper would offer a novel, non-parametric, grid-free method for uncertainty propagation that avoids spatial discretization and its curse of dimensionality. The paper contains explicit algorithmic details, benchmarks against known analytical solutions, and timing measurements. However, the theoretical support provided only covers the inner Sinkhorn fixed-point recursion; the consistency of the outer splitting scheme with the JKO flow is not established. This missing analysis currently limits the significance of the central claims, although the numerical evidence suggests the approach may have merit.

major comments (3)

[III-C and overall] The paper's abstract and introduction claim that the proposed scheme is 'theoretically equivalent to solving the underlying transport PDEs' in the small time-step limit, but the only convergence result proved (Theorem 3 and Corollary 4) concerns the inner fixed-point iteration for fixed h, epsilon, and C_k. No theorem or error estimate connects the two-step update (Euler-Maruyama for positions, then Algorithm 1 for weights) to the JKO scheme or to the FPK/McKean-Vlasov PDE as h tends to zero. The numerical experiments use a single time step h = 10^{-3} and do not report any error-versus-h study, so they do not fill this gap.
[III-A, Eq. (27)] The entropic regularization parameter epsilon is taken fixed and is not scaled with the time step h. In the objective (27), the term epsilon H(M) is not multiplied by h, whereas hF(rho) is. As h tends to zero with epsilon fixed, the relative weighting changes, and the limit of the proximal step need not be the gradient flow of F. The paper never sends epsilon to zero (e.g., by choosing epsilon = epsilon(h) with a suitable rate) nor proves that the entropically regularized minimizer converges to the unregularized JKO step. Without this, the claimed equivalence of the proximal recursion to the original free-energy gradient flow is not established.
[III, paragraph before Eq. (23)] There is an inconsistency in the meaning of the weights rho^i_k. The text states that rho^i_k 'denotes the value of the joint PDF evaluated at that point,' but Eq. (23) treats rho as a probability vector in the simplex, so the weights are probability masses summing to one. The discrete free energy in (27), equal to <psi + beta^{-1} log rho, rho>, is the negative entropy of the atomic measure, not the continuous free energy integral rho log rho dx. No convergence analysis is provided to show that the discrete weighted point cloud approximates the continuous density as N grows, which is necessary to justify the claim that the algorithm computes the transient joint PDF.

minor comments (4)

[Theorem 1, Eq. (33b)] Equation (33b) is typeset ambiguously; it should read z \odot (Gamma_k^\top y) = \xi_{k-1} \odot z^{-\beta\epsilon/h}. The exponent on z is not clearly displayed.
[Proof of Theorem 3] The statement that the positive linear map Gamma_k is contractive in the Thompson metric 'by Perron-Frobenius theorem' is imprecise; the correct result is Birkhoff's contraction theorem for strictly positive matrices. Since the map theta_1 is already strictly contractive, this claim is not needed for the conclusion.
[Algorithm 1] The initialization z0 is chosen randomly. The authors could note that the Banach contraction principle guarantees convergence from any initial point in the positive orthant, so the random choice is immaterial to the convergence.
[Section IV and V] The computational-time plots (Figs. 9 and 14) do not state the hardware and software environment, which makes the timing results difficult to reproduce or compare.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the algorithm is validated against independent analytical solutions, and the JKO equivalence is cited from external prior work; the only gaps are unproved consistency, not circularity.

full rationale

The paper's claimed derivation chain is: (1) the JKO theorem, cited to the external works [21] and [23], states that the Wasserstein proximal recursion with the free energy functional converges to the FPK PDE; (2) the paper discretizes that recursion into a finite-dimensional entropic-regularized optimal transport problem (eq. 27); (3) dualization yields the fixed-point system (33); (4) Algorithm 1 is a block-coordinate iteration proven to converge to that fixed point (Theorem 3, Corollary 4); and (5) the support points are moved by Euler-Maruyama updates (39), with weights updated by Algorithm 1. At no step is the target density used as an input: no parameter is fitted to the reference PDFs, and the numerical benchmarks are independent analytical solutions for the OU process, the multivariate LTI system, and the McKean-Vlasov flow, plus the known stationary Gibbs density for the nonlinear example. The regularization parameter epsilon is fixed by the user, not calibrated to match the output. The contraction proof is internal to the discrete fixed-point map, and the JKO-to-PDE equivalence is an external mathematical result, not an assumption equivalent to the paper's own prediction. The self-citations [18], [19], and [25] are motivational or supply coordinate transformations in the MIMO linear example; they are not invoked as uniqueness theorems forbidding alternatives, and the transforms are externally checkable prior work. The manuscript itself flags the main open issue: no consistency or error estimate connects the Euler-Maruyama/Sinkhorn split to the JKO scheme as h tends to 0 with epsilon fixed, and the proof of Theorem 3 asserts Perron-Frobenius contractivity of Gamma_k without demonstration. Those are correctness and proof-completeness gaps, not circular reductions of the claimed predictions to the inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The algorithm depends on established variational and numerical analysis results, not on newly postulated physical entities. The main free parameter is the entropy regularization epsilon, chosen by hand. The convergence of the full split scheme is assumed rather than proved.

free parameters (4)

epsilon (entropic regularization) = 5e-2
Chosen by hand for all numerical experiments; controls bias of the Sinkhorn approximation to exact optimal transport and enters the contraction rate in Theorem 3.
time step h = 10^-3 (10^-5 in the satellite example)
Discretization step for the proximal recursion; the claimed equivalence with the PDE holds only as h approaches 0, and numerical accuracy depends on it.
sample count N = 400
Number of points in the point cloud; affects approximation quality of the density, with no analysis of how N must scale with dimension.
initial PDF approximation for Dirac delta (CIR example) = N(5, 10^-4)
Used in Section V-A to approximate rho_0 = delta(x-5); this introduces a modeling approximation not discussed further.

assumptions (5)

domain assumption The proximal recursion (9) with d = W2 and Phi = F converges to the FPK flow as h tends to 0 (Jordan-Kinderlehrer-Otto).
Cited as the theoretical basis for the algorithm; the paper does not reprove it.
standard math The entropically regularized problem (27) has zero duality gap.
Objective is proper convex and lower semicontinuous; used to derive the dual (29).
domain assumption The Euler-Maruyama scheme converges strongly to the SDE solution when the drift is globally Lipschitz.
State positions are advanced with this scheme; stated in Remark 1 with a citation to Kloeden and Platen.
standard math A positive linear map Gamma_k is a strict contraction in the Thompson metric on the positive orthant.
Used in Theorem 3; the paper invokes Perron-Frobenius theory, but the strict contraction property for positive matrices is a nontrivial background result not proved there.
domain assumption The semi-implicit scheme (48) with F(rho_{k-1}, rho) converges to the McKean-Vlasov flow as h tends to 0.
Cited to Laborde [47, Section 12.3]; used for nonlocal interactions in Section IV-C.

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Pith. "Pith review of Gradient Flow Algorithms for Density Propagation in Stochastic Systems." pith.science (2026). https://pith.science/paper/Q5FKVYKE

@misc{pith2026190800533,
  author       = {Pith},
  title        = {Pith review of: Gradient Flow Algorithms for Density Propagation in Stochastic Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q5FKVYKE}},
  note         = {Machine review of arXiv:1908.00533}
}

read the original abstract

We develop a new computational framework to solve the partial differential equations (PDEs) governing the flow of the joint probability density functions (PDFs) in continuous-time stochastic nonlinear systems. The need for computing the transient joint PDFs subject to prior dynamics arises in uncertainty propagation, nonlinear filtering and stochastic control. Our methodology breaks away from the traditional approach of spatial discretization or function approximation -- both of which, in general, suffer from the "curse-of-dimensionality". In the proposed framework, we discretize time but not the state space. We solve infinite dimensional proximal recursions in the manifold of joint PDFs, which in the small time-step limit, is theoretically equivalent to solving the underlying transport PDEs. The resulting computation has the geometric interpretation of gradient flow of certain free energy functional with respect to the Wasserstein metric arising from the theory of optimal mass transport. We show that dualization along with an entropic regularization, leads to a cone-preserving fixed point recursion that is proved to be contractive in Thompson metric. A block co-ordinate iteration scheme is proposed to solve the resulting nonlinear recursions with guaranteed convergence. This approach enables remarkably fast computation for non-parametric transient joint PDF propagation. Numerical examples and various extensions are provided to illustrate the scope and efficacy of the proposed approach.

Figures

Figures reproduced from arXiv: 1908.00533 by the authors.

**Figure 1.** The gradient descent on the manifold of PDFs can be described by successive evaluation of proximal operators to recursively update PDFs from time t = (k − 1)h to t = kh for k ∈ N, and time-step h > 0. as xk = arg min x∈Rn 1 2 k x − (xk−1 − h∇ϕ(xk−1)) k 2 2 = arg min x∈Rn 1 2 k x − xk−1 k 2 2 + hx − xk−1, h∇ϕ(xk−1)i + hϕ(xk−1), (5) where we used the fact that adding and omitting constant terms do not change the arg m… view at source ↗

**Figure 2.** Schematic of the proposed algorithmic setup for propagating the joint state PDF as probability weighted scattered point cloud {x i k , %i k }N i=1. The location of the points {x i k }N i=1 are updated via EulerMaruyama scheme; the corresponding probability weights are updated via Algorithm 1. The dashed arrow shown above is present only when the state dynamics is density dependent, as in (19). applied to the under… view at source ↗

**Figure 3.** Comparison of the analytical and proximal solutions of the FPK PDE for (45) with time step h = 10−3 , and with parameters a = 1, β = 1, = 5 × 10−2 . Shown above are the time evolution of the (left) PDFs, (middle) means, and (right) variances. z ≤ ze. Thompson [42] proved that K is a complete metric space w.r.t. the so-called Thompson metric given by dT (z, ze) := max{log γ(z/ze), log γ(ze/z)}, where γ(z/ze) := inf… view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: Comparison of the analytical (contour plots) and proximal (weighted scattered point cloud) joint PDFs of the FPK PDE for (43) with time step h = 10−3 , and with parameters β = 1, = 5 × 10−2 . Simulation details are given in Section IV-A2. The color (red = high, blue …

**Figure 5.** Figure 5: Comparison of the components of the mean vectors from analytical (dashed) and proximal (solid) computation of the joint PDFs for (43) with time step h = 10−3 , and with parameters β = 1, = 5×10−2 . Simulation details are given in Section IV-A2. B. Nonlinear Non-Gaus…

**Figure 6.** Figure 6: Comparison of the components of the covariance matrices from analytical (dashed) and proximal (solid) computation of the joint PDFs for (43) with time step h = 10−3 , and with parameters β = 1, = 5 × 10−2 . Simulation details are given in Section IV-A2 [PITH_FULL_IM…

**Figure 7.** Figure 7: The drift potential ψ(x1, x2) = 1 4 (1 + x 4 1 ) + 1 2 (x 2 2 − x 2 1 ) used in the numerical example given in Section IV-B. computed via the proximal recursion, tend to the known stationary solution ρ∞ (contour plots in the right bottom sub-figure in [PITH_FULL_IMAGE…

**Figure 9.** Figure 9: shows the computational times for the proposed proximal recursions applied to the above nonlinear non-Gaussian system. Since the proposed algorithm involves sub-iterations (“while loop” in Algorithm 1 over index ` ≤ L) while keeping the physical time “frozen”, the conv…

**Figure 10.** Figure 10: Comparison of the analytical and proximal solutions of the McKean-Vlasov flow for (46) with time step h = 10−3 , ρ0 = N (5, 9), and with parameters a = b = 1, β = 1, = 5 × 10−2 . Shown above are the time evolution of the transient (left) PDFs, (middle) means, and (r…

**Figure 11.** Figure 11: Comparison of the analytical and proximal transient PDFs of the FPK PDE for (50) with time step h = 10−3 , and with parameters a = 3, b = 2, θ = 2, x0 = 5, = 5×10−2 . To approximate the analytical PDFs resulting from ρ0(x) = δ(x−5), the proximal recursions were perf…

**Figure 13.** Figure 13: Univariate marginal PDFs at t = 0.01 s for (59)-(60) computed from the joint PDF at that time obtained via the proposed proximal algorithm for recursion (61) with time step h = 10−5 , and with parameters β = 1 m2 /s2 , γ = 1 s−1 , = 5 × 10−2 , δ = 10−3 , L = 100, an…

**Figure 14.** Figure 14: The computational times needed for proximal updates in the 6-state numerical example reported in Section V-B. Here, the physical time-step h = 10−5 s, and k ∈ N. Figs. 12 and 13 show the univariate marginal PDFs at t = 0.005s and t = 0.010s respectively, associated wi…

**Figure 12.** Figure 12: Univariate marginal PDFs at t = 0.005 s for (59)-(60) computed from the joint PDF at that time obtained via the proposed proximal algorithm for recursion (61) with time step h = 10−5 , and with parameters β = 1 m2 /s2 , γ = 1 s−1 , = 5 × 10−2 , δ = 10−3 , L = 100, a…

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