A transition-density-based operator learning method for Fokker-Planck equations with various initial conditions

Fabio Nobile; Li Zeng; Tao Zhou; Xiaoliang Wan; Yaobin Wang

REVIEW 3 major objections 6 minor 7 references

A transition-density-based operator learning method for Fokker-Planck equations with various initial conditions

T0 review · 3 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The paper claims that the Fokker-Planck solution operator over many initial conditions can be learned once by training a conditional normalizing flow on transition densities, then evaluated for any new initial distribution via the Chapman-K

desk verdict A solid operator-learning framework for FPEs with varying initial conditions; the short-time base-distribution theory needs to be reconciled with the actual experiments. read the letter →

arxiv 2606.09434 v2 pith:LACWBLMZ submitted 2026-06-08 cs.LG

Li Zeng , Xiaoliang Wan , Yaobin Wang , Fabio Nobile , Tao Zhou This is my paper

classification cs.LG MSC 68T0762G0765M99

keywords Fokker-PlanckequationtransitionprobabilitydensityconditionalnormalizingflowoperatorlearningChapman-KolmogorovlinearizedSDEtime-weightedlossimportancesampling

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

The paper sets out to show that the Fokker-Planck equation's solution operator over a family of initial conditions can be captured in a single offline training. The route is to learn the transition probability density p(x,t|x0)—the density generated from a single sharp starting point x0—and then obtain the solution for any initial distribution p0 by the Chapman-Kolmogorov integral p(x,t)=∫p(x,t|x0)p0(x0)dx0. The difficulty is that the transition density starts as a Dirac delta, which no learned map can represent directly; the paper bypasses this by using the explicit Gaussian transition density of the linearized SDE as the base of a conditional normalizing flow, so the flow only has to learn a near-identity deformation at small times. A time-weighted loss and an importance-sampling estimator keep training stable and evaluation accurate across both small and large times. If the claim is right, ensemble forecasting, data assimilation, and other settings where the dynamics are fixed but the initial distribution varies would no longer require repeated numerical solves.

What carries the argument

The central object is the transition PDF p(x,t|x0), the FPE solution with Dirac initial data. The carrying identity is Chapman-Kolmogorov: p(x,t)=∫p(x,t|x0)p0(x0)dx0, which turns operator learning over initial conditions into one conditional density estimate. The mechanism that makes this tractable is the linearized-SDE base distribution: a first-order Taylor expansion of the drift and a zeroth-order expansion of the diffusion give an explicit Gaussian transition density for each x0, and this Gaussian is the base of a conditional normalizing flow. Because the flow is constrained to the identity at t=0 and the base matches the true density to O(t) in total variation at small t, the flow learn

What would settle it

Compute the true transition density at an intermediate time, say t=0.5, for a two-dimensional SDE with a strongly nonlinear drift and multiplicative noise, using a fine Euler-Maruyama histogram; then compare it with the trained conditional normalizing flow's density at the same x0 and t. If the flow yields a near-Gaussian shape or a low residual that still disagrees with the histogram (for example, missing two well-separated modes), the central premise fails in exactly the regime the paper extends to.

Watch

Extended reading notes

Core claim

The central claim is that the FPE solution operator for a whole class of initial conditions reduces to one object: the transition PDF p(x,t|x0), and that this object can be learned by a conditional normalizing flow whose base distribution is the transition PDF of the SDE linearized around x0. The paper proves that as t→0 this base is close to the true transition density—total variation O(t) and KL divergence O(t^2) under constant diffusion and bounded first and second drift derivatives—so the flow's transformation is near the identity at small times and gradually becomes more expressive. With the identity enforced at t=0, the Dirac initial condition is satisfied automatically. The training u

Load-bearing premise

The load-bearing premise is that the Gaussian transition density of the locally linearized SDE stays close enough to the true transition density for every starting point x0 and every time up to tf that the conditional flow needs only a mild deformation—a premise proven in detail only for small times under constant diffusion and bounded drift derivatives, yet relied on at larger times and for state-dependent diffusion.

Editorial extensions

If this is right

One offline training on the transition density produces the whole family of solutions p(x,t)=∫p(x,t|x0)p0(x0)dx0 for all initial distributions in the class, with no retraining.
The normalizing-flow formulation enforces positivity and unit mass by construction, avoiding the special treatment of PDF constraints needed by grid-based solvers.
The linearized base plus time-weighted loss makes the approximation accurate at both small and large times; in the closed-form benchmark the error does not grow with time.
The mixture importance-sampling estimator keeps variance low near t=0 and at larger times, and handles discontinuous initial distributions that finite-difference methods struggle with.
A four-dimensional test shows practicable accuracy, suggesting the approach extends to moderately high-dimensional Fokker-Planck problems.

Reading between the lines

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

A natural stress test is to push the method into a regime where the true transition density is strongly non-Gaussian at moderate t (for example, a double-well drift with multiplicative noise) and check whether the learned map remains accurate; if not, the linearized base may need to be replaced by a family of local bases or a mixture.
If the Chapman-Kolmogorov reduction is sound, the same trick should apply to any Markovian linear or semilinear PDE solvable through a Green's function, giving Green-function-based operator learners without retraining.
The time-weighted loss is a generic cure for singular initial data and could be ported to score-based or particle solvers of Fokker-Planck equations, not just normalizing flows.
The mixture importance-sampling estimator suggests that observables of p(x,t) could be estimated directly rather than the full density, which would lower cost in high dimensions where pointwise density evaluation is expensive.

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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

Dear [name],

Short version: this is a real contribution and deserves a serious referee. The idea is to learn the transition PDF of the SDE once, with a conditional normalizing flow conditioned on x0 and t, then get any FPE solution by Chapman-Kolmogorov. The linearized-SDE base distribution is a smart choice, and the time-weighted loss and importance-sampling proposal are built on actual analysis of the residual and variance, not just tuned ad hoc. The Beneš SDE, nonlinear drift, and state-dependent diffusion experiments show the method works well for the tested 2D and 4D problems, including discontinuous initial conditions.

What's genuinely new: prior flow-based FPE solvers target a fixed initial distribution; this one learns a transition density over x0, so one offline training handles a family of initial conditions. The parallel Neural Galerkin flow [31] is a different training framework. The proposition showing H(σ_t|ν_t) = O(t^2) for the linearized base is proved carefully under constant diffusion and uniformly bounded drift derivative.

Soft spots. The theory and the experiments don't line up. Prop. 3.1 assumes constant diffusion and a global bound on ∇f, but Section 6.2 uses f with a cubic term (so ∇f is unbounded), and Section 6.3 has state-dependent diffusion. So the 'near-identity' anchoring that motivates the base distribution is not actually guaranteed in exactly the regimes where the method is demonstrated. The numerical results are encouraging, but they don't show that the base is doing the claimed anchoring work — the flow's expressiveness could be compensating. That's not fatal, but it should be stated honestly and ideally supplemented with a local-in-time or bounded-domain version of the bound, or at least a diagnostic of the base's closeness. Second, the paper ships no code and no error tables — the figures show trends but not exact numbers. Third, several hyperparameters (α(t), γ_i, the 4D scaling factor, the time-weight exponents) are presented without ablation; a referee should ask for sensitivity checks. None of these sink the central operator-learning claim, which holds up empirically for the tested cases, but they are real gaps.

Who should read it: people working on normalizing flows, PINNs, or FPE solvers, and anyone interested in amortizing cost over initial conditions. I'd bring it to a reading group and would cite it if I work in this area.

Recommendation: send to peer review. Ask for code, quantitative tables, and a more careful treatment of the assumptions behind the base distribution in the state-dependent and unbounded-drift cases. It's a worthwhile paper that needs revision, not a desk reject.

Referee Report

3 major / 6 minor

Summary. The paper proposes a conditional normalizing-flow method for approximating the transition probability density p(x,t|x0) of the SDE (2.1), and then uses the Chapman–Kolmogorov identity (2.7) to obtain FPE solutions for arbitrary initial distributions. The base distribution of the flow is chosen as the transition density of a linearized SDE (3.3), obtained by first-order Taylor expansion of the drift and zero-order expansion of the diffusion at x0. This makes the map reduce to the identity at t=0, avoids the Dirac singularity, and is motivated by Proposition 3.1, which states that for constant non-degenerate diffusion and globally bounded drift Jacobian, both the KL divergence and squared TV distance between the true and linearized transition measures are O(t^2), or O(t^3) with bounded Hessian. The training uses a time-weighted PINN residual with weights t^{d/2+2} and t^{d+2}, derived in Proposition 4.1 from the short-time scaling of the residual of a Gaussian perturbation. After training, solutions for new initial conditions are evaluated by an importance-sampling estimator (5.2) with a mixture proposal. Numerical experiments cover the Benez SDE in 2D and 4D, a nonlinear-drift constant-diffusion example, and a state-dependent-diffusion example.

Significance. If the method works as claimed, it is a useful contribution: one offline training of a conditional normalizing flow gives an approximate solution operator for the FPE over a family of initial conditions, avoiding retraining for each new initial density. The use of the linearized-SDE Gaussian as a time- and x0-dependent base is a sensible and principled device for handling the Dirac initial condition, and the explicit short-time estimates in Proposition 3.1 and the residual scaling in Proposition 4.1 are valuable. The paper also includes a concrete importance-sampling construction that mitigates the small-time variance of the naive Monte Carlo estimator. The numerical results, especially the 2D Benez and nonlinear-drift cases, are encouraging and illustrate the claimed operator behavior. However, the theoretical statements that anchor the key design choices are proved under hypotheses that are strictly narrower than the experiments: the main theorem assumes constant diffusion and globally bounded drift derivatives, while the experiments include unbounded drift derivatives and state-dependent diffusion. The paper would be significantly strengthened by either extending the theory to t

major comments (3)

[§3.1, Prop. 3.1 and Eq. (3.3)] The theoretical anchor for the base distribution assumes constant non-degenerate diffusion and globally bounded Jacobian ∥∇f(x)∥≤C_f. Section 6.2 uses drift f=(2X2, 2X1−0.8X2−0.2X1^3), whose Jacobian contains −0.6X1^2 and is unbounded on R^2; Section 6.3 additionally uses state-dependent diffusion g(x)=(0.5+0.3x1, 0.4+0.1 sin x2) and a drift with an unbounded X1^3 term. For these examples, the claimed O(t^2) closeness between the linearized base and the true transition density is not established. This is load-bearing because the near-identity property of the flow at small times is the stated justification for the architecture and for avoiding the Dirac singularity. Numerical accuracy could be due to the expressiveness of the normalizing flow rather than to the base distribution doing the anchoring work. Please either extend Proposition 3.1 to cover state-dependent diffusion/unbounded dri
[§4, Prop. 4.1 and Eq. (4.6)] The time-weight exponents t^{d/2+2} and t^{d+2} are derived for a Gaussian base with constant diffusion and bounded drift derivative. The same exponents are then used in all experiments, including the constant-diffusion nonlinear-drift case with unbounded Jacobian and the state-dependent-diffusion case. No analogous scaling analysis is given for these cases, so it is not clear that the weighting is asymptotically balanced for the actual residual. If the true residual scaling differs, the weights could over- or under-emphasize small times and affect training stability. At minimum, the paper should state that these weights are a heuristic outside the hypotheses, and ideally test sensitivity of the results to the exponents or measure the residual scaling in the state-dependent-diffusion example.
[§2, Eq. (2.7) and §6] The operator-learning claim is stated for 'a whole range of initial conditions' with G left vague. The numerical validation uses only Beta distributions supported on [−1,1]^d, which is a reasonable test but much narrower than the title's 'various initial conditions'. The paper should either define a precise class G and state which elements are covered by the trained model, or explicitly describe the tested class of initial distributions and avoid over-general phrasing. This is not a fatal flaw, but it affects the precision of the central claim.

minor comments (6)

[Algorithm 4.1] The line describing S2, 'N2 random samples drawn from S', is ambiguous: it should specify whether these are drawn from the previous training set with replacement and whether they retain their indicator η. Clarifying this would make the adaptive strategy reproducible.
[Fig. 1a] The blue and orange curves are said to overlap, but the figure legend/line styles are not described in the text. Please add a legend or explicitly distinguish the base-distribution curve from the vanilla-PINN curve.
[§6.1.2] The residual scale factor 1000 in the 4D experiment is introduced without analysis. Since the residual is linear in p, scaling p by a constant should not change the minimizer if applied consistently, but this point should be stated; otherwise readers may suspect the factor is a tuned hyperparameter affecting the PDE residual balance.
[§6.3] The state-dependent-diffusion experiment uses only 4 adaptive iterations with 1000 epochs each, whereas other examples use 10 or 20 adaptive iterations. The convergence behavior is not discussed; a brief comment on stopping criteria or loss decay would be useful.
[Notation, §3.2] The notation Z_t|Z0=Z0(x0) and the construction of the stacked bijections is heavy and at points confusing; in particular the distinction between the base random variable Z_t and its initial value should be made cleaner. This is a presentation issue only.
[References] The paper cites a parallel work [31] as arXiv:2603.18907. Given the date, this may be appropriate; however, the reader would benefit from a sentence explicitly stating what is shared and what is new relative to [31] beyond the Neural Galerkin framework.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the transition PDF is learned from the FPE residual and the Chapman-Kolmogorov integral is an exact identity; validation uses independent references.

full rationale

The central claim (Eq. 2.7) is the exact Chapman-Kolmogorov identity p(x,t)=∫p(x,t|x0)p0(x0)dx0; it is not an approximation fitted to data. The object learned offline is the transition PDF p(x,t|x0), trained by minimizing the PINN residual r=∂pθ/∂t−L*_{f,g}pθ in (4.2)/(4.6), rather than by regressing on target FPE solutions for particular initial conditions. The base distribution is defined by the linearized SDE (3.3)–(3.5) and its closeness to the true transition PDF is addressed by an original proof (Prop. 3.1, Appendix A) using external results [4,16], not by assuming the method's output. The time-weighted loss exponents follow from Prop. 4.1's analysis of Gaussian-residual scaling, and the importance proposal q is constructed from the same linearized/reversed dynamics (5.3)–(5.4), not fitted to benchmark outputs. Numerical validation uses analytical solutions (Sec. 6.1) and an independent ADI scheme (Secs. 6.2–6.3). Self-citations ([13,47] for adaptive sampling, [37,38] for KRnet, [18] for Fourier features) support implementation choices but are not load-bearing for the operator-learning claim. The skeptical concern that Prop. 3.1's assumptions (constant diffusion, globally bounded ∇f) are violated in Secs. 6.2–6.3 is a legitimate correctness/robustness gap, but it is not circularity: the method's derivation does not presuppose the numerical results it reports.

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

The central method rests on the linearized-SDE Gaussian base being a good short-time approximation, on the flow being able to learn the residual deformation, and on several hand-chosen coefficients (α, γ, scale factor, time-weight exponents). These are not fitted to target data but are also not derived for the full generality of the experiments.

free parameters (5)

proposal mixing decay a in α(t)=exp(−a t) = a=6
Eq. (5.4) uses α(t)=exp(−6t) in all experiments; the value is chosen by hand and affects the importance-sampling variance and accuracy.
4D residual scale factor = 1000
Introduced in §6.1.2 to avoid numerical underflow; it changes the loss landscape for the four-dimensional experiment and is not derived from the theory.
adaptive sampling mixture weights γ1, γ2, γ3 = γ=(0.2,0.6,0.2) or (0.2,0.4,0.4)
User-set hyperparameters in Algorithm 4.1 and Eq. (4.3); they control the relative proportions of uniform, previous-stage, and current-flow collocation points.
time-weight exponents d/2+2 and d+2 = d/2+2 and d+2
Motivated by Prop 4.1 for a Gaussian perturbation under constant diffusion, then applied to the full model and to state-dependent diffusion without re-derivation.
network/training hyperparameters = 8 coupling layers; 32 Fourier features; 2×32 hidden units; lr 0.001; batch sizes 1e4–5e4; epochs 3000–4000
These are hand-chosen and no ablation study is reported, so their influence on the claimed accuracy is not quantified.

assumptions (5)

standard math Itô calculus and the Chapman–Kolmogorov representation p(x,t)=∫ p(x,t|x0)p0(x0)dx0 hold for the SDE (2.1).
Used throughout Section 2, especially Eq. (2.7); this is a standard property of Markov diffusion processes.
domain assumption The drift f is sufficiently smooth and the diffusion is non-degenerate so that FPE solutions exist and the first-order Taylor linearization is valid.
Assumed at the start of Section 3.1; Prop 3.1 additionally requires constant diffusion and bounded ∇f/∇²f.
ad hoc to paper The Gaussian transition PDF of the linearized SDE is close enough to the true transition PDF that the normalizing flow only needs to learn a near-identity deformation.
This is the load-bearing modeling choice. It is proved only for small times and constant diffusion in Prop 3.1, but applied for all t up to 1.5 and to state-dependent diffusion in §6.3.
ad hoc to paper Training on finite bounded boxes with sampled collocation points enforces the FPE on R^d, including decay as |x|→∞, well enough.
The PDE is posed on R^d, but training and validation are restricted to bounded domains such as [−5,5]^2; no boundary or tail error bound is given.
ad hoc to paper The time-weighted loss exponents t^{d/2+2} and t^{d+2}, derived for Gaussian perturbations under constant diffusion, remain appropriate for the actual model and for state-dependent diffusion.
Prop 4.1 is proven for a Gaussian perturbation, and the same weights are applied to the non-Gaussian learned density and to the state-dependent diffusion experiment.

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Cite this review

Pith. "Pith review of A transition-density-based operator learning method for Fokker-Planck equations with various initial conditions." pith.science (2026). https://pith.science/paper/LACWBLMZ

@misc{pith2026260609434,
  author       = {Pith},
  title        = {Pith review of: A transition-density-based operator learning method for Fokker-Planck equations with various initial conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LACWBLMZ}},
  note         = {Machine review of arXiv:2606.09434}
}

read the original abstract

Solving Fokker-Planck equations (FPEs) for multiple initial conditions typically requires repeated computations, leading to substantial computational costs. In this work, we propose a transition-density-based operator learning method to efficiently approximate the solution operator of FPEs with various initial conditions. The core idea is to learn the transition probability density function (PDF) of the underlying stochastic differential equation (SDE), from which the solution associated with a new initial distribution can be obtained through the Chapman-Kolmogorov equation without retraining the model. A major challenge in learning the transition PDF lies in the singular behavior induced by the Dirac initial condition. To address it, we introduce a conditional normalizing flow whose base distribution is given by the explicit transition PDF of a linearized SDE. This base distribution captures the short-time behavior of the target transition PDF and allows the normalizing flow to learn a near-identity transformation at small times. We further incorporate a time-weighted loss function to stabilize training near the initial time and develop an importance-sampling strategy for evaluating solutions associated with general initial conditions. A variety of numerical experiments are presented to illustrate the effectiveness and robustness of the proposed method.

Figures

Figures reproduced from arXiv: 2606.09434 by the authors.

**Figure 1.** Left: Rel(pC-KRnet(·, t|·)) for different loss functions. Right: the decay of Rel(pC-KRnet(·, t|·)) against the adaptive iterations for the multi-weighted case. We present the Rel(pC-KRnet(·, t|·)) (6.1) in Figure 1a, where the validation dataset {x0, x} consists of 105 samples drawn uniformly from [−1, 1]2 × [−5, 5]2 . The blue curve, which overlaps with the orange one, represents the relative error of the base dis… view at source ↗

**Figure 2.** Solutions (top row) and the absolute errors (bottom row) at [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Same as Figure 2 for [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: Same as Figure 2 for [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: Uniform case: Evolution of Rel(ˆpθ(·, t)) over time. The subplots are arranged from left to right with M = 104 , 105 , 106 . (a) q = p0 (b) q = q1 (c) q = e −6t q1 + (1 − e −6t )p0 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: Uniform case: The decay of Rel(ˆpθ(·, t)) with respect to the number of Monte Carlo samples at different times. Left: Draw samples from p0; Middle: Draw samples from q1; Right: Draw samples from the mixture of p0 and q1. We next assess how well the solution of the FPE …

**Figure 7.** Figure 7: The variance of ˆpθ(x, t) over time. Left: x = (0, 0). Right: Average variance of the set {x i} consisting 104 grid points in [−5, 5]2 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: Time evolution of the Rel(ˆpθ(·, t)) for various initial distributions. Left: M = 104 ; Right: M = 106 . Moreover, we report in [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 9.** Figure 9: b presents Rel(ˆpθ(·, t)) for different Beta distributions as initial conditions. For each initial condition, the numerical solution is obtained using the equation (5.2), where q = q2 is defined in the equation (5.4) with α(t) = e −6t and M = 104 . A total of 105 test …

**Figure 10.** Figure 10: Errors for various initial distributions in the nonlinear-drift, constant [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

**Figure 11.** Figure 11: Numerical solutions and the absolute errors at [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

**Figure 12.** Figure 12: Errors for various initial distributions in the nonlinear-drift, state-dependent [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]

**Figure 13.** Figure 13: Numerical solutions and the absolute errors at [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]

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Works this paper leans on

7 extracted references · 2 linked inside Pith

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

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Aldirany, R

[1]Z. Aldirany, R. Cottereau, M. Laforest, and S. Prudhomme, Operator approximation of the wave equation based on deep learning of Green’s function, Computers & Mathematics with Applications, 159 (2024), pp. 21–30. [2]R. Baptista, B. Hosseini, N. B. Kovachki, and Y. M. Marzouk, Conditional sampling with monotone GANs: From generative models to likelihood-...

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[19]Z. Hu, Z. Zhang, G. E. Karniadakis, and K. Kawaguchi, Score-based physics-informed neural networks for high-dimensional Fokker–Planck equations, SIAM Journal on Scientific Computing, 47 (2025), pp. C680–C705. [20]T. Hytonen, J. van Neerven, M. Veraar, and L. Weis, Analysis in Banach spaces, volume i: Martingales and littlewood-paley theory, Ergebnisse...

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[48]L. Zeng, X. W an, and T. Zhou, Bounded KRnet and its applications to density estimation and approximation, SIAM Journal on Scientific Computing, 47 (2025), pp. C1294–C1318. [49]J. Zhai, M. Dobson, and Y. Li, A deep learning method for solving Fokker-Planck equations, in Mathematical and Scientific Machine Learning, PMLR, 2022, pp. 568–597. [50]M. Zhou...

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Sirignano and K

[36]J. Sirignano and K. Spiliopoulos, DGM: A deep learning algorithm for solving partial differential equations, Journal of Computational Physics, 375 (2018), pp. 1339–1364. [37]K. Tang, X. W an, and Q. Liao, Deep density estimation via invertible block-triangular mapping, Theoretical and Applied Mechanics Letters, 10 (2020), pp. 143–148. [38]K. Tang, X. ...

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[11]W. E and Y. B, The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems, Communications in Mathematics and Statistics, 6 (2018), pp. 1–12. [12]N. El Bekri, L. Drumetz, and F. Vermet, Flowkac: An efficient neural fokker-planck solver using temporal normalizing flows and the feynman-kac formula, Transactions on Ma...

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