Sampling in High-Dimensions using Stochastic Interpolants and Forward-Backward Stochastic Differential Equations

Anand Jerry George; Nicolas Macris

REVIEW 3 major objections 3 minor 30 references

Sampling in High-Dimensions using Stochastic Interpolants and Forward-Backward Stochastic Differential Equations

T0 review · 3 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper designs diffusion processes that carry Gaussian samples to a target distribution in finite time by solving a Hamilton–Jacobi–Bellman PDE through forward-backward stochastic differential equations.

desk verdict The theory is mostly sound and the FBSDE training idea is new, but the Eq. (19) loss is not a consistent finite-time surrogate for the HJB and the paper's own high-dimensional benchmarks fail exactly there. read the letter →

arxiv 2502.00355 v1 pith:7DQL37R4 submitted 2025-02-01 cs.LG stat.ML

Anand Jerry George , Nicolas Macris This is my paper · ORCID

classification cs.LGstat.ML MSC 60H1065C3068T0793E20

keywords stochasticinterpolantsdiffusion-basedsamplingforward-backwarddifferentialequationsHamilton-Jacobi-Bellmanequationunnormalizeddensityfinite-timetransportscore-basedgenerativemodelinglog-normalizationconstantestimation

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's central claim is that a diffusion process can carry Gaussian samples to a target distribution in finite time, given only the target's unnormalized density. The construction uses stochastic interpolants—a time-indexed family $x_t=g(t)x_*+r(t)z$ bridging the Gaussian prior to the target—and reduces sampling to solving a Hamilton–Jacobi–Bellman equation for a function whose gradient encodes the needed conditional expectation. The HJB equation is solved through its forward-backward stochastic differential equation formulation with a neural network, a route that avoids the expensive path-gradient computations of earlier diffusion samplers. Experiments on Gaussian mixtures, Neal's funnel, double-well potentials, and a 100-dimensional spin-glass model with a known free-energy benchmark indicate that the trained sampler draws accurate samples and estimates $\log Z$ without importance-weight corrections.

What carries the argument

The load-bearing object is the stochastic interpolant $x_t=g(t)x_*+r(t)z$, a time-indexed random variable whose density bridges the Gaussian prior to the target. The core identity is the representation of the conditional expectation as a gradient: $E[x_*\mid x_t=x]=\frac{r^2(t)}{\beta(t)g(t)}\nabla u(t,x/\beta(t))$, where $u(t,x)=\log[\rho(t,\beta(t)x)/\psi(t,\beta(t)x)]$; this converts the sampling problem into the problem of finding $\nabla u$. The function $u$ is characterized as the solution of the HJB equation (14), and the machinery that solves this PDE is its FBSDE formulation (16), whose processes satisfy $Y_t=u(t,X_t)$ and $Z_t=\sigma(t)\nabla u(t,X_t)$. Training uses the local FBSDE loss (19), which enforces the backward equation at randomly sampled times along a detached ODE trajectory (17); detachment keeps the training domain from depending on the network's gradient path and avoids Neural SDE back-propagation.

What would settle it

Train the sampler on a target with known normalization and exact score (for example, a non-isotropic Gaussian or the $d=100$ spin-glass model with its exact free energy), and monitor the terminal residual $\|\nabla u_\theta(T,X_T)-\nabla\varphi(X_T)\|$ and the Wasserstein distance between $S_T$ and $\pi$ as $\lambda$ and the training steps grow; if neither approaches zero, or if the $\log Z$ estimate shows a bias that persists when $\lambda$ is increased, the finite-time exactness claim is falsified.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that exact finite-time transport is a consequence of the stochastic-interpolant construction. Given a target density $\pi$ on $\mathbb{R}^d$, choose interpolant functions $g,r$ with $g/r$ non-decreasing and define $x_t=g(t)x_*+r(t)z$ with $x_*\sim\nu$ and $z\sim N(0,I_d)$; the terminal condition $\rho(T,\cdot)=\pi$ fixes $\nu$ implicitly. Lemma 3 shows that the tilting function $u(t,x)=\log[\rho(t,\beta(t)x)/\psi(t,\beta(t)x)]$ satisfies the HJB equation (14) with terminal condition (15), and Lemma 4 shows that the FBSDE (16) has solutions $Y_t=u(t,X_t)$ and $Z_t=\sigma(t)\nabla u(t,X_t)$. Since $E[x_*\mid x_t=x]=r^2(t)\nabla u(t,x/\beta(t))/(\beta(t)g(t))$, learning $\nabla u$ on the right domain gives both drift $b$ and score $s$ in (12), and then the SDE (11) has exactly the interpolant density $\rho(t,\cdot)$ at every time, so $S_T\sim\pi$. The empirical version replaces the PDE solution with a neural-network approximation trained through the local FBSDE loss (19), and the paper reports that this yields accurate samples and $\log Z$ estimates on distributions that conventional samplers struggle with.

Load-bearing premise

The central guarantee rests on the assumption that minimizing the local FBSDE loss actually converges to the true solution $\sigma\nabla u$ of the HJB equation, even though the terminal condition is imposed only through a finite penalty and the region where the equation is enforced is defined by the very network being trained.

Editorial extensions

If this is right

If the central claim is correct, a trained network turns the SDE (11) into a finite-time sampler: Gaussian draws at $t=0$ become target draws at $t=T$ with no annealing, rejection, or burning-in.
Because the paper estimates moments and $\log Z$ directly from the samples, the claim implies that the generated samples are good enough for downstream estimators without importance reweighting; the Itô-formula identity (35) gives normalization constants from the same trained $\nabla u$.
The detached training ODE (17) means path gradients are not back-propagated, and the paper reports per-step training times between 98 and 105 ms for targets of dimension 2 through 100.
The half-interpolant sampler generalizes the Föllmer process and reduces to the Path Integral Sampler when $g(t)=f(t)$, $r(t)=\sqrt{f(t)}$, while the full-interpolant two-step version extends finite-time transport to interpolants with $r(T)=0$.

Reading between the lines

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

A sharper test of the exactness claim would be to measure, across many random disorder matrices, the variance of the sampler's free-energy estimate for the spin-glass model near the predicted $\beta=1$ phase transition; a bias that does not vanish with $\lambda$ would locate the effect of the soft terminal constraint.
The local FBSDE loss may transfer to other HJB problems where the evaluation paths differ from the controlled paths, since the loss enforces the PDE residual pointwise in time rather than over the trajectory of a single SDE.
Replacing the soft terminal penalty with a hard terminal parameterization, exploiting the fact that the network is initialized so that $u_\theta(T,\cdot)=\varphi$, would directly probe the hyperparameter sensitivity reported in Table 6 and could yield a sampler with fewer tunable constants.
If exact finite-time transport holds, the sampler should also work as a proposal for annealed importance sampling when reweighting is desired, since the paper itself notes that importance weights are currently unavailable.

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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 paper has a correct HJB/FBSDE skeleton and a genuinely new training idea, but the training loss that is supposed to solve the HJB is not a consistent finite-time surrogate, and the paper's own high-dimensional benchmarks fail in a way that matches that defect. I would not take the central empirical claim as established yet.

What is actually new: the stochastic-interpolants framing with a Gaussian prior, solved through a local FBSDE loss instead of control-cost minimization; the detached probability-flow ODE (17) used to define the training distribution; and the full-interpolant two-stage PDE split. The derivations in Lemmas 1-4, 8-10 check out: the sign in the FBSDE, the Tweedie relation, and the Ito-based log Z estimator are all consistent. The paper also honestly states that PIS/Follmer is a special case of the half-interpolant method for g=f, r=sqrt(f), which is the right thing to do. The spin-glass and low-dimensional examples are plausible positives.

Where it gets soft. The load-bearing step is Eq. (19). The terminal penalty is O(1) in delta, but the FBSDE residual is O(delta) pathwise: the sqrt(delta) Brownian terms cancel for any u_theta because the same Z is used in Y-hat and Y. Its square is O(delta^2), so with lambda=2000/delta the FBSDE term contributes about 10^-2 at delta=5e-6, and the objective is basically a terminal-condition penalty evaluated on a detached path. That is not a consistent surrogate for (14), and no theorem is offered that minimizers of (19) approximate grad u on the support of the sampling SDE. The training domain is itself bootstrapped from the current network and detached from gradients, so the loss can be small without constraining the sampler where it needs it. The empirical failures line up: on the 10-mode MoG at d=10, FIS gives log Z=-2.43 vs 0 and 2-Wasserstein 243 vs 6.2 for GBS; at d=200 log Z is -183.56. Table 6 also shows strong sensitivity to delta and lambda. These are addressable in principle, for example by rescaling lambda or adding a proper FBSDE consistency theorem, but as written the headline claim of finite-time Gaussian-to-target transport is not supported.

Verdict: not a desk-reject-quality paper. The framework is worth a serious referee, and the theory portion is solid enough to publish after fixing or re-scoping the empirical claims. I would send it to review with a specific request to look at the scaling of Eq. (19), and I would not cite the sampler as a working method until code or corrected benchmarks appear.

Referee Report

3 major / 3 minor

Summary. The paper proposes a class of finite-time diffusion samplers for unnormalized densities, built on stochastic interpolants and forward-backward stochastic differential equations (FBSDEs). The authors define half and full interpolants, derive Hamilton-Jacobi-Bellman (HJB) equations for the associated log-density functions, and propose neural-network training based on a local FBSDE residual together with a terminal gradient penalty. After training, sampling is performed by simulating the SDE (or ODE) whose drift and score are obtained from the learned gradient. The theoretical portion (Lemmas 1-4 and the corresponding appendix proofs) is largely coherent: the HJB/FBSDE relation and the Gaussian-to-target transport identity are derived consistently. The empirical portion compares the resulting Full Interpolant Sampler against Langevin Monte Carlo and the Generalized Bridge Sampler on Gaussian mixtures, funnels, double-well potentials, and a spin-glass model, with log-normalizing-constant and Wasserstein metrics. The central empirical claim, however, is not established, because the training objective in Eq. (19) has a scaling defect and the training trajectories are generated by the very network being trained.

Significance. If the proposed training procedure were sound, the paper would offer a useful finite-time alternative to Neural-SDE-based samplers such as PIS, with the practical advantage that no gradients through stochastic paths are required. The mathematical derivation of the HJB/FBSDE connection and the finite-time exact-transport statement in the ideal (exactly solved PDE) case are valid and constitute a genuine theoretical contribution. I checked the core identities in Lemmas 1-4 and the Itô-based relation between the FBSDE (16) and the HJB equation (14), and they are consistent. Nevertheless, the load-bearing step is the claim that minimizers of the empirical loss (19) solve the HJB equation on the support of the sampling SDE. That step is neither proved nor convincingly supported by the numerical evidence, and the loss-scaling issue in §3.1.1 casts serious doubt on the current formulation as a consistent PDE-solving method.

major comments (3)

The FBSDE term in the loss is not a consistent surrogate for the HJB residual. In Eq. (18), the leading O(√δ) Brownian terms in Y^δ and u_θ(τ+δ, X^δ) cancel for any u_θ, because both terms use the same Z = σ∇u_θ: expanding u_θ at (τ+δ, X+(μ+σZ)δ+σ√δ w) gives the same √δ Z^T w as Y^δ. Consequently the pathwise residual is O(δ), its square is O(δ²), and with λ = 2000/δ the FBSDE contribution is O(δ) per sampled time, about 10⁻² at δ = 5×10⁻⁶, while the terminal gradient penalty in (19) is O(1) and is evaluated at a single detached point X_T. Unless the number of FBSDE checks is made to grow as 1/δ — which is not stated and would be computationally burdensome — the terminal penalty dominates and the loss does not enforce the PDE (14). No consistency or convergence theorem is provided for minimizers of (19).
The domain on which the FBSDE residual is trained is defined by the current network and is explicitly detached from the computational graph in Section F.2. The ODE (17) uses ∇u_θ to generate the trajectories X_t, so at any stage of training the loss is minimized over paths that the network itself produces. At convergence to the true ∇u these trajectories would coincide with the sampling ODE (10), but that is exactly the conclusion to be established; nothing guarantees that a small loss on these self-generated trajectories transfers to the support of the sampling SDE (11). This circularity is load-bearing, because the finite-time transport claim requires ∇u to be accurate precisely on the domain sampled by (11), not merely on trajectories of the detached ODE.
The empirical evidence does not support the abstract's claim that the method can effectively draw samples from distributions that conventional methods struggle to handle. On a 10-component Gaussian mixture at d = 10, FIS reports log Z = −2.43 ± 0.02 against the true value 0 and a 2-Wasserstein distance of 243.5 ± 5, whereas GBS gives −0.02 ± 0.03 and 6.21 ± 0.04; at d = 200 the FIS log Z estimate is −183.56 ± 90.24. Table 6 also shows that on a 2D GMM the log Z estimate varies from −0.25 to −0.74 when δ or λ is changed by a factor of 10. These results are consistent with the loss-scaling issue in Major Comment 1 and do not demonstrate that the trained sampler transports the Gaussian prior to the target in finite time.

minor comments (3)

There are several typos: “non-deceasing” in Definition 1, “score funciton” in §3.1, and “paramterized” in §3.1.1. These should be corrected.
The reference to “Bismuth” [Bismut, 1973] is misspelled; it should be Bismut.
The log Z estimator (35) is derived by Itô's formula for the exact solution u. In practice ∇u is replaced by the trained ∇u_θ, and the resulting bias is not analyzed. The paper should state explicitly that the estimator is valid only to the extent that the learned gradient approximates the true gradient on the relevant SDE paths.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the HJB/FBSDE construction is self-contained, with the target density entering only through the terminal condition and the log-normalizer estimate derived by Itô's formula.

full rationale

The derivation chain is not circular. Section 3.1 defines u from the interpolant density (Eq. 13), proves by direct differentiation (Lemma 9) that u satisfies the HJB PDE (14), and imposes the terminal condition (15) equal to the target log-density. The terminal condition is an input of the construction, not a derived prediction; if the PDE is solved exactly, the Fokker-Planck argument (Lemma 1, Eq. 9) ensures S_T has law π. The FBSDE representation (16) and the identification Y_t = u(t,X_t), Z_t = σ∇u (Lemma 10) rely on the standard Pardoux-Peng theorem, which is cited as external mathematical support. The log-normalizer estimator (35) is an Itô identity, not a fitted parameter. No load-bearing self-citation appears: the stochastic-interpolant framework is cited to Albergo et al. and the FBSDE connection to Pardoux-Peng. The self-referential training loop (Eq. 17 uses the current network to define the training distribution, and the process is detached from the computational graph as stated in Section F.2) and the soft terminal penalty with λ = 2000/δ are convergence and consistency risks, not circular reductions, because the loss is a PDE residual rather than a restatement of the target π. The paper's own limitations (Section 5: no importance-weight correction; Table 6: strong hyperparameter sensitivity; Table 5: poor MoG log Z estimates) are correctness concerns that do not make the derivation circular.

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

The central scheme has no fitted physical constants but several hand-tuned design functions and loss weights (beta, lambda, delta, g, r, epsilon). Table 6 quantifies that estimates swing by up to 60% across these settings. The mathematical framework assumes standard FBSDE/PDE regularity and an unproved consistency of the local training loss, so the honest count of what is pulled from prior theory versus what is assumed ad hoc is roughly balanced.

free parameters (5)

interpolant schedules g(t) and r(t) = e.g., sin(pi*t/2), cos(pi*t/2) for full interpolants; sqrt(t) variants for half interpolants
Hand-chosen design choices; Figure 3 and Section D state that keeping r_dot/r close to 0 for small t is better for training. Results depend on this choice.
beta(t) = r(t)/g(t)
Design parameter chosen to keep PDE coefficients well-defined; Eq. (3) and Section 3.1. It rescales the coordinate system of the HJB and affects the learned function's domain.
lambda (FBSDE penalty weight) = 2000/delta = 4e8
Hand-set; Table 6 shows lambda = 0.1*c2 and 10*c2 degrade estimates from -0.26 to -0.74 in log Z, so the reported performance relies on this value.
delta (FBSDE discretization step) = 5e-6
Hand-set; Table 6 shows log Z varies from -0.66 to -0.25 as delta changes from 0.1*c1 to 10*c1, and E||X||2 varies from 18.41 to 28.42.
epsilon(t) (sampling diffusion coefficient) = 1.0
Chosen as a constant during sampling; Table 1. The method allows arbitrary epsilon, and results depend on this choice.

assumptions (5)

standard math Existence and uniqueness of the FBSDE (16) with quadratic generator (1/2)||Z||^2 and coupled forward drift mu + sigma Z
Invoked via Pardoux-Peng and Pardoux-Tang in Section 2. Quadratic-growth BSDE theory is required but not discussed.
domain assumption The HJB PDE (14) has a smooth classical solution u for the terminal condition phi = log(pi(beta(T)x)/psi(T,beta(T)x))
Needed for Lemmas 3 and 4 and for grad phi to be well-defined. All test targets are smooth with full support on R^d, but this is never stated.
standard math Tweedie's formula E[z|x_t=x] = -r(t)s(t,x) for the Gaussian z
Used in Lemma 8 to relate b and s; standard for Gaussian conditional expectations.
ad hoc to paper The ODE (17) discretization error does not affect training accuracy
Stated in Section 3.1.1 without proof. The ODE is Euler-discretized and its trajectories determine the domain over which the FBSDE loss is enforced.
domain assumption Density pi is positive everywhere so log pi and its gradient are finite
Required for terminal condition (15); not true for compactly supported targets, which the method cannot handle.

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Cite this review

Pith. "Pith review of Sampling in High-Dimensions using Stochastic Interpolants and Forward-Backward Stochastic Differential Equations." pith.science (2026). https://pith.science/paper/7DQL37R4

@misc{pith2026250200355,
  author       = {Pith},
  title        = {Pith review of: Sampling in High-Dimensions using Stochastic Interpolants and Forward-Backward Stochastic Differential Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7DQL37R4}},
  note         = {Machine review of arXiv:2502.00355}
}

read the original abstract

We present a class of diffusion-based algorithms to draw samples from high-dimensional probability distributions given their unnormalized densities. Ideally, our methods can transport samples from a Gaussian distribution to a specified target distribution in finite time. Our approach relies on the stochastic interpolants framework to define a time-indexed collection of probability densities that bridge a Gaussian distribution to the target distribution. Subsequently, we derive a diffusion process that obeys the aforementioned probability density at each time instant. Obtaining such a diffusion process involves solving certain Hamilton-Jacobi-Bellman PDEs. We solve these PDEs using the theory of forward-backward stochastic differential equations (FBSDE) together with machine learning-based methods. Through numerical experiments, we demonstrate that our algorithm can effectively draw samples from distributions that conventional methods struggle to handle.

Figures

Figures reproduced from arXiv: 2502.00355 by the authors.

**Figure 2.** Estimates of log Z as a function of training steps along with 95% confidence intervals. Soft Spherical Spin Glass model : We consider a spin glass model whose probability density is given by π(x) ∝ exp{ √ β 2d x T Ax − β 2 4d ∥x∥ 4 − 1 2 ∥x∥ 2 }, where A ∈ R d×d with i.i.d. Gaussian entries. We use our sampler for estimating different quantities associated to the above distributions. We estimate the log normalizatio… view at source ↗

**Figure 3.** Examples of linear interpolants. Top row: half interpolants. Bottom row: full interpolants. [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: shows the training loss as a function of training steps. The plot on the left is for GMM with d = 2, and the plot on the right is for MoG (d = 100) [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

**Figure 5.** Figure 5: KDE plots of Gaussian mixture, and Rings distributions. [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

**Figure 6.** Figure 6: Estimates of E|X|1 and E∥X∥ 2 as a function of training steps [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

**Figure 7.** Figure 7: Estimate of 1 d log Z for Gaussian Spin Glass model with d = 100 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

**Figure 8.** Figure 8: KDE plots of GMM samples generated by Langevin diffusion-based algorithm and full interpolant-based [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]

**Figure 9.** Figure 9: Estimates of 1 d log Z as a function of training steps given by FIS and GBS for different targets and dimensions [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...

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S., Boffi, N

Albergo, M. S., Boffi, N. M., and Vanden-Eijnden, E. (2023). Stochastic Interpolants : A Unifying Framework for Flows and Diffusions . arXiv:2303.08797 [cond-mat]

arXiv 2023

[3] [3]

B., Dembo, A., and Guionnet, A

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