Neural Guided Diffusion Bridges
Authors: Gefan Yang
Frank van der Meulen
Stefan Sommer
Venue: ICML 2025 poster (2025)
Keywords: Diffusion bridge
variational approximation
change of measure
Primary area: probabilistic_methods->variational_inference

Abstract: We propose a novel method for simulating conditioned diffusion processes (diffusion bridges) in Euclidean spaces. By training a neural network to approximate bridge dynamics, our approach eliminates the need for computationally intensive Markov Chain Monte Carlo (MCMC) methods or reverse-process modeling. Compared to existing methods, it offers greater robustness across various diffusion specifications and conditioning scenarios. This applies in particular to rare events and multimodal distributions, which pose challenges for score-learning- and MCMC-based approaches. We propose a flexible variational family for approximating the diffusion bridge path measure which is  partially specified by a neural network. Once trained, it enables efficient independent sampling at a cost comparable to sampling the unconditioned (forward) process.

Neural Guided Diffusion Bridges
Gefan Yang Frank van der Meulen 2
Stefan Sommer
Abstract
registration problems (Arnaudon et al., 2022; Baker et al., 2024b; Yang et al., 2025). Additionally, diffusion bridges play a crucial role in Bayesian inference and parameter esti- mations based on discrete-time observations. (Delyon & Hu, 2006; van der Meulen & Schauer, 2017; 2018; Pieschner & Fuchs, 2020)
We propose a novel method for simulating condi- tioned diffusion processes (diffusion bridges) in Euclidean spaces. By training a neural network to approximate bridge dynamics, our approach eliminates the need for computationally intensive Markov Chain Monte Carlo (MCMC) methods or score modeling. Compared to existing meth- ods, offers greater robustness across various diffusion specifications and conditioning scenar- ios. This applies in particular to rare events and multimodal distributions, which pose challenges for score-learning- and MCMC-based approaches. We introduce flexible variational family, par- tially specified by neural network, for approxi- mating the diffusion bridge path measure. Once trained, it enables efficient sampling of indepen- dent bridges at a cost comparable to sampling the unconditioned (forward) process.
Simulation of diffusion bridges in either Euclidean space or manifolds is nontrivial since, in general, there is closed- form expression for transition densities, which is key to con- structing the conditioned dynamics via Doob's 's ht-transform (Rogers & Williams, 2000). This task has gained a great deal of attention in the past decades (Beskos et al., 2006; Delyon & Hu, 2006; Schauer et al., 2017: Whitaker et al., 2016; Bierkens al., 2021; Mider et al., 2021: Heng et al., 2022: Chau et al., 2024; Baker et al., 2024a). Among them, one common approach is use proposed bridge process (called guided proposal) as an approximation to the true bridge. Then either MCMC or Sequential Monte Carlo (SMC) methods are deployed to sample the true bridge via the tractable likelihood ratio between the true and pro- posed bridges. Another solution is to use the score-matching technique (Hyvarinen, 2005; Vincent, 2011) to directly ap- proximate the intractable score of the transition probability using gradient- optimization. Here, neural network is trained with samples from the unconditioned process (Heng et al., 2022) or adjoint process (Baker et al., 2024a), and plugged into numerical solving schemes, for example, Euler- Maruyama. Several recent studies deal with the extension of bridge simulation techniques beyond Euclidean spaces to manifolds (Sommer et al., 2017: Jensen & Sommer, 2023; Grong et al., 2024; Corstanje et al., 2024). All of these rely on either a type of guided proposal or score matching.
1. Introduction
Diffusion processes play a fundamental role in various fields such mathematics, physies, evolutionary biology, and, re- cently, generative models. In particular, diffusion processes conditioned to hit a specific point at fixed future time, which are often referred to as diffissio bridges, are of great interest in situations where observations constrain the dy- namics of a stochastic process. For example, generative modeling, stochastic imputation between two given images, also known as the image translation task, uses diffusion bridges to model dynamics (Zhou et al., 2024; Zheng al., 2025). In the area of stochastic shape analysis and computa- tional anatomy, random evolutions of biological shapes of organisms are modeled as non non-linear diffusion bridges, and simulating such bridges is critical to solving inference and
and score-learning-baseee bridge simulation methods have certain limitations: the guided proposal requires a careful choice of a certain "aux- iliary process". (Mider et al., 2021) provided various strate- gies, but it is fair say that guided proposals are mostly use- ful when combined with MCMC or Sequential Monte Carlo (SMC) methods. In case of a strongly nonlinear diffusion or high-dimensional diffusion, the simulation of bridges us- ing guided proposals combined with MCMC (most notably the preconditioned Crank-Nicolson (pCN) scheme (Cotter et al., 2013)) or SMC may be computationally demanding. On the other hand, score mctre-natching relies on sampling un-
Both
DDpprrmeen of Computer Science, University of Copenhagen, Universiletsparken 1, 2100 Kobenhavn, Denmark 'Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1111, 111,, 108 Amsterdam, The Netherlands. Correspondence to: Gefan Yang <gy@di.ku.kkk
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Neural Guided Diffusion Bridges
conditioned processes, and it performs poorly for bridges conditioned rare events, as the unconditioned process rarely explores those regions, resulting in inaccurate esti- mation. Additionally, the canonical score-matching loss requires the inversion of aa T with a denoting the diffu- sion coefficient of the process. This rules out hypo-elliptic diffusions, where aa is singular. It also poses computa- tional challenges for high-diman nigh-dimensional diffusions, further exacerbating the diffieulty of obtaining stable and accurate minimization of the loss.
elliptic diffusions and later extend to partially observed hypo-elliptic diffusions in (Bierkens al., 2020).
Another class of methods approximate the intractable tran- sition density using machine learning or kernel-based tech- niques. (Heng et al., 2022) applied score-matching to define a variational objective for learning the additional drift in the reversed diffusion bridge. (Baker et al., 2024a) pro- posed learning the additional drift directly in the forward bridge via sampling from an adjoint process. (Chau et al., 2024) leveraged Gaussian kernel approximations for drift estimation.
To address these issues, we introduce a new bridge sim- ulation method called neural guided diffusion bridge. It consists of the guided proposal introduced in (Schauer et al., 2017) with an additional correction drift term that is parametrized by a learnable neural network. The family of laws on path space induced by such improved proposals provides a rich variational family for approximating the law of the diffusion bridge. Once the variational approximation has been learned, independent samples can be generated at a cost similar to that of sampling the unconditioned (forward) process. The contributions of this paper are as follows:
The method we propose is a combination of existing ideas. It used the guided proposals from (Schauer et al., 2017) to construct conditioned process, but learns an additional drift term parametrized by a neural network using variational inference.
The diffusion bridge prob- lem addressed in this paper may appear similar to the diffu- sion Schrödinger bridge (DSB) problem due to their names, but they are fundamentally different. A diffusion bridge is a process conditioned to start and end at specific points, often used for simulating individual sample paths under endpoint constraints. In contrast, Schrödinger bridge connects two marginal distributions over time by finding the most likely stochastic process (relative to a reference diffusion) that matches these marginals. While both modify the original dynamics, the diffusion bridge imposes hard constraints on endpoints, whereas the Schrödinger bridge enforces them in distribution. Although DSB has gained attention for ap- plications in generative modelling (Thornton et al., 2022; De Bortoli et al., 2021; Shi et al., 2024; Tang et al., 2024), it is important to recognize the distinctions between these problems.
Diffusion Schrödinger bridge:
We propose a simple diffusion bridge simulation method inspired by the guided proposal framework, avoiding the need for score modelling or intensive MCMC or SMC updates. Once the network has been trained, obtaining independent samples from the vari- ational approximation is trivial and computationally cheap:
Unlike score-leaming-bassee based simulation methods, which rely on unconditional samples for learning, our method is grounded to learn directly from conditional samples. This results in greater training efficiency, es- pecially for learning the bridges that are conditioned on rare events.
Neural SDE: Neural SDEs generalize neural ODEs (Chen et al., 2018) by introducing stochasticity, enabling the model- ing of systems with inherently random dynamics. Research in this area can be broadly divided into two categories: (1) modeling terminal state data (Tzen & Raginsky, 2019a;b), and (2) modeling entire trajectories (Li et al., 2020; Kidger et al., 2021). Our approach falls into the latter category, leveraging trainable drift terms and end-point constraints to capture full trajectory dynamics.
We validate the method through numerical experi- ments ranging from oimeeinnall linear to high- dimensional nonlinear cases, offering qualitative and quantitative analyses. Advantages and disadvantages compared to the guided proposal (Mider et al., 2021) and two score-learning-ber based methods, (Heng et al., 2022) and (Baker et al., 2024a), are included.
3. Preliminaries: Recap on Guided Proposals
2. Related Work
3.1. Problem Statement
Diffusion bridge simulation: This topic has received con- siderable attention over the past two decades and it is hard to give a short complete overview. Early contributions are (Clark, 1990; Chib et al., 2004; Delyon & Hu, 2006; Beskos et al., 2006; Lin et al., 2010; Golightly & Wilkinson, 2010). The approach of guided proposals that we use here was intro- duced in (Schauer et al., 2017) for fully observed uniformly
Let FPP be a probability space with filtration [Fiheelo,ry W a du.ddimensional IP-Wiener process, b [0, x R2d + Rd and o [0. T] x IR - Rdxd the drift- and diffusion coefficients. A d-dimensional [F.]-adapted diffusion process X with the law of IP is defined as the
2

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Neural Guided Diffusion Bridges
strong solution the stochastic differential equation (SDE):
Here, y is considered the parameter that gets assigned the prior density p(T, y 0, and is the observation. Therefore, E is the posterior distribution of y and (3) shows that sampling of the conditioned process can be done by first sampling the endpoint ær from distribution Ç, followed by sampling a bridge connecting To and xr.
dX2, = bll, X,)dt + o(,, x, Xo - zo E R.d. (1)
The coefficients b, a assumed to be Lipschitz continuous and of linear growth to guarantee the existence of strong solution X, (Øksendal, 2014, Chapter 5.2). In addition, we impose the standing assumption that X admits smooth transition densities p with respect to the Lebesgue measure on (R4 ))Rd)), where ((RR)) is the Borel algebra of IR... That is, PXX, E A X, = x) la p(t, y s,d)) for
Throughout the paper, we will consider q(v: y) (b; Ly, E), where L E Rd'xd xd with d' I d and L of full (row) rank. Here, (1;;; Σ) denotes the density of the N (u, E)diitrribution, evaluated at x. For example, for two-dimensional diffusion y = [y ya] observing only the first component 3/1 corresponds to L [1 0] as Ly = y1. In our simulation experiments, we will assume Σ I, for small value of e, which is close to observing without error. Taking strictly positive stabilizes numerical computations.
0 SS 8 < S T, A IRd.
Notation 3.1. Let Po; ,P' and F' be measures on F), denote the laws of X on C( T). Rd) under ipo, P* and F byIL LL, I and I respectively. For notational ease, the expectations under po, IP* and F (and similarly IL L and L") are denoted by E", E= and E" respectively. The process X under L L' and LL is sometimes denoted by x²,X* and X respectively. For any measure Q on (2, we always denote its restriction to Ft by Qt-
3.2. Guided Proposal
If were known in closed form, then the conditioned process could be directly sampled from Equation (2). This is rarely the case. For this reason, let X be an auxiliary diffusion process that admits transition densities p in closed form. Let h(t,z) = fga p(T, y t, z)z((( ())( Define
The following proposition combines Proposition 4.4 and Example 4.6 in (Pieper-Sethmacher et al., 2025). It shows how the dynamics of X change under observing certain events at time T.
Proposition 3.2. Fix T. Let y E Rdd and q( (y) be a probability density function with respect to a finite measure v. Let h(t, x²- = Jg P(T, y t,, z) y)y(dy). and define the measure PP; on F3 by dP; = N(t.X.) the new measure P7. the process X solves the SDE
E, = X.) exp (%
+ A)h (s. x.) ks)ds (6)
i((0, xa)
h
=(0,a0) aI,,, Then under
where A is the infinitesimal generator of the process X, i.e. for any I in its domain Af(x) = E, b1( (1. + 1/ En, aig (t, (),x)) z). Let < T. Under weak condi- tions (see e.g. (Palmowski & Rolski, 2002, Lemma 3.1)), B[E.] 1. Using h we can define the guided proposal. Definition 3.4. (Schauer al., 2017) If we define the change of measure dl? E,P,, then, under P2, the pro- cess X solves the SDE
dx, = 6" (d +() b" x)) = ((,,)))(=sxx))) + a(s,,))((s
Xo = To.
where r(s, 2) = v log h(s, x),, a(s, = x)) (s) and W is -Wennr process.
Furthermore, for any bounded and finntralleeenncioon and 0 S t1 S < tm <T
function g
dx, (s X,) ds ol, X.) dw. 6 (ss x) b(s,)) + als z)(ss)), Xo zo:
E" X1.)l
(7)
= -/e (Xe X..)) XT= = E(dy),
where E [0. F(1, = Va log log and Wo is a P-- Wiener process. The process X under the law Pi is known as the guided proposal.
where& & is the measure defined on (IRd, B(IRd)) via
P(T.y 0, (,) (dy(
E(dy) Jns p(T,v) y 0, 20)9(1 y)v(dy))
(4) Intuitively, X is constructed to resemble the true condi- tioned process X* by replacing r by F. Crucially, as its drift and diffusion coefficients are known in closed form, the guided proposal can be sampled using efficient numerical SDE solvers such as Euler-Maruyama.
Remark 3.3. A Bayesian interpretation of Equation (4) can be obtained by considering the following hierarchical model:
The definition of the guided process can be extended to [o, T] by continuity. Whereas P. < P. for t < T the mea- sures will typically be singular in the limit t 1 T. Neverthe- less, P; < P. may still hold under this limiting operation
y ~ q(vl v): y ~~ pTP(T, y 0,
(5a)
(5b)
3

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Neural Guided Diffusion Bridges
and this is what matters for our purposes. In (Schauer al., 2017) and (Bierkens et al., 2020), precise conditions are given under which P < Pq. In the case of condition- ing on the event {LXr = 2} -so there is no noise on the observation- this is subtle. We postpone a short discussion on this to Section 3.4 to argue that all numerical examples considered in Section 5 will not break down in case the noise level on the observation, e, tends zero. We then get the following theorem from (Bierkens et al., 2020, Theorem 2.6) that states the change of laws from I to L*
2.4)):
dt(t) L(t) dt, L(T) = L,,
(15a)
dM = (1) (T(T) - = (15b)
du(t) = ((t))((t) dt, u(T) =0
(15c)
Remark 3.6. A simple choice of X is a scaled Brownian motion, i.e. X, = oW, If āā T is invertible, then
Theorem 3.5. If certain assumptions (Bierkens al., 2020, Assumptions 2.4, 2.5) hold, then
Vz log (T,, u 3, (āā
u T T-8
(16)
Therefore, the guided proposal has drift b(s, x) + ass, z) (5, \- - s). If oss x) this reduces to the guiding term proposed in (Delyon & Hu, 2006).
dIL h(0, 20) (X), dLL (X) 100
(8)
where
(T(X)==xx) = exp (1-00 A)h
(de + (s, X.)ds)
3.4. Choice of Linear Process
h
The linear process is defined by the triplet of functions
(3, B, 8) In choosing this triplet, two considerations are of importance:
3.3. Guided Proposal Induced by Linear Process
The choice of the auxiliary process X, which determines and hence the guided process, offers some flexibility, as long as the conclusion of Theorem 3.5 applics. We now specialize to the case where the process X solves a lin- ear SDE, as in this case h can be obtained by solving fnite-dimensoonal system of ordinary differential equations (ODEs). Specifically, we assume X solves
1. In case of conditioning on the event (LX,- u) -so no extrinsic noise on the observation- the triplet needs to satisfy certain "matching conditions" (see (Bierkens al., 2020, Assumption 2.4)) ensure 1+ < F For uniformly elliptic diffusions, this only affects ö. In case L = La. so the conditioning is on the full state, ō should be chosen such that ā(T) = aT,, (T,T))) Hence, for this setting, we can always ensure absolute conti- nuity. For the partially observed case, it is necessary to assume that a of the form a(t, x) = s(1, Lx) for some matrix values map s. In that case, suffices to choose â such that Lā case the diffusivity does not depend on the state, natural choice is to take ā = to guarantee absolute continuity.
dX, = b(t, X.) dt + a(t dW. b(t, = (t)) + B(t),, X, = zo.
(10)
Let A denote the infinitesimal generator of the process X.
Since solves (d: + A)h = O, we can replace (0. + A)h
Ls(T, v) In
by (A - Å)h. This gives
Vx(X)= = exp (f(oss.))ds)) SS., (11)
For hypo-elliptic diffusions, the restrictions are bit more delicate. On top of conditions on o, it is also re- quired to match certain properties in the drift by choice of B. With the exception of the FitzHugh-Nagumo (FHN) model studied in Section 5.3, in all examples that we consider in Section 5 we have ensured these properties are satisfied. The numerical simulation re- sults for FHN model presented in (Bierkens et al., 2020) strongly suggest that the conditions posed in that paper are actually more stringent than needed for absolute continuity. For this reason, we chose the aux- iliary process just as in (Bierkens et al., 2020).
G(s, = (b(s.x) x),
1 tr (la(s,z) a(s)] [1(s) (FF )ss (12)
Here, (s) is the negative Hessian of log h( r). which turns out to be independent of x, ā(s) = (ōā (s). Under the choice of q(v) y) = vbu; Ty, Σ) and X, H and are given by
(t) = LT (()M((())))))) F(t,z) = LT (t)M(())())) u(t) L(t)x).
(13)
(14)
2. Clearly, the closer b to and to a, the more the guided proposal resembles the true conditioned pro- cess. This can for instance be seen from log V =
where M(t = (t) and L. M and u satisfy the sys- tem of backward ODEs (See (Mider al., 2021, Theorem

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Neural Guided Diffusion Bridges
I" LA-A)h (s, X2) gds,, which vanishes if the coeffi- cients are equal. As proposed in (Mider et al., 2021), a practical approach is to compute the first-order Tay- Ior expansion at the point one conditions on, i.c. Jo(t, a)(x v), where Js
Then by Girsanov's theorem, the process W = W Jo do(s, X,)ds is - process. We now define a new diffusion process X under P
Definition 4.2. The neural guided diffusion bridge is a diffusion process that is defined as the strong solution to the SDE:
B(t) b(t, v), B())x is the Jacobian of b(t, æ) with respect to x. Compared to simply taking scaled Brownian motion, this choice can result in a guided proposal that better mimics true conditioned paths.
dX. (.)
(19)
(s, = b (s. (0)
Xo 20,
3.5. Strategies for Improving upon Guided Proposals
where 10 (8, x) + () x) is defined in Equa- tion (7).
Although the guided proposal takes the conditioning into account, its sample paths may severely deviate from true conditioned paths. This may specifically be the case for strong nonlinearity in the drift or diffusivity. There are different ways of dealing with this.
A unique strong solution X to Equation (19) is guaranteed due Assumption 4.1.
We propose to construct th as learnable neural network, whose goal is to approximate the difference of drift coeffi cients. When dg = a (r - f). the discrepancy between IP* and IP" vanishes. Lipschitz continuity of the neural net can be achieved by employing sufficiently smooth activation functions and weight normalization. Gradient clipping can prevent extreme growth on ar. In numerical experiments, we use either tanh or LipSwish (Chen et al., 2019) activa- tions and gradient clipping by the norm of 1.0 to fulfill such conditions.
If we write the guided path as functional of the driving Wiener process, can update the driving Wiener in- crements using the PCN within a Metropolis-Hasting algorithm. Details are provided in Appendix A. see also the discussion in (Mider et al., 2021).
Devising better choices for B, B and ō.
Adding an extra term to the drift of the guided pro- posal by a change of measure, where neural network parametrizes this term. We take this approach here and further elaborate on it in the upcoming section.
To learn the map dg. we propose a loss function derived from a variational approximation where the set of mea- sures { 0 e O} provides variational class for approxi- mating TP The following theorem shows that minimizing 0 - DKL. (PT is equivalent minimizing L as defined below.
4. Methodology
4.1. Neural Guided Diffusion Bridge
Theorem 4.3. If we define the loss function by
For a specific diffusion process, it may be hard specify the maps B, 3 and o, which may lead to a guided pro- posal whose realizations look rather different from the ac- tual conditioned paths. For this reason, we propose to adjust the dynamics of the guided proposal by adding a learnable bounded term o 2s,x))p(s, x) to the drift. Specifically, let Og: [0, T] x Rd - Rd be a function parameterized by 0 E θ, where O denotes the parameter space. Define:
L( = E (
(2 Bols, (o(ss Xx) -G) ds, (20)
then
DKL (PA = L(0) - log h(, xo) 1((0, x0)
(21)
with G( as defined in Equation (12).
x = exp (S
1 2 /./ [|oos, [1)oss
The proof is given in Appendix A.2. Note that under P+. 0 and therefore the dependence
the law of X depends of L on 0 is via both On and the samples from X under d- parameterized F.. If 6opt is local minimizer of L and h(U.ao)
(17)
We impose the following assumption on dg: Assumption 4.1. The map 0p is bounded and x + (s, x) Lipschitz continuous, uniformly in s [0, T7.
[((ppt)) = log k(0.an)> then Oopt is a global minimizer. This implies DKL (IP 8_.. IIP)) 0 from which obtain Pa Gage Pp. (Heng et al., 2022) applied similar varia- tional formulation. However, contrary to our approach, in this work the drift of the bridge proposal is learned from unconditional forward samples. Not surprisingly, this can be inefficient when conditioning on a rare event.
The Lipschitz continuity ensures x, is a martingale with E[xer] Define a new probability measure P. on (2, Fr) by dp == xP
(18)
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Neural Guided Diffusion Bridges
It can be seen that I is low bounded by log R(0.mo) In gen- k(0,zz)) eral, h is not known in closed form, but in some simple settings it is. In such settings, we can directly inspect the value of the lower bound and assess whether the trained neu- ral network is optimal. In Section 5.1, we provide examples for such sanity checks.
Algorithm Neural guided bridge training
Input:
Discrete time grid T == {&m m. initial
0, number of iterations K
Solve Equation (15) on T backwards, obtain and store
{ [ (lm) (F(mmm)))) using Equations (13) and (14).
repeat
for n = 1,..., N do
4.2. Reparameterization and Numerical Implementation
(n)) on TT Sample w (m) = {wim
Solve Equation (19) on T with w((x)) = (wit
Optimizing by gradient descent requires sampling from parameterized distribution I' and backpropagating the gradients through the sampling. To estimate the gradient, We use the reparameterization trick proposed in (Kingma & Welling, 2022). Specifically, the existence of a strong solution X to Equation (19) means that there is a measur- able map Φg : C([O, T], ) + C((0, 71. IR,,, such that X = <o("W") Here, we have dropped the dependence of de on the initial condition To as it is fixed throughout. The
obtain
{=tm (m))
end for Approximate L( by Equation (23).
Backpropagate (6 and update th by gradient de-
scent. until Iteration count > K
5. Experiments
5.1. Linear Processes
objective Equation (20) can be then rewritten as:
We consider one-dimensional linear processes with ana- lytically tractable conditional drifts, including Brownian motion with constant drift and the Cemstein-Uhlenbee pro- cess. For these models, the lower bound of 0 + L(0) can explicitly computed, serving as a benchmark to assess whether the neural network reaches this bound. Additional details are provided in Appendix B.2.
[000) -G(, do (W:))
T(0) E.
dt.
(22)
Choose a finite discrete time grid T == f'm Im=u.1.....
with lo 0, 4M = Let Xim, Win. be the evaluations of X W at tm respectively, {
((m) (2)((1)) }. (wi.. },n
1. ...... N be collections of samples of X².. W tm. and Eta
Consider one-dimensional Brownian motion with constant drift: dX, ydl + adW, As its transition density p(t, | 8, x.)) ) is Gaussian, the fully- observed process conditioned on {Xr = v}, satisfies the SDE:
Brownian bridge:
(n) <o(w>t))) ((m)) zo- Then Equation (22) can be
approximated by the Monte Carlo approximation:
N
M
L(0)= ~0
[tm (00 ))12
dx;
V Xi di odW.
(24)
N n= m=
- G( 22 (wrt ( {) 86.
(23)
We construct the guided proposal using the auxiliary process X, = oW, It is easy to see that X solves the SDE:
dx; = {
V -- X2 + +000(1, xx) x di + odW, (25) T-t
In practice, Itmm Φo(w;, (n) (()) implemented as a numeri- that takes the previ-
(n)
cal SDE solver fo(w (n) fm-11, 2t (n)
By comparing X with X it is clear that the optimal map i is given by Do (1,, x)) = 7/a. Additionally, the lower bound on L, log X(O..o))
additional arguments. As ottm (n) also depends on 0, the gradient with respect to 0 needs to be
",-1 [(n)
ous step (1m-11,
h(0..o) is analytically tractable since the
computed recursively. Leveraging automatic differentiation frameworks, all gradients can be efficiently recorded in an acyclic computational graph during the forward integration, enabling the backpropagation for updating 0. While the complexity of backpropagation scales linearly with M and quadratically with d-a property inherent to gradient-basse optimization methods approach remains highly effi- cient for moderate-dimennionaa problems and provides robust foundation for further scalability improvements. Fur- ther details about the gradient computation can be found in Appendix A.3 and the numerical algorithm is shown as Algorithm
transition densities 5 of X are Gaussian. In Figure 6a, we track how the training varies over iterations under different settings of y and a, which leads to different lower bounds. can be seen that all the trainings converge to corresponding theoretical lower bounds. Figure 2 compares the trained map úp with the optimal map De,,. The neural network matches the optimal map in regions well supported by the training data, but the approximation error grows outside these re- gions common limitation of neural network training. Figure 3 shows the empirical marginal densities of the neu- ral bridge alongside the analytical densities obtained from
6

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Neural Guided Diffusion Bridges
independent simulations of Equations (24) and (25). The learned and analytical distributions are in close agreement.
process, which we refer to as "normal" events; (2) rare events; and (3) events that cause trajectories to exhibit multi- ple modes, where the marginal probability at certain times is multimodal. We compare our method to (a) the guided pro- posal (Schauer et al., 2017); (b) bridge simulation via score matching (Heng et al., 2022); and (c) bridge simulation using adjoint processes (Baker et al., 2024a). The topleft panel of Figure shows realisations of forward samples of the process. The other panels show performance of the neural guided bridge upon conditioning on various events, that we detail below. Further details on these experiments are presented in Appendix B.3.
Ornstein-Uhlenber bridge: We now consider the
Bastein-Uhlenbeck (OU) process:
dX,- = -u Xt)dt + odW;
(26)
which requires a t, z-dependent u. to correct the guided
proposal. Upon choosing X. oW,, the neural bridge
satisfies the SDE:
V- X T + ()) dt+odW, dx;- (10 x9)+
(27) As with Brownian motion, the OU process has Gaussian transition densities and is therefore analytically tractable. Using this property, we derive the optimal map dyyy and the corresponding lower bound on L. given in Equations (46) and (47). We vary the parameters % f4, and a and plot the resulting training-loss curves alongside the analytical lower bound in Figure 6b; in every case the loss converges to its respective bound. Figure 4 compares the outputs of the trained network with the optimal map, and Figure 5 shows the empirical marginal densities of the learned neural bridge with those of the analytical bridge obtained from independent simulations of Equations (27) and (44).
13-
(a) Unconditioned
(b) Normal event
-10
From the above two experiments, neural network is able to learn the optimal drift with the proposed loss function and that the neural guided bridge is very close to the true bridge in terms of KL divergence (reflected by the very small differences between training losses and analytical lower bounds). In the remaining nu- merical examples no closed form expression is available for h(0, 20) and therefore performance can only be assessed qualitatively.
conclude that the
(c) Rare event
(d) Multi-modality
Figure I. Top left (a): 30 realisations from the unconditioned cell diffusion model (Cf. Equation (28)). Top right (b): 30 realisations from the learned conditional process "normal event" v [2.0 -.1] Bottom left (c): 30 realisations from the learned conditional process on a rare event right (d): 30 realisations from the learned conditional process on event v = [-1.0 -1.0] that causes multi-modality Except
[1.0 -0.1]" Bottom
in panel (d), all processes start from zo =0011 -0.1] and
5.2. Cell Diffusion Model
run up to time T = 4.0. In (d). the process starts from mg
(Wang et al., 2011) introduced model for cell differentia- tion which serves as a test case for diffusion bridge simu- lation in (Heng et al., 2022; Baker al., 2024a). Cellular expression levels X,- = [xe,1 Xx2]] are governed by the 2-dimensional SDE:
[-1.0 -1.0] and runs up to time T' = 5.0.
In all the experiments in the following we take o = 0.1.
Normal event: We set zo [0.1 -0.1]", T 4 and
2.0 -0.1] T From the topleft panel in Figure I it is clear that this corresponds to normal event: balls around v at time T get non-negligible mass under the un- conditioned process. In Figure 1b, we show 30 sample paths obtained from sampling the trained neural bridge and Figure 7 shows a comparison to the three baseline meth- ods mentioned above. Since the true conditional process is analytically intractable, generated 100, 000 samples from the forward (unconditional) process Equation (28), and obtained 172 samples that satisfy ||LX - ell s 0.01, and only show first 30 samples in the figure. Those samples can be treated as samples close to true bridges. Overall, all four methods successfully recover the true dynamies. The perfor-
x
X1,1
2xx,
dX2
dt odd,, (28)
+
+x,,,
2-1
X122
driven by 2-dimensional Wiener process W. The highly nonlinear drift makes this a challenging case. The guided neural bridge is constructed as an OU process:
dx,- = (1 [1-X,,ddt- + adW,
(29)
We study three representative fully- observed cases that result in distinct dynamics for the condi- tional processes: (1) events that are likely under the forward
with 1 = [1
7

---

Neural Guided Diffusion Bridges
mance of all four methods considered is comparable.Note that the adjoint bridge sample paths appear slightly more dispersed.
described by the SDE:
X.,1 8-X211
dx.- =
Xx +
X
dt
+
dW;
Rare event: We set z0 [0.1 -0.1] T and T = as
a
o
(30)
before but now we take e = [1.0 -0.1] T. which is a rare event. Unlike the "normal event" case, the true dynamies cannot be recovered by forward sampling from the uncondi- tioned process, as it is highly improbable that paths end up in small ball around v. In Figure Ic, we show 30 sample paths obtained from sampling the trained neural bridge and Figure shows a comparison to the three baseline methods mentioned above. All methods except the adjoint forward approach capture the correct trajectory dynamies, consistent with findings in (Baker et al., 2024a). Among the remaining three, trajectories from the neural bridge align more closely with those from the guided proposal than those from score matching. In the score matching method, samples of X,,1 show higher variance prior to t = 3.5, suggesting less ac- curate score estimates. This is expected, as score matching learns its drift from samples of the unconditioned process which rarely reaches small balls around u, leading to poor approximation quality.
We condition the process by the value of its first component by setting L = [1 0] and hence condition on the event LXx = v E R. We construct the guided proposal as pro- posed in (Bierkens et al., 2020) using the Taylor expansion x² = 2t 3u2xx at æ v. Thus, X satisfies the SDE
dx, {
1-3,a 3
2 +
x
x
X2
X
dt
+
aW,
7
-1
α
(31)
Suppose &x,8,7,,,o) = (0.1,0,,5,088,0000 amine the conditional behaviour of X within E [0, 2.0] under two scenarios: (1) conditioning on a normal event e = 1.0; and (2) conditioning on rare event == 1.1. As the score-matching and adjoint-process methods have not been proposed in the setting of partially observed state, we only compare our method to the guided proposal.
We ex-
Normal event: Figure 11 compares sample trajectories generated by the neural bridge and guided proposal. As a reference, paths obtained by unconditional sampling are added, where any path not ending close to the endpoint has been rejected. Both methods capture the key features of the conditioned dynamics, closely matching the reference trajectories. However, the neural bridge achieves this using independently drawn samples after training, whereas the guided proposal requires a long MCMC chain. The simi- larity between the neural bridge and the guided proposal confirms that the neural bridge effectively approximates the conditioned distribution, indicating accurate modeling of the dynamics under normal event conditioning.
Multi-modality: In both of the previous cases, each com- ponent's 's marginal distribution at times (0, 7'] is unimodal. However, with some special initial conditions, multimodal- ity can arise, which poses a challenging task where one would like to recover all modes. Specifically, let zo
[-1.0 - -1.0] T_ T = 5.0 and v = [-1.0 -1.0] In Figure Id, we show 30 sample paths obtained from sampling the trained neural bridge and Figure 7 shows a comparison to the three baseline methods mentioned above. Both the adjoint bridge and score matching fail to model the dynam- ics accurately. In contrast, the neural bridge and guided proposal yield similar marginal distributions. However, good performance of the guided proposal may take many MCMC iterations, or possibly the use of multiple (interact- ing) chains. Once trained, the neural bridge can generate independent samples, a cost comparable to unconditioned forward simulations, while maintaining sampling quality close to that of the guided proposal.
Rare event: In Figure 12 we redo the experiment under conditioning on rare event. Both the neural bridge and the guided proposal successfully capture the bimodal structure in the trajectories, as reflected by the two distinct clusters in the X paths, particularly evident after = 0.5. How- ever, the guided proposal matches the reference distribution more closely, especially in the concentration and spread of trajectories beyond time t = 1.5.
Across all three tasks, the neural bridge shows strong adapt- ability and achieves performance comparable to the guided proposal, with the added benefit of faster independent sam- pling. In contrast, the other two methods exhibit limitations under specific settings.
5.4. Stochastic Landmark Matching
Finally, we present high-dimensionall stochastic nonlinear conditioning task: stochastic landmark matching, see for instance (Arnaudon al., 2022). We consider a stochastic model that describes the evolution of distinct landmarks in Rd of a closed nonintersecting curve in RRd, The state at time t, X,, consists of the concatenation of each of the n landmark locations at time t. Hence, X, takes values in Rdn, The stochastic landmark model defined in (Arnaudon et al., 2022), for ease of exposition considered without mo-
5.3. HarHHugh-Nagumi Model
We consider the FitzHughNNagume model, which is a pro- totype of an excitable system, considered for example in (Ditlevsen & Samson, 2019; Bierkens et al., 2020). It is
8

---

Neural Guided Diffusion Bridges
OU (d = 1)
Cell (d = 2)
FHN (d = 2)
Landmark
(d = 20)
Methods
#Params
Time
#Params
Time
#Params
Time
#Params
Time
Adjoint forward (Baker et al., 2024a)
21, 969
162.68s
22, 114
265.06s 06.
N/A
N/A
114, 744
543.80s
Score matching (Heng ct al., 2022)
26, 353
65.55s
26, 498
103.38e
N/A
N/A
26, 766
353.05s
Neural guided bridge (ours)
921
44.12s
20306
94.79s
3, 362
113.77s
15, 188
122.48s
Table 1. Benchmarks with two other deep-learnng based methods. Neither adjoint forward nor score matching is available for the FHN partially ohserved conditioning case (denoted as "N/A", not applicable in the table). All the experiments conducted with the parameters described in Appendix B.
mentum variables, defines the process (X,, t E [0, TT) as the solution to the SDE
5000 pCN iterations and plotted the final iteration. From this, one can see that these iterations provide another way to improve upon the guided proposal in the top row. The other columns repeat the same experiment with different Wiener process initializations. Due to the nigh-dimension of the problem, repeated simulation required for pCN steps may be computationally expensive.
dX2,- = Q, Q(X:),) = kxx,), x Ia. (32)
Here Ia is the d-dimensional identity matrix, W is a dn- dimensional Wiener process and & is kernel function Rd x Rd - R. In our numerical experiments chose the ||=-v| The kernel
Gaussian kernel k(s,y) y) a exp
We benchmarked our method on the four test cases described above, comparing it to two other deep learning-based ap- proaches in Table 1. The comparison considers the number of network parameters and total training time, with all meth- ods trained using the same number of gradient descent steps and batch sizes. Our method outperforms both alternatives in terms of model size and training efficiency.
2xd
parameters are chosen to be a 0.3, K = 0.5 to ensure a strong correlation between wide range of landmarks. Note that the diffusion coefficient Q is state-dependent. To demonstrate the necessity of constructing such state- dependent diffusion coefficient, consider the process X solving the SDE
6. Conclusions and Limitations
di, = Q
(33)
We propose the neural guided diffusion bridge, novel method for simulating diffusion bridges that enhances guided proposals through variational inference, eliminat- ing the need for MCMC or SMC. This approach enables efficient independent sampling with comparable quality in challenging tasks where existing score-learning-bane baasdd meth- ods struggle. Extensive experiments, including both quanti- tative and qualitative evaluations, validate the effectiveness of our method. However, as the framework is formulated variationally and optimized by minimizing DKI.(Pp) [IP*), it exhibits mode-seeking behaviour, potentially limiting its ability to explore all modes compared to running multiple MCMC chains. Despite this limitation, our method provides computationally efficient alternative to guided proposals, particularly in generating independent samples from the conditioned process.
where the diffusion constant is constant. As a result, the system becomes linear. The processes defined by Equa- tion (32) and Equation (33) are fundamentally different. In particular, Equation (32) guarantees that t + X, gener- ates a stochastic flow of diffeomorphisms (Sommer et al., 2021). This diffcomorphic setting preserves the shape struc- ture during evolution, whereas the linear process defined by Equation (33) does not. As illustrated in Figure 13, a vi- sual comparison (using identical driving Wiener processes) reveals that the linear process disrupts the shape topology, leading to overlaps and intersections. The diffusion process defined by Equation (33) can however be used as auxiliary process in the construction of guided proposals. We opted for this choice in our numerical experiments.
In our numerical experiments, we chose one ellipse as the starting point and another ellipse the endpoint of the bridge. Each ellipse is discretized by 50 landmarks, leading to the dimension of X, being 100. We took T = 1.0. In the leftmost column of Figure 14 we fix a Wiener process and show on the top- and middle row the guided proposal and neural bridge using this Wiener process. We observed that due to the very simple choice of auxiliary process, the guided proposal has difficulty reaching the final state. In fact, we had to increase e² to 2e- 3 for not running into numerical instabilities. Here, one can see that the additional learning by the neural bridge gives much better performance. In the bottom row, we used the guided proposal with the same Wiener process as initialization, augmented by running
Our approach focuses on better approximating the drift of the conditioned process while keeping the guiding term that ensures the process hits v at time T relatively simple. In future work, an interesting direction to obtain improved re- sults consists in trying to jointly learn de and the parameters of the linear process (ō. B. .). Also, as the gradient updat- ing relies on backpropagating through the whole numeri- cal SDE solvers, techniques such as the stochastic adjoint sensitivity method (Li al., 2020) and adjoint matching (Domingo-Eie Enrich et al., 2025) can be introduced to improve the efficiency of the computation. Another venue of future research consists of extending our approach to conditioning on partial observations at multiple future times.

---

Neural Guided Diffusion Bridges
Acknowledgements
Bradbury, J., Frostig, R., Hawkins, P., Johnson, M. J., Leary, C., Maclaurin, D., Necula, G., Paszke, A., VanderPlas, J., Wanderman-Milne S., and Zhang, Q. JAX: Composable transformations of Python+NumPPy programs, 2018. URL http://g http//gttttbbb//. cow/jax-m/ jax.
The authors thank anonymous reviewers and Thorben Pieper-Sethmache (TU Delft) for their valuable comments and suggestions, which have helped improve the clarity and quality of this work. The work presented in this paper was supported by the Villum Foundation Grant 40582, and the Novo Nordisk Foundation grant NNF180C0052000.
Chau, H., Kirkby, J. L., Nguyen, D. H., Nguyen, D., Nguyen, N., and Nguyen, T. An efficient method to simulate diffusion bridges. Statistics and Computing, 34(4))131, 2024. ISSN 0960-3174, 1573-1375. doi: 10. 1220007/11122-224440
Impact statement
This paper presents work whose goal advance the field of Machine Learning and Statisties. There are many poten- tial societal consequences of our work, none of which we feel must be specifically highlighted here.
Chen, R. T., Rubanova, Y., Bettencourt, J., and Duvenaud, D. K. Neural ordinary differential equations. In Advances in neural information processing systems, volume 31, 2018.
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Neural Guided Diffusion Bridges
A. Theoretical Details
A.1. Preconditioned Crank-Nicolson
Algorithm is special case of Algorithm 4.1 of (Mider et al., 2021).
Algorithm 2 Preconditioned Crank-Nicolson scheme for guided proposals
1: Input: Discrete time grid T == 0,1,.... tuning parameter v [0, 1), number of required samples K
2: Solve Equation (15) on T, obtain (H(mm) (f(tm. +)) using Equations (13) and (14).
3: Sample - [wr,.. J on T.
4: Solve Equation (7) T with [w obtain y [y
5: repeat
6: Sample new innovations 2 = f2 on T independently.
7: Set w mpw + /1 1j22.
8: Solve Equation (7) on 'I with 2 = { obtain y" = {ve )- Compute A = (y))) (1) with {ve and {ye... } using Equation (1 1).
9:
10:
Draw U ~ (0,
11:
U < A then
y - y" and w w
13:
end if
14:
Save y-
15:
until Sample counts > K.
A.2. Proof of Theorem 4.3
Proof. Consider the KL divergence between Pg and IP*:
dILg
dILa dLL
dIL
DKL [T
PP*)
E
log
(X)
E
log
dIL (x)
cIL*
dlLa
dL
E
log
dIL o (x)
E
log
dL (X)
(34a)
By Girsanov's theorem,
CLL
(x))
log dP cPp
E
log
(35a)
dL
1 2 [100 x
{
T
O1 x.)dw,
2
1 T
//
E
Oo Xt)dW:
E
(t)(t,X))l''
(35b)
2
where the stochastic integral vanishes because the martingale property of the Itô integral. The first equality follows from Equation (18). By Equation (8)
dL* J.
E. [ G(I, + log 100,
(0,
E
log
(x)
(36)
Substituting Equation (35b) and Equation (36) into Equation (34a) gives
/-" T
{ |oo(t, x2)l - G(t, X.) dt log
70, xo) h(0, 20)
L(0) -log 70 20)
DKL IP*) = T
2 0,
(37)
h(0,
with T((0) as defined in Equation (20).
13

---

Neural Guided Diffusion Bridges
A.3. SDE Gradients
[w]"") As discussed, (n)) Go Wh (m)) is implemented numerical SDE solver tt ,, ((n), -m- 2 that takes the previous step Fim
We derive the gradient of Equation (23) with respect on a fixed Wiener realization w*()) {wi,
(())
(1m-1, 1, T (20) as additional arguments. As It
(n) also depends 0, the gradient with respect to needs be computed
recursively. Specifically, with Fim (())
fa fo .m 1, mm
(m)
(x2)
(4m- (w (10) 212 Vo 2
- (100 so
(38a)
Vodo (1mm J0 -1]]
10(0m 1m
(38b)
@0 10 10
00 00 10.
Vofamm-
do 11
(38c)
00
0smm-1
@0o J0,m-
@0a(mm 1,
05
05
]²
Vofo
(38d)
00
jfo.m-1
08
0fa,m-2
]
1"
@0 so,m-1
Ja,m-1)
@10.m-1
OSo.j
a5.i
+
10
Oo
00
0f
08
ffo
08
1=1
(38e)
Similiarly, the gradient of G with respect 0 can also be computed recusively:
(0 @G(tm-11 f0.m-1)
O @fe.m
m
m
f
Ofes
+
Σ
(39)
afo.m-1
00
2 afoj-
00
j=it1
The gradient of I(0)) can be approximated by:
N
M
VoL(O)
V
1 1 (1m- (o( (m)) ))13)
VeG(tm-1, (o(w)) (o 8t.
(40)
1=1
m=1
The realization of fo depends on the chosen numerical integrator. We choose Euler-Maruyama as the integrator used for all the experiments conducted in Section 5. Under this scheme, fo is:
fo(we!n) (m))
(1)) (()) 1m-11, xmm-))
I (--- + (b + aF + (00) ((m) + a (1m-1, 2
(n.) (x) Fm W
(41)
(n) with wi..
NO0 (Im - 1m-11mm11 lm). The derivatives can be computed accordingly:
0 fo o @1m--1, x (n)
Ltm-1
(42a)
20
00
0
1
0b + ar + 000)
(1m-1,
It
(n)
Or (n)
(m) (tm-1,
(n)
(42b)
f.
On (n)
Ox
The automatic differentiation save all the intermediate Equation (42a) and Equation (42b), which enables to compute (o)(
B. Experiment Details
B.I. Code Implementation
The codebase for reproducing all the experiments conducted the paper is available in https: http://ggttthccccce com/ bookdiver/neeurabbri..e
14

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Neural Guided Diffusion Bridges
B.2. Linear Processes
Brownian bridges: If dX,, ydt + adW, then
(v x - 7(T - 1))2
log h1(, = (Tg (,):) log(2 (T -7
(43)
20 t)
If dX, odW, then log h(t, x)) ) is obtained by taking y 0 in the preceding display.
Therefore, in this case we can compute the lower bound on the loss: L(Ø) 2 log k(0.mo) the optimal map De.g. is given by 18.y. (1, = o(2, log h((,,)) 0₂ log (1, .)) =
(0-2-37) (v-2= Moreover,
h(D,ro)
2
In the numerical experiment, we took = 10 5, The map dg is modeled by a fully connected neural network with 3 hidden layers and 20 hidden dimensions for each layer. The model is trained with 25,000 independently sampled full trajectories of
X The batch size was taken be 50 and the time step size òt 0.002, leading to in total M 500 time steps. The network was trained using the Adam (Kingma & Ba, 2017) optimizer with learning rate 0.001.
10,
7 20
7 -20
-04
0.0
0.0
0.0
0.0
1.0 0.6
0.2
0.4
0.6
0.8
3.0 0.0
0.2
0.6
0.6
1.0
0.0
0.2
0.4
3.5
0.8
1.0
Figure Evaluations of trained neural networks (t .) against the optimal maps 0.. (f, under different settings of y, a, where the background colour intensities indicate the absolute error [0 -. In each panel, 10 independent samples from the guided proposal shown in grey to indicate the sampling regions where one expects the error to be smallest.
1.0
Lingl DAX laglee 10o
sapl 2A
CITY
2008
(tensty
-1.0
0.8
1.0
6.55 3.0
6.5
0.0
3.0 3.0 0.0
1.0
0.0
Figure Comparison of marginal distributions different time slices the learned neural bridge (n.b.) and analytical true Brownian bridge bt.b.) under the setting 1.0, a 1.0, conditioned on v = 0.0. The purple bars show the absolute error (abs. error) of each bin between learned neural bridge and the histograms obtained by forward sampling from the true bridge process, which are expected be low when two distributions are close in terms of their shapes. The histograms are made from 50,000 independent trajectory samples.
Ornstein-Uhlent bridge: When conditioning Equation (26) on the conditioned process X* satisfies the SDE
ax; {1(m ( X7) 2-2-(I-0) (v- 1)) xT (x; m)] dt odW,
x1) + 27e (T--))
(44)
15

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Neural Guided Diffusion Bridges
which is obtained from the transition density given byEquation (26) is:
1
(u mxr(
p(T.y (lylt,z)
oxp
2
(45a)
/ 2
(2, = 1 (x- 1i)e r(t
(45b)
Σg 0²²
(1- (
(45c)
2
Since the auxiliary process chosen the same as the Brownian case, one can easily show the optimal value of de (1, x) to be
Dgn (t, o(1 - e 2x(-- (V- (V- - - - (x -1)]
((T--)
(V-x) x) o(T - t)
(46)
Therefore, the lower bound on / L(() is given by
7(0,, ro) log h(0, zo)
x)) 1 1 log ( 2T -) +2 log((=
(v m4 (x)))
(47)
2
We repeated the same numerical and experimental settings as the previous Brownian example, except for training with 50,000 samples obtain better results.
1.7.u 10.0 0.
0G 10,00 0.3
17 18. 10
17, 00,0 03
1.0
1.5
1.0
0.5
0.2
Figure 4. Evaluations of trained neural networks Oa(t, 2)) against the optimal maps Da,t (t, x) under different settings of y, where the background colour intensities indicate the absolute error (0o &agil each panel, 10 independent samples from the guided proposal are shown in grey to indicate the sampling regions where one expects the error to be smallest. Except the rightmost setting with y = 1.7, μ = 0.0, a = 0.3, all the processes are conditioned on v = 1.0, whereas in the rightmost case, the process is conditioned on 0.0.
1- 0.20
1- 0.40
[-0000
L- 0,50
am2 dos
-0.50-0.25 0.03 0.25 0.50 0.75
1.00
-0.50-025 0.01 0.25
0.50
1.00 1.25
1.50
-4.50-0.55 9:00
0.25
0.50
0.75
1.00
1:21
1.50
-0.50-0.25 0.00
0.20 0.50
0.25
1:00 1.20 1.00
Figure 5. Comparison of marginal distributions at different time slices of the learned neural bridge (n. and analytical true OU bridge (t.b.) under the setting ? 1.7, μ 1.0. 0.3, conditioned 1.0. The purple bars show the absolute error (abs. error) of each bin. The histograms are made from 50,000 independent trajectory samples.
B.3. Cell Diffusion Process
For the benchmark tests, we adapt the published guided proposal implementations of the corresponding methods to fit into our test framework with possibly minor modifications. Specifically, the original guided proposal codebase implemented with
16

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Neural Guided Diffusion Bridges
1.0,0 1.0
Lopt
Lopt
10.00
Lopt (0 0.5)
0.0.
1.0,
0.3
Topt E7 0.63
2.0,0 1.0
Toopt (1 2.0)
10,0
Lapt (e 1.0)
-20,
Lope ( -2.03
D.0,
0.3
Lupt (p 0.00
Loss
Loss
Ieeations
Iterations
(a) Brownian bridges
(b) OU bridges
Figure 6. Loss curves of training Brownian and OU bridges. For Brownian bridges, the SDE parameters are taken as 1.0, 1.0, for the OU bridges, the parameters taken as 1.7, 1.0, 0.3 unless any of them is specified.
Julia in !, we rewrite it in JAX (Bradbury et al., 2018); the score matching bridge repository is published in additionally, as also reported by the authors, (Heng et al., 2022) introduces two matreemathhin---- based bridge simulation schemes, reversed and forward simulation, and the forward simulation relies on the reversed simulation, and learning from approximated reversed bridge can magnifies the errors due to progressive accumulation. Therefore, we only compare our method with the reversed bridge earning to avoid error accumulations; the adjoint bridge implemented in
Normal event: We took 2² 10-10 and dg is modeled fully-connected network with hidden layers and 32 hidden dimensions per layer, activated by LipSwish. We trained the model for 5,000 gradient descent updating iterations. In each step, we sampled batch of 100 independent trajectories of X under the current O. The numerical sampling time step size is öt 0.01. Therefore, in total M = 400 discrete steps of single trajectory. The Adam optimizer with learning rate of le 3 was used for optimization. For the guided proposal, we set 0.98, and ran one MCMC chain for 10,000 iterations. This resulted in an acceptance percentage of 21.38%. The initial 5, 000 iterations were considered as burn-in samples. After burn-in, we collected the samples and subsampled them by taking every 133 samples to obtain 30 samples. the following experiments, explicitly stated, all the samples from the guided proposal are similarly obtained from one MCMC chain by subsampling from the outputs. For the score matching and adjoint forward methods, deployed the given network structures provided in their codebases. As reference, we sampled from the forward process until we obtained 30 samples satisfying the inequality ||X vll < 0.01. These samples are shown in grey.
8
64
06
43
(a) neural bridge
(b) guided proposal
(c) score matching
(d) adjoint forward
Figure 7. Visualization of 30 simulated bridge trajectories under normal event conditioning, using various sampling methods. All samples independently drawn, except those generated by the guided proposal. As reference, in grey, added trajectories from the unconditional forward process, with samples that satisfy the condition LLXy - u|l > 0.01 rejected.
Rare event: The setups for conditioning on rare events of the neural guided bridge are replicated from the previous normal event case, except for the MCMC is rumning with sightly increased tuning parameter 7 0.999 and for 20,000 iterations and the first 10,000 iterations discarded the burn-in period. The acceptance percentage is 22.29%. For the score matching,
ptettss:/jjliappcccee htw.l/aaacckses..coo////r//eoeee
Phttps: //tpps::/giie thub. com homm//eeremmmmennnn///iii onbridge hettps://githie com cwt.ccmmm///// http:///gggggooooooo///a/a///eee bridge
17

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Neural Guided Diffusion Bridges
since we only use the reversed bridge, where learning is independent of the event condition on directly deploy the trained score approximation from the previous For the adjoint forward, we fix the neural network architecture and training scheme, changing only the conditioned target.
As reference, we sampled the forward process 100, 000 times. This time however, none of the samples ended near the point we condition on, confirming we are dealing with a rare event.
(a) neural bridge
(b) guided proposal
(c) score matching
(d) adjoint forward
Figure 8. Visualization of 30 simulated bridge trajectories under event conditioning, using various sampling methods. All samples are independently drawn, except those generated by the guided proposal. Contrary to Figure 7, no reference trajectories were added as extensive forward simulation of the process did not yield any samples satisfying LX,, v|l 0.01.
Multi-modality: All neural network architectures and training settings match those used in previous examples. We set öt = 0.01 and M = 500. For the guided proposal, a single chain is run for 50, 000 iterations with 0.9, discarding the first 20, 000 as burn-in. The acceptance percentage is 26.81%. As in the rare event case, no valid samples were found from forward simulating 100, 000 times the (unconditioned) forward process.
Figure 10 shows marginal distributions for both unconditioned and conditioned processes, using the guided proposal and the neural bridge. At 3.0 and t 4.0, multiple peaks appear in the marginal densities. suggesting multi- multi-modality. This is consistent with the unconditioned sampling, where multiple modes are also visible. However, neither score matching nor the adjoint forward method captures these modes. In contrast, the neural bridge and guided proposal yield similar marginals that accurately reflect the multi-modal nature of the process.
-13
-33
(a) neural bridge
(b) guided proposal
(c) score matching
(d) adjoint forward
Figure 9. Visualization of 30 simulated bridge trajectories under multi-modal event conditioning, using various sampling methods. All samples are independently drawn, except those corresponding to the guided proposal.
B.4. FitzHugh-Nagumo Model
Normal event: We set öt = 0.005, which leads to M 400 time steps. We took c² 1e 8. For the neural guided bridge, (l, constructed as a fully connected neural network with hidden layers and 32 hidden dimensions at each layer, activated by LipSwish functions. The training done for 5,000 gradient descent steps. each step, we generated N 100 independent samples from the current X for Monte Carlo estimation. The network is optimized by Adam with learning set 77 0 suggested in (Bierkens et al., 2020) and ran chain for 50, 000
rate of 1e 3. For the guided proposal, iterations with burn-in of 20, 000 steps, obtaining 64.41% acceptance percentage. The reference is obtained by sampling the (unconditional) forward process filtering samples with LXX vll < 0.01, as v = [1.0] is an event around which the process is likely to reach small ball, one can expect to easily obtain sufficient samples that meets the filtering condition, turns out that we only need to sample the forward process for 10, 000 to obtain 450 vaild samples, and we only show the first 30 the figure.
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Neural Guided Diffusion Bridges
2.3
1.0
0.3
0.9
10.0
Xx. Dereits
1.0
Figure 10. Marginal distributions of X selected time points for three settings: the unconditioned process (unc.), the neural bridge (n.b.), and the guided proposal sampled with pCN (g-p.). All trajectories start at xo [-1 -1] and, The neural bridge and the guided proposal conditioned to reach [-1 -1] the terminal time 7 5. Each histogram is based 10,000 camples-independee draws for unc. and n.b. and subsampled MCMC draws for g.p. The first and second components of X are displayed in the top and bottom rows, respectively.
1.0
X1,
1.0
-0.5
0.5
-0.5
-1.0
100
0.00
0.55
0.50
1.00
1.30
1.30
0.00
0.35
0.50
0.75
1.00
1.50
3.00
0.03
0.50
0.50
0.95
1.00
0.50
1.25
2.00
(a) neural bridge
(b) guided proposal
(c) reference
Figure 11. Visualization 30 simulated bridge trajectories under normal event conditioning from the learned neural bridge and pCN sampling of the guided proposal. The reference trajectories in (c) obtained by forward sampling the unconditioned process, keeping only samples that satisfy the condition LXx ell s 0.01.
Rare event: We duplicate the neural network training setting before, including the used network structure and training hyperparameters. For the guided proposal we set the PCN-parameter to f 0.9 and ran one chain for 50,000 iterations which yielded a Metropolis- Hastings acceptance percentage equal to 22.46%. The initial 20,000 iterations are considered burnin samples. As reference, we forward sampled 200, 000 paths the unconditioned process. Of those, 35 samples satisfied LX,, - ell < 0.01, which about 0.02% of the samples. As expected, in the "normal event" case, this percentage was higher (4.5%).
19

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Neural Guided Diffusion Bridges
1.0
6.-
0.0
-0.5
0.5
-0.
100
+10
0.25
9.00
5.00
(a) neural bridge
(b) guided proposal
(c) reference
Figure 12. Visualization of 30 simulated bridge traeectories under event conditioning from the learned neural bridge and pCN sampling of the guided proposal. The reference trajectories in (c) are obtained by forward sampling the unconditioned process, and filtered the results by the condition || ell I 0.01.
B.5. Stochastic Landmark Matching
1.0
Lineur
0.6
-1.0
Time
Noninear
00
-0.5
D0
Figure 13. Comparison of unconditioned samples from the linear process (Equation (33)) and the nonlincar process (Equation (32)). Top row: 4 independent samples from Equation (33). Bottom row: corresponding samples from Equation (32), each using the same Wiener process realization as in the top row.
The observation noise variance deploy the neural network architecture suggested in (Heng et al., 2022) to model ds. whose encoding part is two-layer MIP with 128 hidden units at each layer, and the decoding part is three-layer MI.P with hidden units of 256, 256, and 128 individually. The network is activated by tanh, and trained with 240,000 independent samples from X with batch size N 8, optimized by Adam with an initial learning rate of 7.0e and a cosine decay scheduler. For the guided proposal, we run chain for 5, 000 iterations each and drop the first 1, 000 iterations as the burn-in. with 77 = 0.95 and obtain 12.62% acceptance percentage on average over the chains.
as c² 2e 3, as we find small values of will cause numerical instability, We
20

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Neural Guided Diffusion Bridges
Guided propoeal
1.-
0.8
Neural bridgee
0.4
Guideel proppeen (w.. pCNN
04
0.0
Figure 14. 4 realizations of sampling from the guided proposal, the trained neural bridge and the final iteration after updating the guided proposal with 5, 000 ptN-steps. Top row: samples the guided proposal using independent Wiener process realisations; middle row: 4 samples from the trained neural bridge using the same Wiener realisations as the top row; Bottom row: the final outputs of 4 independent chains updating the guided propsals using PCN-steps. The chains are initialised with the same Wiener realisations as the top row.
21


Reviews:
[{"review_id": "bfuBtKkbAO", "reviewer": "Reviewer_ysG7", "summary": "This paper introduces a novel variational method for simulating conditioned diffusion processes (diffusion bridges) by proposing a more expressive guided proposal framework of Schauer et al. (2017) with a learnable drift correction term parameterized by a neural network. By leveraging variational inference, the proposed method overcomes key limitations in existing approaches: guided-proposal-based methods require a careful, non-trivial choice of an auxiliary process and rely on computationally intensive MCMC/SMC updates, while score-learning-based methods often struggle with inadequate exploration of rare event regions and the numerical challenges of inverting nearly singular diffusivity matrices. \n\nThe method is validated on a diverse set of problems. The experiments, which include comparisons with state-of-the-art guided proposals, score matching, and adjoint-process methods, demonstrate that the neural guided bridge is more flexible and adaptable.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Claims and Evidence / Methods and Evaluation Criteria**\n* While neural bridges and MCMC-guided proposals differ methodologically—complicating direct cost comparisons—their forward simulation costs are comparable: for example, in the landmark process ($d=100$), the forward simulation time is 9.81ms (neural) vs 6.85ms (guided). Training complexities diverge as  $O(Nd^3)$ (score-matching/adjoint) vs  $O(N^2d^2)$ (neural), where $N$ is the number of time steps. Due to the space limitation, we can not include benchmark results here, but please refer to our reply to Reviewer 8sR1, and we also have added a more comprehensive summary into the manuscript per suggestion.\n\n**Other Comments or Suggestions**\n\nWe have fixed the equation, thank you!\n\n**Questions for authors**\n1. We are unsure what you mean by ``discrete densities''. Proposition 3.2 requires $\\nabla_x \\log h(s,x)$ to be well defined. The simplest way to ensure this  is to assume existence of smooth transition densities. Note however that we can write $h(t,x)= \\int q(v\\mid y) P(T, d y\\mid t,x)$, where $P(T, d y\\mid t,x)$ is the Markov kernel. This expression also makes sense if the kernel does not admit (smooth) densities with respect to some dominating measure. Informally, the case of conditioning on a state without noise corresponds to taking $q(v\\mid y)$ like a Dirac mass in $v$ and then the above display should be interpreted as $ h(t,x) = p(T,v\\mid t,x).$\nClearly, existence of $\\nabla_x \\log h(s,x)$ requries that $h$ is strictly positive and that its gradient exists. Throughout the paper we have assumed that the distribution of $v$ conditional on $y$ is Gaussian. It is however possible to relax this assumption.\n2. The non-noisy case is most challenging. Intuitively, the larger the noise, the less the process needs to be guided in a certain direction. So we tested the approaches on the more difficult case. \nWe formulated Proposition 3.2 deliberately for the non-noisy case;  in case we condition on the full state without noise, we can simply take $h(t,x) = p(T,v\\mid t,x)$. However, in case of a partial observation, such as with the FitzHugh-Nagumo model, the form of $h$ reads more pleasantly when noise is assumed. We refer to Section 1.3.2. of Bierkens et al. for the somewhat more involved formulas that appear when no noise is assumed. So the motivation is merely to present a ``clean'' statement. In the examples, we assume $q(v\\mid y) = \\psi(v; L y, \\epsilon^2 I)$, where $\\epsilon$ is very small. For example, in the FHN-example $L=[1, 0]$, which corresponds to only observing the first component. Taking $\\epsilon$ nonzero makes guided proposals also numerically better behaved near the time of conditioning.  \n3. The drift of the bridge behaves in a very specific way near the point we condition on. This is most easily seen in the uniformly elliptic case and when we condition on the full state $v$ at time $T$. Essentially, the drift of the true bridge behaves for $t\\approx T$ as $(v-x)/(T-t)$. Failure to replicate this behaviour breaks absolute continuity between true and proposal bridge laws. Our guided proposal network explicitly replicates this asymptotic behavior. The neural bridge's auxiliary drift (confined to $\\sigma$'s range and bounded) enhances proposals on $[0, T-\\eta)$ for small $\\eta$, preserving absolute continuity when added. Empirical validation appears as Fig. 6 in Bierkens et al.\n4. In general yes. The additional learned drift in our method is the difference between the true score $\\nabla_x\\log h(s, x)$ and the proposed score $\\nabla_x\\log \\tilde{h}(s, x)$ (with $\\sigma$ as a scaling). The objective function is based on a closed-form expression for the  likelihood ratio between the target measure $\\mathbb{P}^{\\star}$ and the proposal measure $\\mathbb{P}^{\\circ}$  (As Eq. (10) in the paper). The key challenges in applying our method are to select $\\mathbb{P}^{\\circ}$ that satisfies two criteria: the likelihood ratio between the target $\\mathbb{P}^{\\star}$ and $\\mathbb{P}^{\\circ}$ is tractable, and sampling paths under $\\mathbb{P}^{\\circ}$ can be done efficiently.\n5. Yes, mode collapse occurs in both cell diffusion and FitzHugh-Nagumo (FHN) experiments, showing multimodal marginal distributions. This stems from the forward KL divergence's mode-seeking objective. We've also considered reverse KL alternatives, but it will introduce problematic stochastic integrals and unstable optimization. Additionally, we see three possible mode collapse explanations: (i) local optimum convergence; (ii) misspecified endpoint guidance from $\\tilde X$; (iii) Oversimplified $\\vartheta$. We conjecture (i) to be most likely. We implemented strategies:  (i) optimizer/hyperparameter sweeps (Adam/RMSprop/SGD); (ii) dropout regularization. Neither approach yielded significant improvements. We hypothesize that the presence of $G(t,x)$ in the objective function introduces significant non-convexity, rendering the optimization landscape inherently challenging.", "claims_and_evidence": "- The paper claims that its method can generate independent conditioned samples at a cost comparable to simulating the unconditioned process—thus avoiding computationally intensive MCMC or SMC updates—by employing a variational inference approach; **however, it does not provide a direct comparison of the neural network training time complexity with that of MCMC/SMC or score optimization methods.**\n- Extensive experiments in Section 5—including tests on the Brownian bridge, Ornstein-Uhlenbeck process, a cell diffusion model, the FitzHugh-Nagumo system, and stochastic landmark matching —offer both quantitative and qualitative evidence that the proposed method is competitively robust compared to guided proposals, score matching, and adjoint-process approaches.", "methods_and_evaluation_criteria": "While I appreciate the thorough evaluation of the proposed method, I encourage the authors to provide a direct comparison of the neural network training time complexity with that of MCMC/SMC or score optimization methods.", "theoretical_claims": "The main contribution of the paper is empirical where many ideas are borrowed from (and referred to) Bierkens et al. (2020). I have skimmed through the derivations in the appendix and they look correct to me.", "experimental_designs_or_analyses": "The experimental design is comprehensive and well-structured, as it evaluates the proposed method across a variety of diffusion bridge scenarios—from simple one-dimensional cases (Brownian and Ornstein-Uhlenbeck bridges) to more complex, nonlinear, and high-dimensional problems such as cell diffusion, the FitzHugh-Nagumo model, and stochastic landmark matching. \n\nThe authors complement qualitative visual comparisons with quantitative analyses (e.g., tracking training loss curves and comparing empirical distributions against known lower bounds in simpler cases), which provides a strong basis for assessing the method's performance relative to guided proposals, score matching, and adjoint-process approaches.", "relation_to_broader_scientific_literature": "The paper highlights several challenges with current methods for simulating diffusion bridges. In Guided-Proposal-Based Methods, the authors note that a careful, often complex, choice of an auxiliary process is required. Moreover, when these methods are combined with MCMC or SMC updates, the computational cost can become prohibitive, especially for strongly nonlinear or high-dimensional diffusions. \n\nWhereas, Score-Learning-Based Methods, the other popular approach, rely on samples from the unconditioned process, which often do not adequately cover regions corresponding to rare events, leading to suboptimal performance. Additionally, the canonical loss function in these methods involves inverting the matrix $\\sigma\\sigma^\\top$, a task that is particularly challenging for hypo-elliptic and high-dimensional diffusions where the matrix may be nearly singular, thus complicating stable and accurate optimization.\n\nFurthermore, the literature distinguishes these problems from related areas such as the diffusion Schrödinger bridge and neural SDEs, where the focus is on connecting fixed marginal distributions or modeling entire data trajectories under stochastic dynamics.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The idea presented in the paper is novel and innovative and is particularly relevant as it addresses key challenges in existing methods. Given the growing interest in diffusion-based models across machine learning community, the proposed method has significant practical relevance for applications requiring efficient sampling of conditioned stochastic processes.", "other_comments_or_suggestions": "Minor\nL166- fix the Typo", "questions_for_authors": "1. How crucial is the assumption that the stochastic process  X has smooth transition densities for the validity of your theoretical results, and what would be the impact of relaxing this assumption to discrete densities?\n2. Why is the non-noisy case emphasized in the paper? The authors primarily discuss scenarios in their experiments where observations are noise-free, but I would appreciate insights into how their approach performs when observations contain noise.\n4. Section 3.4:- Why did the authors opt to add an additional learnable term to the drift rather than attempting to learn the entire drift function, and do they have empirical evidence to support the claim that the guided proposal’s sample paths significantly deviate from the true conditioned paths?\n5. Is it possible to use drift correction in existing approaches such as score optimization based methods? If yes, can the authors discuss the challenges associated with using drift correction in existing approaches.\n6. Given that mode collapse is a well-known challenge for such problems, did you observe any instances of mode collapse during your optimization? If so, could the authors elaborate on the nature of these occurrences and describe the specific strategies or modifications that they implemented to mitigate mode collapse?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "N/A"}}, {"review_id": "XCIiRlcv3j", "reviewer": "Reviewer_8sR1", "summary": "The paper presents a novel method for simulating conditioned diffusion processes, called diffusion bridges. This approach trains a neural network to approximate the bridge dynamics, providing a more robust and efficient alternative to traditional methods like MCMC or reverse-process modeling, particularly for rare events and multimodal distributions. By learning a flexible variational approximation of the diffusion bridge path measure, partly defined by a neural network, the method enables efficient independent sampling similar to simulating the unconditioned process. The paper validates this \"neural-guided diffusion bridge\" through various numerical experiments, comparing its performance against existing techniques in challenging scenarios.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Methods and Evaluation Criteria**\n1. All $\\vartheta_{\\theta}$ implementations use fully-connected networks with $(1+d)$-dimensional inputs (time $t$ and state $x$). Time integration varies by dimensionality: direct concatenation for low-dimensional systems (Brownian/OU/cell/FHN) versus sinusoidal time embeddings with feature-wise modulation (Perez et al.) in high-dimensional landmark processes. Architectures employ tanh activations to ensure Lipschitz continuity (critical for Assumption 4.1 via gradient clipping). Layer counts/hidden dimensions were determined via loss-minimizing parameter sweeps.\n2. To our knowledge, no established quantitative metrics exist for high-dimensional performance evaluation. Only visual assessment by brute force unconditional sampling (cell and FHN)—prohibitively inefficient for rare events, are available. In contrast, real-world applications (e.g., image/video) permit rigorous benchmarking through domain-specific metrics like FID/IS scores used in translation tasks.\n\n**Experimental designs or analyses**\n1. We took two representative examples to compare the computational cost of the methods considered in more detail in the following table.\n\n||OU||Cell|||\n|-|-|-|-|-|-|\n|Methods|#Params|Time|#Params|Time|Complexity|\n|Adjoint Bridge|21969|93.53s|22114|2465.06s|$\\mathcal{O}(d^3)$|\n|Score Matching|26353|335.96s|26498|1496.87s| $\\mathcal{O}(d^3)$|\n|Neural Bridge|1341|29.21s|3362|186.02s|$\\mathcal{O}(d^2)$|\n\n2. Our proposed loss eliminates matrix inversion of $a=\\sigma\\sigma^T$. Precomputable terms $L(t)$, $M(t)$, and $\\mu(t)$ are solved once before training, while $\\tilde{r}(t,x)$ (Eq. 11b) and $G(t,x)$ (Eq. 10b) are computed during integration - all without requiring $a^{-1}$. This contrasts with the canonical score matching loss (Heng et al.) which contains $\\Sigma(t,x)=a(t,x)$. Such inversions prove numerically unstable for near-singular $a$, and critically fail in our FitzHugh-Nagumo example where $a$ is singular. Our method remains applicable where score matching becomes undefined. To show more insight why the (neural) guided bridge works for hypo-elliptic diffusions, consider the SDE $$dY_t = \\{b(t, Y_t) + \\sigma(Y_t)f(t, Y_t)\\}dt + \\sigma(Y_t)dW_t,$$ and denote the law of $Y$ to be $\\mathbb{Q}$. $\\mathbb{P}^{\\star}$ is absolutely continuous with respect to $\\mathbb{Q}$ if there exists a bounded solution $\\eta$ to the equation $$b^\\star(t,x)- b(t,x)-\\sigma(x) f(t,x) = \\sigma(x) \\eta(t,x).$$ Recall that  $b^\\star(t,x)=b(t,x) + \\sigma(x) \\sigma(x)^T \\nabla_x \\log h(t,x)$. Then the preceding display can be rewritten to $$\\sigma(x) \\left( \\sigma(x)^T\\nabla_x \\log h(t,x) -f(t,x)\\right) = \\sigma(x) \\eta(t,x).$$ Hence, one can easily see such an $\\eta$ exists by the specific form of the additional drift. In this way, we circumvent inversion of $\\sigma$. This also explains why the additional drift in the neural guided bridge contains the $\\sigma$ premultiplication.\n\n**Questions for Authors**\n1. We cannot ensure this, similarly to other applications of Variational Inference. But some heuristic methods can be helpful in our case. For example, we can initialize the neural bridge differently, and test whether it learns the same drift term.\n2. This \"matching condition\" appeared first in Theorem 1 of Schauer et al. This paper deals with the fully observed, uniformly elliptic case. It says that $\\tilde\\sigma$ should be $\\tilde{a}(T)=a(T,v)$, see also our discussion in Section 3.3 of our paper. The claim that all examples in Section 5 satisfy the matching conditions needs adjustment. It is true, except for the Fitzhugh-Nagumo (FHN) model. Thank you for drawing our attention to this: we adjusted the formulation. \nThe matching condition specifically refers to Assumption 2.4 in Bierkens et al. It consists of verifying 4 inequalities. As the diffusivity is constant, the 4th of the these is trivially satisfied. The first and third assumptions can be verified similar to Example 3.2 in Bierkens et al. (with $\\Delta(t)=(T-t)^{-1}$). The second assumption in there is on the difference $b(t,x) -\\tilde{b}(t,x)$. Inspecting the proof, it suffices that the second assumption only needs to hold true for $t=T$ and those $x$ for which $Lx=v$. With our choice of $\\tilde b(t, x)$, it reads $b(T,x) -\\tilde{b}(T,x)=0$, therefore the second assumption also suffices.\n3. Yes, please refer to our reply to Review ysG7's Question 5.\n4. We now simply choose $B$ and $\\beta$ in a rather simple way to ensure the neural bridge will end up in the right point.  We don't see any direct complication from parametrising these functions by a neural net, apart from computational resources. Presently, it is unclear if the additional training time is worth the effort.\n5. Guided proposals in the case of multiple partial observations are discussed in Mider et al. We would start from this approach, and simply add a drift term on each of the segments between observations times, as we propose in the present paper for only one segment.", "claims_and_evidence": "yes, the claims made in the submission are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "1. Could you provide more details on the specific neural network architecture used to parameterize the drift correction term v(t,x) in the different experiments? For example, what was the rationale behind selecting the number of hidden layers, the dimensionality, and the activation functions?\n\n2. For the \"Stochastic landmark matching\" task, since the true bridge is intractable, it’s challenging to make a quantitative comparison. Are there alternative metrics or qualitative analyses, beyond visual inspection, that could help assess the performance of the proposed method relative to the guided proposal in high-dimensional settings?", "theoretical_claims": "yes, all the theoretical claims made by the authors are correct.", "experimental_designs_or_analyses": "1. The paper emphasizes that the proposed method learns directly from conditional samples, offering improved training efficiency compared to score-learning methods that rely on unconditional samples. Could you provide a more quantitative comparison of the training times and computational resources needed for your method versus the score-matching approach from (Heng et al., 2022) and the adjoint method from (Baker et al., 2024a) in one or more of the experimental settings?\n\n2. The paper notes that the canonical score-matching loss involves inverting $\\sigma\\sigma^T$, which can be challenging for high-dimensional and hypo-elliptic diffusions. Could you elaborate on how your method avoids these particular challenges and how it may offer advantages in such scenarios?", "relation_to_broader_scientific_literature": "The proposed method aims to improve the efficiency of guided proposal-based simulation by replacing computationally expensive MCMC/SMC steps with a learned neural network, enabling fast, independent sampling. It also offers an alternative to score-learning methods that learn directly from conditional information, potentially leading to better performance in challenging scenarios like rare events.", "essential_references_not_discussed": "All relevant references are discussed.", "other_strengths_and_weaknesses": "Strengths: \nDemonstrates robustness across different diffusion specifications and conditioning scenarios.\n\nEffective in handling rare events and multimodal distributions.\n\nEnables efficient independent sampling after training, with a cost similar to the unconditioned forward process.\n\nScales well to relatively high-dimensional problems, outperforming MCMC-based methods.\n\nWeakness:\nAlthough the neural-guided bridge and the guided proposal demonstrated comparable performance, both methods could only capture part of the modes in a multimodal distribution resulting from specific initial conditions. The paper notes that, while the neural bridge might not recover all modes, it offers faster sampling as a trade-off. It also suggests that additional MCMC updates could be applied to the trained neural bridges for better-quality samples.\nThe \"FitzHugh-Nagumo model\" experiment (Section 5.3), which involved a rare event, also highlighted a limitation in capturing all modes. Although the reference-conditioned process was bimodal, the neural-guided bridge only sampled paths from one of the two modes.", "other_comments_or_suggestions": "The paper is well-written and easy to follow.", "questions_for_authors": "1. In the Brownian bridge and Ornstein-Uhlenbeck bridge experiments, the lower bound of the loss function L(θ) could be computed analytically and used as a benchmark. For more complex, non-linear examples where this is not feasible, how do you ensure that the trained neural network is performing optimally and not converging to a suboptimal solution?\n\n2. In Section 3.3, it is mentioned mention the importance of \"matching conditions\" for the linear auxiliary process, particularly for hypo-elliptic diffusions, to ensure P*≪ P. Could you elaborate on these conditions and explain how they were satisfied in the hypo-elliptic FitzHugh-Nagumo model discussed in Section 5.3?\n\n3. In the \"Multi-modality\" experiment (Section 5.2), both the neural guided bridge and the guided proposal were able to cover only part of the modes. Considering the mode-seeking nature of variational inference, did you explore any strategies, or could you suggest approaches, to better capture the full multimodality of the conditioned process without relying on multiple MCMC chains?\n\n4. Could you discuss the potential advantages or disadvantages of jointly learning $\\vartheta_\\theta$ and the parameters of the auxiliary linear process (e.g., $B$, $\\beta$, $\\sigmã$) using variational inference, as proposed for future work? What challenges do you anticipate in implementing this joint learning approach?\n\n5. Section 6 suggests extending the approach to conditioning on partial observations at multiple future times. Could you outline the key modifications needed in your current methodology to accommodate such conditioning scenarios?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ujld1U1VcN", "reviewer": "Reviewer_JEjz", "summary": "This paper introduces Neural Guided Diffusion Bridges, which added variational inference to Guided Proposal. \nNeural-guided Diffusion Bridges show bridge sampling without MCMC. They perform competitively in many experiments. They can handle rare events, which is very hard for other methods. \nThough such good performances, it has some limitations; it could become mode-seeking in multimodality, it occasionally samples from only a single mode in FitzHugh-Nagumo model's rare event, etc.\n\n\n## update after rebuttal\n\nThe rebuttal clarified my concerns, and I continue to see this paper as leaning towards acceptance.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Theoretical Claims**\n\nThe formulation was imprecise and we propose to reformulate it as follows:\n> If $\\theta_{\\rm opt}$ is a local minimizer of $L$ and $L(\\theta_{\\rm opt})=-\\log \\frac{\\tilde{h}(0,x_0)}{h(0,x_0)}$, then $\\theta_{\\mathrm{opt}}$ is a global minimizer. This implies from which we obtain $D_{KL}(\\mathbb{P}^{\\bullet}_{\\theta_{\\mathrm{opt}} | \\mathbb{P}^{\\star})=0$, from which we obtain $\\mathbb{P}^{\\bullet}_{\\theta_{\\mathrm{opt}}} = \\mathbb{P}^{\\star}$.\n\n**Experimental designs or analyses**\n\nWe cannot explicitly choose the target mode to sample from. Theoretically, the variational class $\\{\\mathbb{P}^{\\bullet}_{\\theta};\\theta\\in\\Theta\\}$ is large enough to contain the target $\\mathbb{P}^{\\star}$, and the global optimal $\\mathbb{P}^{\\bullet}_{\\theta_{\\mathrm{opt}}} = \\mathbb{P}^{\\star}$ can be guaranteed. \n\nHowever practically, the inherent non-convexity of the objective function induced by $G(s, x)$ predisposes $\\vartheta_{\\theta}$ to converge to local minima, as empirically evidenced by sample concentration within individual modes. However, directing the optimization trajectory within this complex landscape remains nontrivial, and we currently do not have a reliable method steer the process towards preferred local minima.\n\n**Questions for Authors**\n\nWe don't fully understand what you mean by \"It would be better if you could give proof or reasoning about the experiments' result.\". We kindly ask you to clarify so we can answer your question accurately.", "claims_and_evidence": "The claims are explicit using mathematical proof. Also, the claims are supported by appropriate experiments.", "methods_and_evaluation_criteria": "The methods are explicit. In addition, the authors use various examples to demonstrate their methods' competitive performances in the experiments part.", "theoretical_claims": "I have one question for lines 271~272. How could the authors use 'local minimizer' and 'lower bound' at the same time? So, what I want to know is that, is there any condition, we could get a lower bound by choosing a local minimizer?", "experimental_designs_or_analyses": "In lines 402-405, the authors say, \"The neural guided bridge samples paths from only one of the two modes, though the sampled paths appear very similar to the actual conditioned paths. \" \nQuestions \n1. Could we choose which modes to sample among the two modes?\n2. Is there any reason for that result? I want to know the authors' thoughts", "relation_to_broader_scientific_literature": "The contributions could be utilized for generative modeling, especially sampling.\n\nAlthough sampling is very important for the generative models, it needs a hard cost for accurate sampling. If the contributions would be helpful to overcome those limitations,", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "**Strengths**\n\nThis paper shows great math skills, resulting in neat development for the main idea.\n\n**Weaknesses**\n\nIt would be better to show a realistic example, such as generated outputs(like images, video, and text) and metrics related to those outputs.", "other_comments_or_suggestions": "No", "questions_for_authors": "Similar to **Experimental Designs or Analyses**,  is there any proof or reasoning for the experiments' result? \nI see only results and explanations, except for proof or detailed reasoning about the experiments' result provided in the paper.\nIt would be better if you could give proof or reasoning about the experiments' result.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DZkPhWZ7WM", "reviewer": "Reviewer_XLMC", "summary": "The paper considers the challenging and widely-applicable problem of conditioning a reference diffusion process to sample rare events or desired outcomes.   Building on \"guided proposal\" approaches which construct the conditioned process for a cleverly-chosen tractable process, the authors propose to learn an additional control drift, parameterized using neural networks and minimizing a stochastic optimal control or mode-seeking KL objective via backpropagation through trajectories and the reparameterization trick.\n\nThe authors demonstrate the efficacy of the proposed approach on simple linear examples, along with cell dynamics, a FitzHugh-Nagumo excitable system, and a stochastic landmark matching problem.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Claims and Evidence**\n\nThe guided bridge relies on training the neural network. Once trained, independent samples are obtained. The quality of the guided proposal depends much on the nonlinearity of the diffusion and number of pCN-steps required in the MCMC-algorithm. This is case dependent. For the performance evaluation, please refer to the table in our reply to Reviewer 8sR1.\n\n**Methods and Evaluation Criteria**\nInclusion of the guiding term from the guided proposal  is necessary to ensure the guided neural process will be a bridge. Without it, the neural net has to learn the unbounded term which behaves near the endpoint like $(v-x)/(T-t)$ (in case of a full observation and a uniformly elliptic diffusion). \n\n**Theoretical Claims**\n1. A proof that $E_t$ in Eq. (5) is a local martingale is given in Palmowski & Rolski: it follows from partial integration and involves Dynkin's martingale. We believe this to be out of scope for this paper as it heavily depends on stochastic calculus.  \n2. For Eq. (10): the ``weight'' is obtained by integrating $$\\frac{({\\mathcal{A}}-\\tilde{\\mathcal{A}})\\tilde{h}}{\\tilde{h}}=\\sum_i(b_i-\\tilde{b}_{i})\\frac{\\partial_i\\tilde{h}}{\\tilde{h}}+\\frac{1}{2}\\sum_{i,j} (a_{ij}-\\tilde{a}_{ij})\\frac{\\partial^2_{ij}\\tilde{h}}{\\tilde{h}}$$ over $[0, t]$.\n3. Eq. (11) and (12) are taken from Section 2 of Mider et al. \n\nDue to the space limitation, we can't include more details about 2 and 3 in this reply, but we are happy to proceed them in the conversation.\n\n**Experimental Designas or Analyses**\n\n* \"Cell dynamics\": Yes, this is true. Empirical results show multimodal marginal distributions emerge after $t=2.0s$ (Fig. 6), with neural-guided bridges evolving primarily unconditionally beforehand. Peak splitting coincides with intensified drift forcing toward $v$, matching guided proposal dynamics. Neither approach fully replicates the three unconditional sampling modes, though all three become distinct during $t=2.0–3.0s$ under weakening constraints – despite potential intermediate convergence failures toward $v$.\n\n**Supplementary Material**\n\nWe acknowledge the importance of additional experimental results, and will prioritize space allocation for key figures and tables while maintaining methodological rigor during the revision of the manuscript.\n\n**Essential References Not Discussed**\n\nWe agree that Proposition is known in the literature. However, we did not find another reference where this is spelled-out for our purposes in detail. We simply want to have one clean statement that explains the change-of-measure to $\\mathbb{P}^\\star$ for this particular $h$-function, in the setting of SDEs.\n\nThanks for pointing out the ``non-essential recent work'':\n1. Denker et al.'s diffusion framework (Proposition 2.2) shares our Proposition 3.2's $h$-function concept (termed \"generalized $h$-transform\"), though methodologically distinct through denoising score matching. We will add reference to contextualize prior appearances of this construct.\n2. While our focus remains on KL divergence's mode-seeking properties, we acknowledge log-variance divergence's potential benefits for mode collapse mitigation. This comparison will be expanded in future research.\n3. The adjoint matching method shows promising scalability for high-dimensional applications (images/point clouds). We plan to investigate adapting our objective function into this framework.\n\n**Other comments or suggestions**\n1. (a) We agree that it is useful to the reader to remind them of the definition of $b^\\circ$ in Eq. (6). There is no $\\tilde{b}^\\circ$ in our paper. \n\n    (b) We agree with your suggestion and will add a remark at the end of Section 3.2 to explain this. \n\n    Indeed, if $\\sigma$ is invertible, then for \n    $\\tilde{X}_t = \\sigma W_t$ transition densities $\\tilde{p}$ are Gaussian and \n    $$ \\nabla_x \\log \\tilde{p}(t,x; T,v) = (\\sigma\\sigma^T)^{-1} \\frac{v-x}{T-t}.$$ Therefore, the bridge has drift $(v-x)/(T-t)$. We added this to the text.\n2. We will clarify this in the revision of the manuscript. The following hierarchical model is meant: $$ v\\mid y\\sim q(v\\mid y), y\\sim p(T,y \\mid 0,x_0).$$ Here, $y$ is the parameter that gets the prior $p( T,y \\mid 0,x_0)$ assigned; $v$ is the observation. In this model, the likelihood can be written as $L(y; v)=q(v\\mid y)$.    \n3. We are unsure what is meant by the remark on $b(t,v)$. Regarding the reason it is useful for $d'<d$: we may only observe certain components of the diffusion, for example, if we have a 2 dimensional example and observe $v \\sim N(x_1, \\sigma^2)$, then the dimension of the observation is lower than that of the diffusion. The Fitzhugh-Nagumo example is an illustration of this. \n4. Yes, we will stress this in the revision and clarify the explanation given in lines 347-355. Indeed, for the guided proposal we always use MCMC with pCN-steps, contrary to the other methods.\n5. We think we have addressed this comment in the previous item.", "claims_and_evidence": "The method is well-justified and incorporates useful conditioning information via the guided proposal.   As mentioned below, I encourage the authors to clarify the implementation and performance differences between the guided proposal and the proposed neural-network improvement, as the guided proposal provides a strong baseline in most of the experiments considered.", "methods_and_evaluation_criteria": "The authors consider a range of applications for validating the efficacy of the proposed method.\n\n\nTo ablate the necessity of the guided proposal, it might be interesting to consider the Neural Guided Bridge learning from the base drift directly (i.e. no guided proposal).  While this may fail in more complex settings, it would be interesting to evaluate for the Brownian Bridge and OU-Process in Sec. 5.1.", "theoretical_claims": "I would like to see more explicit reasoning to justify Eq. 5 and 10-12, just to give the reader more intuition for (i) the evolution of the auxiliary h-function and the (ii) appearance of particular terms in Eq 10 (particularly the Hessian).   \n\nI have confirmed the correctness of of the proposed method.", "experimental_designs_or_analyses": "The authors compare to existing methods such as score matching and adjoint bridges, along with the vanilla guided proposal.   While score matching and adjoint bridge appear to struggle with simple examples, the authors probe examples where the neural guided bridge improves over the vanilla guided proposal.\n\n*Cell Dynamics*\n- For cell dynamics with multi-modality in the conditional paths, the neural guided bridge appears to beter match the marginals of the unconditioned process in Fig 7.   \n-Is this the right evaluation?  (assuming the Original entry is  obtain from the unconditional process).   One could imagine that some modes of intermediate marginals do not result in the process hitting the desired v.\n\n*FHN Model*\n- the neural guided samples in Fig 9-10 are somewhat mode-seeking, but do not produce samples which interpolate between the modes (as in the guided proposal).   Here the reference process samples are appropriately filtered according to consistency with $v$.", "relation_to_broader_scientific_literature": "The paper builds on the guided proposals of Schauer et. al 2017, Mider et. al 2021 using neural network control drifts.    The latter idea has been an active area of recent research in diffusion model finetuning and diffusion bridge literature, along with transition path sampling applications in computational chemistry.   However, these works often use the base drift of the original problem without including additional drift terms from the auxiliary, tractable bridge process.\n\nHence, I expect the paper to be interesting to the ICML community.", "essential_references_not_discussed": "The authors might consider more fundamental citations regarding Prop 3.2, which appears to be well known.   If I am missing technical conditions relevant to the setting the authors have in mind (which require this recent result), then these might be stated.   \n\n*Non-essential recent work:*\n- Denker et. al (NeurIPS 2024),  \"DEFT: Efficient Fine-tuning of Diffusion Models by Learning the Generalised h-transform\".   Of particular note  is the Stochastic control objective using Log Variance divergence (their conditional score matching training is outside the setting of this paper).\n- Seong et. al (ICLR 2025):  \"Transition Path Sampling with Improved Off-Policy Training of Diffusion Path Samplers\":   context of Transition Path sampling in computational chemistry using a control objective and Log Variance divergence\n- on the surface, Log Variance losses for stochastic control problems (i) leverage off-policy samples and (ii)  while this is an additional design choice, clever choice of exploration samples may mitigate mode-seeking behavior associated with the KL.   These considerations would be left for future work.\n\nThe authors might also be interested in recent work improving upon the adjoint method for backpropagation through trajectories (and references therein).\n- Domingo-Enrich et. al 2024, \"Adjoint Matching\"", "other_strengths_and_weaknesses": "See other comments", "other_comments_or_suggestions": "1) There is enough notation appearing that it might be useful to remind the reader of several aspects.\n- In Eq 15, it would useful to be able to denote $\\tilde{b}^{\\circ}$ as a conditioned process with reference drift given by the linear $\\tilde{b}_t$ in Eq 9, or simply rewrite the full form of Eq 15.\n- In reading the paper, I had to remind myself that the Brownian bridge \"prior information\" $\\frac{v-X_t^{\\bullet}}{T-t}$ was not an ad-hoc decision, but rather is captured by $\\tilde{b}^{\\circ}$ (even for $\\beta(t)=0, B(t) =0$ in choosing $\\tilde{b}_t$).   It might be useful to state that the auxiliary process / guided proposal produces this term.  \n\n\n\n2) I had trouble parsing $q(v|y)$ as the likelihood at first glance, and it is written as $\\ell(y|v) = q(v|y)$ in line 150L.   Please mention the desired role of $v$ before Prop 3.2 (or even provide the full Bayesian view).\n\n\n3) Why does it make sense to use $b(t,v)$ as conditioning at intermediate time points, given the fact that $r(s,x)=∇ \\log h(s,x) = ∇ \\log \\int p(T, y|t,x) q(v|y) \\nu(dy)$.   Further, why is this meaningful when $v \\in \\mathbb{R}^{d^\\prime}$ with $d^\\prime < d$?", "questions_for_authors": "4) Is the pCN used for experiments with the guided proposal?  I think this is difficult for the uninitiated reader to follow, particularly when it comes to Lines 347-355 R comparing the \n\n5) If MCMC vs. no-MCMC is a distinction between the guided proposal and neural guided proposal, then this should be more clearly emphasized to distinguish the presented method and demonstrate its efficacy over the guided proposal.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The reviewers agree that this is an interesting and solid paper addressing an important and topical problem. The consensus is that this would make a useful addition to ICML. The reviewers have made a number of suggestions to improve the clarity of the work, and the authors have also offered some changes/additional empirical results. Please go carefully through the discussion below and incorporate these changes before the camera-ready version is due.
Decision: Accept (poster)