Adjoint Schrödinger Bridge Sampler
Authors: Guan-Horng Liu
Jaemoo Choi
Yongxin Chen
Benjamin Kurt Miller
Ricky T. Q. Chen
Venue: NeurIPS 2025 oral ()
Keywords: Boltzmann distribution
diffusion sampler
Schrödinger bridge
Primary area: probabilistic_methods

Abstract: Computational methods for learning to sample from the Boltzmann distribution—where the target distribution is known only up to an unnormalized energy function—have advanced significantly recently. Due to the lack of explicit target samples, however, prior diffusion-based methods, known as _diffusion samplers_, often require importance-weighted estimation or complicated learning processes. Both trade off scalability with extensive evaluations of the energy and model, thereby limiting their practical usage. In this work, we propose **Adjoint Schrödinger Bridge Sampler (ASBS)**, a new diffusion sampler that employs simple and scalable matching-based objectives yet without the need to estimate target samples during training. ASBS is grounded on a mathematical model—the Schrödinger Bridge—which enhances sampling efficiency via kinetic-optimal transportation. Through a new lens of stochastic optimal control theory, we demonstrate how SB-based diffusion samplers can be learned at scale via Adjoint Matching and prove convergence to the global solution. Notably, ASBS generalizes the recent Adjoint Sampling (Havens et al., 2025) to arbitrary source distributions by relaxing the so-called memoryless condition that largely restricts the design space. Through extensive experiments, we demonstrate the effectiveness of ASBS on sampling from classical energy functions, amortized conformer generation, and molecular Boltzmann distributions. Codes are available at https://github.com/facebookresearch/adjoint_samplers

Adjoint Schrödinger Bridge Sampler
Guan-Horng Liu Jaemoo Choi?., Yongxin Chen², Benjamin Kurt Miller', Ricky T. Q. Chen
FAIR at Meta, "Georgia Institute of Technology, "Core contributors
Abstract
Computational methods for learning to sample from the Boltzmann distribution- where the target distribution is known only up to an unnormalized energy function- have advanced significantly recently. Due to the lack of explicit target samples, however, prior diffusion-based methods, known as diffusion samplers, often require importance- sepportanee-eiighed estimation or complicated learning processes. Both trade off scalability with extensive evaluations of the energy and model, thereby limiting their practical usage. In this work, we propose Adjoint Schrödinger Bridge Sampler (ASBS), a new diffusion sampler that employs simple and scalable matching- based objectives yet without the need to estimate target samples during training. ASBS is grounded on mathematical model- the Schrödinger Bridge-which wwich enhances sampling efficiency via kinetic-optimal transportation. Through a new lens stochastic optimal control theory, we demonstrate how SB-based diffusion samplers can be learned at scale via Adjoint Matching and prove convergence to the global solution. Notably, ASBS generalizes the recent Adjoint Sampling (Havens et al., 2025) to arbitrary source distributions by relaxing the so-called memoryless condition that largely restricts the design space. Through extensive experiments, we demonstrate the effectiveness of ASBS on sampling from classical energy functions, amortized conformer generation, and molecular Boltzmann distributions. Codes are available at https: //gi thub. wtw.ttt.....o.eeeeeeeeteeteeeete
adj ointt_sttt_eeose paanttssmmless..._
Introduction
Sampling from Boltzmann distributions is fundamental problem in computational science, with widespread applications Bayesian inference, statistical physics, and chemistry (Box and Tiao, 2011: I: Binder et al., 1992; Tuckerman, 2023). Mathematically, we aim to sample from a target distribution v(x) known up to a unnormalized, often differentiable, energy function E(x) X' C Rd + R,
F(x))
where 7:= - B(x)d dr
v(x)
Z
x
an intractable normalization constant. For instance, the energy funetion F(xx) of a molecular system quantifies the stability of chemical structure based on the 3D positions of particles. A lower energy indicates a more stable structure and hence higher likelihood of its occurrence, i.e., v(x) x e BB())
Classical methods that generate samples from v(x) rely Markov Chain Monte Carlo algorithms, which run a Markov chain whose stationary distribution is v(x) (Metropolis et al., 1953; Neal, 2001; Del Moral et al., 2006). These methods, however, tend to suffer from slow mixing time and require extensive evaluations of energy function, limiting their practical usages due to prohibitive complexity.
To improve sampling efficiency, modern samplers focus on learning better proposal distributions (Noé et al., 2019; Midgley et al., 2023). Among those, recent advances in diffusion-basse generative models (Song et al., 2021; Ho et al., 2020) have given rise to a family of Diffusion Samplers, which
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Table I: Compared to prior diffusion samplers, Adjoint Schrödinger Bridge Sampler (ASBS) offers the most flexible design for diffusion samplers (2), while learning the drift un, via scalable matching objectives that do not rely on computation of importance weights (IWs).
Design condition for (2)
Learning method for of
Method
Non-memoryless Arbitrary prior Matching objective' No reliance on IWs
PIS (Twvng and Ches 2007y DDS Niegs al, 2013)
x
x
LV-PIS & LV-DDS (Ritee Berser 2009)
x
x
X
PDDS miliiga 2000 iDEM (oihorol 2004) AS (Havens et al., 2025)
X
/
X
/
Sequential SB eemmnn 2019)
/
X
X
Adjoint Schrödinger Bridge Sampler (Ours)
V
consider stochastic differential equations (SDEs) of the following form:
dx; = [X((X,) + o.u dt + o,dW,, Xo ~ (Xo,,
(2)
where f((x): : [0. 1] x X' X' the base drift, o, -] the noise schedule, and uux) the oz. initial source distribution. Given (fi, Tz. p),, the diffusion sampler learns parametrized drift uf (x) transporting samples to the target distribution vx(x) at the terminal time t = 1.
Computational methods for learning diffusion samplers have grown significantly recently (Zhang and Chen, 2022; Vargas et al., 2023; Berner et al., 2024; Chen et al., 2025). Due to the distinct problem setup in (1), the target distribution is defined exclusively by its energy E(x),, rather than by explicit target samples. This characteristic renders modern generative modeling techniques for scalability- particularly the score matching objectives! applicable. As such, prior matching- based diffusion samplers (Phillips et al., 2024; Akhound-Sadegi et al., 2024; De Bortoli et al., 2024) often require computationally intensive estimation of target samples via importance weights (IWs).
Recently, Havens et al. (2025) introduced Adjoint Sampling (AS), new class of diffusion samplers whose matching objectives rely only on on-policy samples, thereby greatly enhancing scalability. By incorporating stochastic optimal control (SOC) theory (Kappen, 2005; Todorov, 2007), AS facilitates the use of Adjoint Matching (Domingo-Enrich et al., 2025), a novel matching objective that imposes self-consistency in generated samples, effectively eliminating the needs for target samples.
The efficiency of AS, however, is achieved through a specific instantiation of the SDE (2) to satisfy the so-called memoryless condition. This condition-formalll discussed in Section 2-restricts its source distribution be Dirac delta ((x:) 6, precluding the use of common priors such as Gaussian or domain-specific priors such as the harmonic oscillators in molecular systems (Jing al., 2023). Notably, the memoryless condition underlies all previous matching- diffusion samplers, restricting the design space of (2) from other choices known to enhance transportation efficiency (Shaul et al., 2023). While the condition has been relaxed in non-matching-base at extensive computational complexity (Richter and Berner, 2024; Bernton et al., 2019), no existing diffusion sampler-to to our best understanding has successfully combined matching objectives with non-memoryless condition. Table 1 summarizes the comparison between prior diffusion samplers.
methods
In this work, we propose Adjoint Schrödinger Bridge Sampler (ASBS), a new adjoint-mathhing- based diffusion sampler that eliminates the requirement for memoryless condition entirely. Formally, ASBS recasts learning diffusion sampler as a distributionally constrained optimization, known as the Schrödinger Bridge (SB) problem (Schrödinger, 1931, 1932; Léonard, 2013; Chen et al., 2016):
base) = Ex-p' U. 1llu (x:))
min DKL (p bla00
(3a)
s.t. dx,, - (
dt o,dW,
Xu ~ p(Xo).
X1 ~ x(x)))))
(3b)
Here, nd denotes the path distribution induced by the SIDE in (3b), whereas passe == n== denotes the path distribution induced by the "base" SDE when u, == 0. By minimizing their KL divergence, the SB problem (3) seeks the kinetic-optimal drift u an optimality structure well correlated
'The matching objective is simple regression loss, ElI (Xx) ex, X1)lP, w.r.t some tractable et.

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with sampling efficiency in generative modeling (Finlay et al., 2020; Liu et al., 2023). Since the SOC problem in AS corresponds specific case of the SB problem with (fr: p)) := (0. 8). ASBS extends AS to handle non-memoryless conditions by solving more general SB problems (see Theorem 3. Computationally. ASBS retains all scalability advantages from AS by utilizing an adjoint- objective that removes the need for estimating target samples. It also introduces corrector- corrector-matching matching objective to correct nontrivial biases arising from non-memoryless conditions. We prove that alternating optimization between the two matching objectives is equivalent to executing the Iterative Proportional Fitting algorithm (Kullback, 1968), ensuring global convergence of ASBS to of (see Theorem 3.2). Though extensive experiments, show superior performance of ASBS over prior diffusion samplers actoss various benchmarks on sampling multi-particle energy functions.
In summary, we present the following contributions:
We introduce ASBS, an SB SB-based diffusion sampler capable of sampling target distributions using only unnormalized energy functions, by solving general SB problems with arbitrary priors.
We base ASBS on new SOC framework that removes the restrictive memoryless condition, de- velop a scalable matching-
algorithm, and prove theoretical convergence to global solution.
We show ASBS's superior performance over prior methods on sampling Boltzmann distributions of classical energy functions, alanine dipeptide molecule and amortized conformer generation.
2 Preliminary
We revisit the memoryless condition introduced by Domingo-Enrin et al. (2025) and examine its impact the constructions of SOC-based diffusion samplers (Zhang and Chen, 2022; Havens et al., 2025), which are closely related to our ASBS. Additional review can be found in Appendix A.
Stochastic Optimal Control (SOC)
The SOC problem (4) studies an optimization problem:
V.
min E X
1 (X.))l + 9(Xi)
s.t. (2),
(4)
which, unlike the SB problem (3), includes additional terminal cost g(z) X R at the terminal time = and considers the SDE without the terminal constraint X1 ~ V. The primary reason for studying this specific optimization problem is that the optimal distribution is known analytically by²
p" (Xo. = hrpp"""s( (Xo. X1)e +(X)))+o(X.)) where Vo(x) log
Pilo (ylæye
base
-a()) -a(td dy (5)
is the initial value function. That is, the optimal distribution p" is an exponentially tilted version of the base distribution, p base ;= p===0, Specifically, p base is tilted by the terminal cost "Xx("")" and the initial value function Vo (Xo). which is intractable. Consequently, to ensure its marginal p" (X1) follows the target distribution v(X)), we must climinate the initial value function bias from (X()..
Memoryless condition & SOC-based diffusion sampler A common approach to climinate the aforementioned initial value function bias, adopted by most diffusion samplers, is to restrict the class of base processes to be memoryless. Formally, the memoryless condition assumes statistical independency between Xo and X, in the base distribution:
base (Xo. X1) memoryless
p base
(Xu) buse (X1).
(6)
This memoryless condition (6) simplifies the optimal distribution at the terminal time t and, upon choosing a proper terminal cost (I)), recovers the target distribution v,
pz (X1) memoryless
[ | (Xo)p base VoX)) d.xo x p
base (Xi)e x((x)) v(Xi),
where the last equality is due to setting the terminal cost to g(x) = log memoryless condition (6) is enforced by careful design of the base distribution p hasse or, equivalently.
(a)
wia) Typically, the
(c)
Equation (5) can be obtained by rewriting (4) (Ku(D"Ipp"T) + Epy [o(X))) and then computing 2(x1)
the analytic solution (Xo) x (X)] -o)(X)))))( x((1) and normalization / (Xi -
(Xo) See Appendix A.1 for details.
3

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Memoryless SOC
Non-Memoryless SOC
ASBS (ours)
Terget vix)
time X,
Source X - -A)
Source X₂ N(O,
Source X -A00, 1)
Figure I: Effect the memoryless condition on learning SOC-based diffusion samplers. We consider Gaussian prior p(x) == (2; 0, 1) with (ft,a,) set to VP-SDE for the first plot and (0, 0.2) for the rest, see Appendix A. for details. The memoryless condition injects significant noise (left) correct the otherwise biased optimization (middle), whereas ASBS can successfully debias any non-memoryless processes (right).
the parameters (fe. tt (4) in (2). For instance, the variance-preservee process (VP; Song et al., 2021) considers linear base drift ft, noise schedule a, that grows significantly with time, and Gaussian prior μu; see Figure 1. Alternatively, one could implement (6) with Dirac delta prior μ(x) = (z)
and ft == 0, leading the following SOC problem (Zhang and Chen, 2022):
min E Ex V. 1 (Xi))l + log
base (X1)
s.t. dX,, ox a,W,, Xo-0. (7)
v(Xi)
Based on the aforementioned reasoning, solving (7) results in a diffusion sampler that transports samples the target distribution t=1, with Adjoint Sampling (Havens al., 2025) the only scalable method of this class. Despite encouraging, the SOC problem in (7) is nevertheless limited by
its trivial source, precluding potentially more effective options for sampling Boltzmann distributions.
Adjoint Schrödinger Bridge Sampler
We introduce new diffusion sampler by solving the SB problem (3). where the target distribution (=x) is given by its energy function (xx) rather than explicit samples. All proofs are left in Appendix B.
3.1 SOC Characteristics of the SB Problem
The SB problem (3)-as an optimization problem with distribution constraints-is widely explored in optimal transport, stochastic control, and recently machine learning (Léonard, 2012; Chen et al., 2021; De Bortoli et al., 2021). Its kinetic-optimal drift u* satisfics the following optimality equations:
(e(())= = frhp base
Pije (ulz) (o(x) p(x)
(8a)
(()) = or V log (I(),,
where
(c(x)) =
buse Pejo (#lu) (1())(((x) =vx)
(8b)
and Paj (y|æ)) = hase (X.=ylX,=x) is the transition kernel of the base process for observing at
time given x at time s. The SB potentials (p)( (x) E C²22((0, 1), Rd) are then defined (up to
some multiplicative constant) solutions forward and backward time integrations w.r.t. Pr]s
buse has an analytical solution Equation (8) are computationally challenging to solve- even when Pils
due to the intractable integration and coupled boundaries at / = and 1. Our key observation is that the first equation (8a) resembles the optimality condition of the SOC problem (4) (see Appendix A. This implies that the optimality conditions of SB hints an SOC reinterpretation. which, as we will demonstrate, is more tractable than solving (8) directly. We formalize our finding below.
Theorem 3.1 (SOC characteristics of SB). The kinetic-optimal drift is in (8) solves an SOC problem
V/ I1 (l log (x1)
min E App
s.l. (2).
v(Xi)

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Theorem 3.1 suggests that every SB problem (3) can be solved like an SOC problem (4) with the terminal cost g(z) == log ex(s), Comparing to the formulation Adjoint Sampling (Havens et al.. 2025), the two SOC problems, namely (7) and (9), differ in their terminal costs costs-where pl is replaced by ف1 and the relaxation of the source distribution from Dirac delta Xo source (Xxo)
general
How (.) debiases non-memoryless SOC problems Taking closer look at the effect of Φ1. notice that the optimal distribution of the SB problem-accorteee according to Theorem 3.1 and follows
p Xi) base (Xo,
"Xp exp (-log X1)
(x(X1) v(X1)
log (o(()))
(10)
where "l yo " the equivalent initial value function. One can verify that the marginal at the terminal time t = 1 indeed satisfies the target distribution,
px(()) = (
(10) v(Xi)
P1(X1] P hase (Xo: X1) Po(X)) dXo
(11)
(80 ((X))
hase (XilXo))o(X)))dl
(8b) v(X)).
(1
That is, the optimality equations in (8), in their essence, construct a specific function (.(() that eliminates the initial value function bias associated with any non-memoryless processes, thereby ensuring that the optimal distribution satisfies the target v at
3.2 Adjoint Sampling with General Source Distribution
We now specialize Theorem 1. to sampling Boltzmann distributions (I), where vxx) e- and hence the terminal cost of the new SOC problem in (9) becomes log (r(=) E(x) + log & (1) To encourage minimal transportation cost (Chen and Georgiou, 2015; Peyré and Cuturi, 2017), we consider the Brownian-moticn base process with a degenerate base drift ft := 0. Applying Adjoint Matching (AM; Domingo-Enrich et al., 2025) to the resulting SOC problem leads to
u* arg min paper
18,, [1l,,
+ a EVE- + Vlog (1) (X:]I ū stopgrad(u). (12)
Note that the AM objective in (12) functions as a self-consistency loss-in that both the regression and its expectation depend on the optimization variable u. This makes (12) particularly suitable for learning SB-based diffusion samplers, unlike previous matching- Liu et al., 2024), which all require ground-truth target samples from X1 ~~ v.
SB methods (Shi et al., 2023;
Computing the AM objective in (12) requires knowing V log (z), which, as we discussed (11), serves corrector that debiases the optimization toward the desired target. Notably, this corrector function V log (z) also admits a variational form (Peluchetti, 2022, 2023; Shi et al., 2023):²
V log क arg min P81 [[))(X,) Va, log huve (X [ |
(13)
To summarize, Equations (12) and (13) characterize two distinct matching objectives that any kinetic- optimal drift on of SBs must satisfy. When the source distribution degenerates to Dirac delta X, == (13) is minimized at V log pase, and (12) simply recovers the objective used in Adjoint Sampling (Havens et al., 2025). In other words, (12) and (13) should be understood as generalization of Adjoint Sampling to handle aubirray-includdng non-memoryless-Se source distributions.
3.3 Alternating Optimization with Adjoint and Corrector Matching
Building upon the theoretical characterization in Section 3.2, aim to design a learning algorithm that finds a diffusion sampler satisfying (12) and (13), which correspond to two simple matching- based objectives. However, these matching objectives cannot be naively implemented due to their interdependency: Solving (12) for the kinetic-optimal drift u² requires knowing V log Φ1. Like- wise, solving (13) for the corrector function log & requires samples from u*. We relax the interdependency with an alternating optimization scheme. Specifically, given an approximation of
v log & R (k-1) from the previous stage k - 1, we first update the drift y (k) with the AM objective:
Formmall. V log (x) is the kinetic- kinetic-optimal drift along the reversed time coordinate := 1 f, and (13) is its variational formulation, i.e. the Markovian projection 0; see Appendix A.2 for details.
5

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Algorithm Adjoint Schrödinger Bridge Sampler (ASBS)
Require: Sample-able source Xo μl, differentiable energy E(x), parametrized uo(l, x)) and ha(a)
1: Initialize (0) :-
2: for stage l: in ..... do
3: Update drift ug (k) by solving (14)
> adjoint matching
4: Update corrector h (k) by solving (15)
corrector matching
5: end for
ASBS Generated Samples (upper) and Grad. of Terminal Cost (bottom) at each Stage K
Target Samples y exol-E)
202
Av3
2u:0 (fiww)
X Sar
AM
CM
Energy Grad -Q m QE
TE 20
Figure 2: Illustration of ASBS 2D example. By alternatively minimizing the Adjoint Matching (AM) objective (14) and the Corrector Matching (CM) objective (15), ASBS progressively learns better corrector A that debiases the SOC problem for the control (h) Note that since the corrector is initialized with N ; O, the first AM stage simply regresses whth to the energy gradient VE, E.
u (k) = arg min E
pepe P8.11
[l,x(X,) + 0e V.VVV (h 1((=-))(x))11
u = stopgrad(u). (14)
Then, we use the resulting drift (R) to update (A) by minimizing the following matching objective, which-in light of the corrector role of V log ś0 -- we refer as the Corrector Matching objective:
(k) = arg min (.))) [[I((X,)
Vzz log hase (X11X))113).
(15)
Po.1
Equation (15) should be distinguish from the tridge-matching objectives in data-driven SB methods (Shi et al., 2023; Somnath al., 2023), where X; must be drawn from the target distribution v. In contrast, the matching objectives in (14) and (15) depend only on model samples at the current stage X1 p e
Xo), hence can used to learn SB-based diffusion samplers at scale.
The alternating optimization between (14) and (15) creates sequence of updates, (u("), ()) (k(k), (k)) ...., that may be thought of as running coordinate descent between the control and the corrector h. Intuitively, each stage k first find the control (k) that best aligns with the corrector from previous stage, (k=1),, then update the corrector (a) accordingly to reflect the "memorylessness" of the current control (8). We summarize our method, Adjoint Schrödinger Bridge Sampler (ASBS), in Algorithm 1, while leaving the full details with additional components, such replay buffers, in Appendix C. Finally, prove that this alternating optimization indeed converges to the kinetic-optimal drift ut in (8).
Theorem 3.2 (Global convergence of ASBS), Algorithm 1 converges to the Schrödinger bridge solution of (3), provided all matching stages achieve their critical points, i.e..
lim u (k) u.. k0000

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Theoretical Analysis
We provide the proof of Theorem 3.2 and highlight theoretical insights throughout. While ASBS is specialized to a degenerate base drift ft := 0, all theoretical results here apply to general ft. To simplify notation, we omit the parameters 0, Φ and reparametrize the corrector by h(4) V log (k))
All proofs are left in Appendix B.
Our first result presents a variational characteristic to the solution of the AM objective in (14).
Theorem 4.1 (Adjoint Matching solves a forward half bridge). Let pa be the path distribution induced by the drift (*) in (14) at stage k. Then, pd solves the following variational problem:
pult) aargmin (DKL(Plla") 5:th (ilgi"""")) " : 11):
(16)
where g²' ( is the path distribution induced by a "backward"" SDE on the reversed time coordinate 1 defined by the corrector from the previous stage (k(k-))
dY, --f,)_ +o Vog (Y.)) ds + a.dWs, 22(1)
base (0
(17)
with the boundary conditions Yo ~ and (y(y) (1)
Theorem 4.1 suggests that any SOC problems with the terminal cost g(x) = log "(a) can be reinterpreted as KL minimization w.r.t. specific backward SDE (17) that is fully characterized by v-which serves its source distribution-and (((4)hhick defines its drift through the function (v). The objective in (16) differs from the one in the original SB problem (3) by disregarding the target boundary constraint, X1 Consequently, (16) only solves a forward half bridge.
Next, we show that the CM objective (15) admits similar variational form, except backward in time. Theorem 4.2 (Corrector Matching solves a backward half bridge). Let h() be the corrector in (15) at stage Then, the path distribution
g/()) solves the following variational problem:
arg min {DKL(P" (K)
114) 91 v²
(18)
Unlike (16), the objective in (18) disregards the source boundary constraint y instead, thereby solving a backward half bridge. Theorems 4.1 and 4.2 imply that our ASBS in Algorithm implicitly employs an optimization scheme that alternates between solving forward and backward half bridges. thereby instantiating the celebrated Iterative Proportional Fitting algorithm (IPF; Fortet, 1940; Kullback, 1968). Combining with the analysis by (De Bortoli al., 2021) leads our final result in Theorem 3.2.
5 Related Works
We provide additional clarification on SB-related works and leave the full review to Appendix A.3.
Data-driven Schrödinger Bridges The SB problem has attracted notable interests in machine learning due to its connection diffusion-basee generative models (Wang et al., 2021). Earlier methods implemented classical IPF algorithms (De Bortoli et al., 2021; Vargas et al., 2021; Chen et al., 2022), with scalability later enhanced by bridge matching- based methods (Shi et al., 2023; Liu et al., 2024). Unlike ASBS, all of them focus on generative modeling and assume access to extensive target samples during training, making them unsuitable for sampling from Boltzmann distributions.
SB-inspired Diffusion Samplers Notably, in the context diffusion samplers, the SB formulation has been constantly emphasized a mathematically appealing framework for both theoretical analysis and method motivation (Zhang and Chen, 2022; Vargas et al., 2024; Richter and Berner, 2024; Havens al., 2025). None of the prior methods, however, offers general solutions to learning SB-based diffusion samplers, instead specializing to either the memoryless condition or non- matching based objectives, which largely complicate the learning process (see Table 1). Conceptually, our ASBS stands closest to SSB (Bernton et al., 2019) by learning general SB samplers. However, the two methods differ fundamentally in scalability: SSB is a Sequential Monte Carlo-based method (Chopin, 2002) augmented with learned transition kernels using Gaussia--aproximated SB potentials. As with many MCMMCauugmnted samplers (Gabrié al., 2022; Matthews et al., 2022), SSB requires extensive evaluations on the energy E(x), in contrast to ASBS, which is much more energy-efficient.
7

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Table 2: Results on the synthetic energy functions for n- n-particle bodies with their corresponding dimensions d. Following (Chen et al., 2025; Havens et al., 2025), we report Sinkhorn for MW-5 and the Wasserstein-2 distances w.r.t samples, W2, and energies, E1,) for the rest. All values are averaged over three random trials. Best results are highlighted.
MW-5 (d=5)
DW (d = 8)
LJ-13 (d 39)
LJ-55 (d= 165)
Method
Sinkhorn +
W2
B(-) W2 4
W22 4
E(-) W2
W2 4
B(-) W2
PDDS (Phillips et al. 2024)
0,92
0,58
4,66
56.01
SCLD (Chen et al., 2025)
0.44
g.30
a.40
2.93
27.98 1:28
PIS Whang and Chen, 2022)
0.65 =0:2
0.68
0,65
1,93
18.02
L12
4.79
228.70
10127
DDS (Nargas et al,, 2003)
0.63-024
0.92
0.90+au77
1.99
24.61 + 8.99
4.60
173.09 18.01
LV-PIS
1.04_.0
1..9
iDEM (Alloual Salezh et il., 2024)
0. a.70
0.55
1.61 1.61=0.0
30.78
4.69
93.53 16.31
AS (laavess 2025)
0.32
0.62 0.62
0.55
1.67=00m
2.40 125
4.04
30.83
ASBS (Ours)
0.15 o0002
0.43
0.20
1.59
1.99
1.
4.00
28. 20: 8.35
Nomoltze Dersiy
DW-4
U:11
Complexity par Update
ASSS joural Greand
1
266
-22
-08 Energy EC0
-.4
-2
45
-3
100
1.5
Energy EUo
Average KFE
Figure 3: The energy histograms of DW-4 and LJ-13 from Table 2. ASBS generates samples whose energy profiles closely match those of the ground-truth samples.
Figure 4: Complexity w.r.t. the number of function evaluation (NFE) on LJ-13 potential.
Experiments
Benchmarks
We evaluate our ASBS on three classes of multi-particle energy functions E(x).
Synthetic energy functions These are classical potentials based on pair-wise distances of an n-particle system, where E(x) is known analytically. Following (Akhound-Sadegh) et al., 2024; Chen et al., 2025), we consider 2D 4-particle Double-Wel. potential (DW-4), 1D 5-particle Many-Wes Well potential (MW-5), 3D 13-particle Lennard-Jones potential (LJ-13) and a 3D 55- particle Lennard-Jones potential (LJ-55). For the ground-truth samples, sample analytically from MW-5 and use the MCMC samples from (Klein et al., 2023) for the rest of three potentials.
Alanine dipeptide This is molecule consisting of 22 atoms in 3D. Specifically, we consider the alanine dipeptide in implicit solvent and aim to sample from its Boltzmann distribution at temperature 300K. Following prior methods (Zhang and Chen, 2022: Wu et al., 2020), we use the energy function E(x) from the OpenMM library (Eastman al., 2017) and consider more 60. The ground-trute -tuuh samples contain 10? configurations, simulated from Molecular Dynamics (Midgley al., 2023).
structural internal coordinate with the dimension
Amortized conformer generation Finally, we consider new benchmark proposed in (Havens et al., 2025) for large-scale conformer generation. Conformers are locally stable configurations located at the local minima of the molecule's 's potential energy surface (Hawkins, 2017). Sam- pling conformers is essentially a conditional generation task, targeting a Boltzmann distribution v(zlg) & - E(*lo) at a low temperature т < 1, conditioned on the molecular topology E G. The training set Gunin contains 24,477 molecular topologies from SPICE (Eastman al., 2023), represented by the SMILES strings (Weininger, 1988), whereas the test set Grest contains 80 topologies from SPICE and another 80 from GEOM-DRUGS (Axelrod and Komez-Bombarellt 2022). As with (Havens et al., 2025), we consider Exx g) foundation model eSEN from (Fu al., 2025), which predicts energy with fumssty-functtioal-tie computational cost. We use CREST conformers (Pracht et al., 2024) as the ground-truth samples.
accuracy at a much lower
Baselines and evaluation We compare ASBS with a wide range of diffusion samplers, including PIS (Zhang and Chen, 2022), DDS (Vargas et al., 2023), PDDS (Phillips et al., 2024), SCLD (Chen
8

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Table Comparison between diffusion samplers on sampling the molecular Boltzmann distribution of the alanine dipeptide. We report the KL divergence DkI. for the ID marginal across five torsion angles and the Wasserstein-2 W22 on jointly (é) v), known as Ramachandran plots (see Figure 5).
Best results are highlighted.
D&L each torsion's 's marginal
W/ on joint 1
Method
21
%/
(e, 19)
PIS (Zhang and Chen, 2022)
0.38
5.61
4.49:0003
4.60
1.27
DDS (Vargus et al., 2023)
0.03++0.01
0.16+
2.44:0006
0.03
0.03:00.00
0.68
AS (Havens et al., 2025)
0.09
0.04
0.17
0.56=0000
0
0.65
ASBS (Ours)
0.02
0.01
0.03 0000
0.02 :0.00
0.02
0.25
Table 4: Results on large-scale amortized conformer generation, evaluated on two test sets, SPICE and GEOM-DRUGS, both with and without and Absolute Mcan RMSD (AMR) of the recall at the threshold 1.0A. Note that RDKD warmup" refers to warm-starting the model ug using RDKit conformers; see Appendix D for details. Best results without and with RDKit warm-up are highlighted separately.
relaxation. We report the coverage (%)
without relaxation
with relaxation
SPICE
GEOM DRUGS
SPICE
GEOM DRUGS
Method
Coverage
AMR 4
Coverage
AMR 1
Coverage
AMR 4
Coverage
AMR
RDKit ETKDG
56, aoo
1.04
50.81
1.15
70.21
0.79
62.55
a,93
AS (Havens et 2025)
56.75
0.96
36.23 13
1.20.so
g254
0.68
64.26 (3457
0.89
ASBS w/ Gaussian prior (Ours)
73.04
0.83.
50,23
1.05 0
88.26 12:57
0.60
72.32 20.0
0.77.no
ASBS w/ harmonic prior (Ours)
74.05 131.
0.82 6:23
53,14
1.03 c
88.71 1i
0.59 ..3
72.77
0.78 1006
AS +RDKit warmup diamm si. 2029
72.21 soum
0.84
$2.19
1.02
$7.84=1133
e.60
73.88
0.76
ASBS +RDKit warmup (Ours)
77.84
0.79 -023
57.19 000.14
0.98
88.08 11:34
0.58 -...-
73.18
0.76
et al., 2025), LV (Richter and Berner, 2024), iDEM (Akhound-Sadeg) et al., 2024) and finally Adjoint Sampling (AS; Havens et al., 2025). For the conformer generation task, we include additionally domain-specific baseline, RDKit ETKDG (Riniker and Landrum, 2015), which relies on chemistry- based heuristics. The evaluation pipelines are consistent with prior methods, where adopt the SCLD setup for MW-5, the PIS setup for alanine dipeptide, and the AS setup for all the rest; see Appendix D for details.
ASBS models For all tasks, we consider degenerate base drift ft := O, as discussed in Section 3.2, and set d3 a geometric noise schedule. For energy functions that directly take particle systems as inputs- suuut--suchaa as DW, LJ, and eSEN we parametrize the models ug, h& with two Equivariant Graph Neural Networks (Satorras et al., 2021) and consider a domain-specific source distribution-the harmonic prior (Jing et al., 2023). Formally, for an n-particle system x = (a; { 1= the harmonic prior /hharmonic (I) is a quadratic potential that can be sampled analytically from an anisotropic Gaussian:
Pharmonic (I:) x exp(- - Ex, ||,; 12).
(19)
For other energy functions, we use standard fully-connected neural networks and consider Gaussian priors. All models are trained with Adam (Kingma and Ba, 2015) and, following standard practices (Havens et al., 2025; Akhound-Sadee Sadegh et al., 2024), utilize replay buffers; see Appendix C for details.
Results Table 2 presents the results on synthetic energy functions. Notably, ASBS consistently outperforms prior diffusion samplers across all energy functions. In Figure 3, we compare the energy histograms of DW-4 and IJ-13 potentials between the ground-truth MCMC samples and those from ASBS. It is evident that ASBS generates samples that closely resemble the target Boltzmann distribution v(x) x e E(x), resulting in energy profiles E(x) that are almost indistinguishable from the ground truth. Computationally. Figure 4 shows the average number of evaluation required on the energy E(x) and the model (t, 2) for each gradient update. ASBS is much more efficient than most diffusion samplers, with slight overhead compared AS due to the additional network ha(x))
Table 3 summarizes the results for alanine dipeptide. Following standard pipeline (Zhang and Chen, 2022), we generate model samples X1 E 1200 and extract five torsion angles-inccludngg the backbone

---

Ground
X
X1
Figure 6: Example of ASBS generative process on amortized conformer generation. Given unseen molecular topology E Grett from the test Set-COCSc1sc2cccoo this case case-ASBS transports samples from the harmonic prior Xo Aharmonic to generate conformers X1.
Figure 5: Ramachandran plots
[n+] [0-] in
for the alanine dipeptide between ground-trute couund-truth and ASBS samples.
SPICE
SPICE relox
GLOM DRUG
GLOM-DRUG relax
100
Covarge Rocal
60-
50
60
40
40
00
65.55 01055
Formiss
0.5
2.2
3.5
2.0
0.5
2.5
3.5
2.0
0.5
1.0
1.5
2.0
0.5
1.0
1.5
Threshold (Areshoiddknngsriii
Threshold (Argström)
Thresholc (Angstrim)
Threshold (Angetnim))
Figure 7: Recall coverage curves amortized conformer generation the SPICE and GEOM- DRUGS test sets without RDKit warm-start. Note that Table reports the recall coverages at the threshold 1.0A.
angles d, 1 and methyl rotation angles "/, 72. 73 all of them exhibit multi-moda distributions. Notably, ASBS achieves lowest KL divergence to the ground ground-truth marginals across all five torsions. Figure 5 further compares the joint distributions of (ò, v), known as the Ramachandran plots (Spencer et al., 2019), between ground-truth and ASBS. While ASBS identifies all high-density modes in the region Φ E [-x, -x,00, 0]. it misses few low-density modes. This mode-seeking behavior, inherit in all SOC-based diffusion samplers, could be improved with important weighting. We provide further discussions in Appendix D.4.
Table 4 presents the recall for amortized conformer generation compared to ground-truth samples. For prior diffusion samplers, we primarily compare AS (Havens al., 2025) due to the benchmark's scale. Following AS, we ablate a warm-start stage using RDKit conformers, which are close but not identical to ground-trute truth samples, and include results with relaxation for post-generation optimization. Since AS is a specific instance of ASBS with a Dirac delta prior- as discussed in Section 3.2 any performance improvements from AS ASBS highlight the added capability to handle arbitrary priors and, consequently, non-memoryless processes. Remarkably, without any warm-start, ASBS with the harmonic prior (19) already matches and, in many cases, surpasses the RDKit-warm-up AS. With warm-start, ASBS achieves best performance across most metrics. This highlights the significance of domain-speciftt priors, aiding exploration as effectively as warm-start with additional data, which may not always be available. Finally, we visualize the generation process of ASBS with harmonic prior (19) in Figure 6 and report the recall curves Figure 7. practice, we observe that ASBS achieves slightly better results with harmonic prior compared to Gaussian prior, with both significantly outperforming AS (Havens et al., 2025). See Appendix D.4 for further ablation studies.
7 Conclusion and Limitation
We introduced Adjoint Schrödinger Bridge Sampler (ASBS), a new diffusion sampler for Boltz- mann distributions that solves general SB problems given only target energy functions. ASBS is based on a scalable matching framework, converges theoretically the global solution, and performs superiorly across various benchmarks. Despite these encouraging results, further enhancement with importance sampling techniques is worth investigating to mitigate the mode collapse inherent in SOC-inspired diffusion samplers. Exploring its effectiveness in sampling amortized Boltzmann distributions would also be valuable.
10

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Acknowledgements
The authors would like thank Aaron Havens, Juno Nam, Xiang Fu, Bing Yan, Brandon Amos, and Brian Karrer for the helpful discussions and comments. JC and YC acknowledge support from NSF Grants
DMS- 2M0-522 and CMMI-2450378.
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21

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A Additional Preliminary and Reviews
A.1 Stochastic Optimal Control (SOC)
In this subsection, we expand Section 2 with details. Recall the SOC problem in (4):
/ 1 (la) + s(Xi)
min Ex-p"
(20a)
s.t. dx, = (X()) + ( di + a,WW, Xo ~
(20b)
Similar to (8), the optimal control to (20) can be characterized through an optimality equation:
(x(x) = - log /
un (z) = - (V,(2),
where
Pile (u|z)e Vi(w) (i dy,
base
Vz = =
(21)
the value function known to satisfy the Hamilton-Jacom We provide further characterization below.
(HJB) equation (Bellman, 1954).
Optimal distribution that the entropy-regularrized objective in (20) can be reformulated as:
The optimization problem in (20) is known analytically. Specifically, notice
DKL (xL)p(X)lp huse (x)) + xx) [x(1)]
DKL. (Ip
buse (Xo)) + (xx)
DKL. (p( (x) buse (X + Ex(x))) [x(X1)]
Xo)
DEL hase + Bp(xo)
DKL (o)xXxo))lz hase (X (X
(22)
where we shorthand X = [ and denote phuse the base distribution induced by (20b) with Le., the uncontrolled distribution. Minimizing (22) w.r.t. yields
1 base (X|Xo)e-ooAs) o(X),, p" (X| Z(Xo)
pX(o) = poso (Xo)
(23)
where Z(xu) is the normalization term defined by
Z(Xo) := f
f
/ hase (x(xo)e-((xidd) dx / p ( (X1 x(xx) - d.X.
(24)
which is exactly e (XXo) due (21). Combing (23) and (24) leads the the optimal distribution in (5), which we restate below for completeness:
(X) hase (X)e (o(m))
p"(oo, hase X1oe -x(x))))eexX)+++++- +6())
(25)
Adjoint Matching (AM) Scalable computational methods for solving (20) have been challenging,
naively back-propagating through (20) induces prohibitively high computational cost. Instead,
Adjoint Matching (Domingo-Entee Ennich et al., 2025) employs matching- objective, named Adjoint Matching (AM):
2² = arg min Exapa [X,) + o ù stopgrad(u).
(26a)
u
where da, = (((,),,, a Vg(Xi)
(26b)
is the backward dynamics of the (lean) adjoint state a, a(t; X(..)). It has been proven that the unique critical point of (26) is the optimal control u", implying new characteristics of the optimal control ut using the adjoint state:
(x) -o [ae] Xx, = 2/.
(27)
Adjoint Sampling (AS) Recently, Havens al. (2025) introduced an adaptation AM tailored to sampling Boltzmann distribution v(z) x e E(x) by considering
base (x)
f ;= 0, μ(x) = 60(2),
g(x) = log
p1
(28)
1
22

---

That is, AS considers the following SOC problem with degenerate base drift, a Dirac delta prior, and a specific instantiation of the terminal cost g(x) = log Pi (x),
m(ar)
min E Ex-p" V. 21 ue (X,) + log
pl (X1)
s.t. dX, = (x)((x))t Xo=0. (29)
v(x1)
Notably, this SOC problem (29) admits a simplified adjoint state aj and a degenerate initial value function Vo(x:)
(26b)
(X(1)
22
V log
base (X1) + VE
v [0,
(30)
Vo (2) log
base
dy = log 1 = 0,
(31)
which further implies that the optimal distribution p" is reciprocal process (Léonard al., 2014):
v(11)
(*(X)
(31) p huse (X)e (1(X))
(8) base
10 huse (x(x))))xx)
(32)
P1 (X1)
base
Combining the adjoint characteristics of the optimal control (27) with the simplified adjoint state or in (30) and optimal distribution p" (32) motivates the following Reciprocal Adjoint Matching (RAM) objective used in AS, where the unique critical point remains to be the optimal control u" in (21).
u arg min E papin
(X., + or (VE E
logPi
(XI)
ū = stopgrad(u). (33)
Remark on reciprocal representation The reciprocal representation of the optimal-controlle distribution p* in (32) extends to general SOC problems (20) with non-trivial base drifts and source distributions. Specifically, any optimal-controlled distribution that solves (20) can be factorized by
(x) --""e(x Xo. X1)p" (Xo, X.).
(34)
We leave formal statement in Theorem B.3 and Corollary B.4.
AS with linear base drift and Gaussian prior (Figure 1) Here, we discuss an alternative instantiation of AM for sampling with linear base drift and Gaussian prior, which reproduces the leftmost plot in Figure Consider
(x)): Be0, Bex,
p(z) (=))==((x:)) Ne;;=,((;0,,1,
o, = :B g(z) log
pi
passe())
base (x)
(35)
v(x)
where 3e is chosen such that (fr, ut σ.) fulfill the memoryless condition. For instance, Figure adopts the VPSDE (Song et al., 2021) setup:
B (1 maxx t.miin
Bmax 20,
8min 0.1.
(36)
Similar to (30), the resulting SOC problem admits simplified adjoint state ag::
(X()))
Ky-( VV-((Vlog plase (Ex)+VE(X;)),
Keg := e 1 ft B-dr (36)
c (1-0
(37)
and the RAM objective becomes
u = arg min E Jone
Pä,, (X)) + oeke (VE + V log paae) (x))113. T stopgrad (u).
(38)
Note that hase can be sampled analytically: P
Re - xi) N(,,;
Kx k2?
(1 (1- x²) R2
base Pejo.
(X, X1)
Xo +
X1.
1), (39)
1 R2
- 1o Brdr (36)
where is defined in (37) and
23

---

A.2 Schrödinger Bridge (SB)
In this subsection, we provide additional clarification on SB and specifically the derivation of (13). Recall the optimality equations of SB in (8):
Pc(z) = f
Pije ( (ul=) po(x))o(z) = p(z) (40a)
huse
(x) = or log (x)), where
(+(2) = f
Pijo (olv) (1(x)(1 = v(x) (40b)
huse
Just like how the value function of an SOC problem fully characterizes the optimal control and its corresponding optimal distribution, so does the SB potential (x)):
p (X) base (X) (X)(
Po((o) p base (X) ())))((X)))ool
(41)
where the last equality is due p (
(X) pase (X (Xo) and then invoking (40a). Note
that (41) recovers (10) by marginalizing over E (0. 1). Due to the construction of (x) and (x) in (40), the marginal optimal distribution admits a strikingly simple factorization:
(x) paaee (X, X = ==XN))ss
1 hoso (X1 2)p hase (X, = (X
- -( hace p x)oo)oo(X))dX)) (f
hose (X, 2){
(x))
(42)
or, more generally,
Ps. Pr (U, Ptis (=)1)) S
hase
(43)
Derivation of (13) based on path measure theory (Shi et al., 2023):
We now provide simpler derivation of (13) compared to its original derivation
V log (z)
(40b)
Vx
Pejo (x (zle)
base
S (x)
1
/
V log logo huse (x (x hase (cll))))
Se (x)
Ve log pogr (xly)) (yl
(44)
where the last equality follows by
(v,
Prlo (xll)) (w))))@(()))p(())
base base ( (sll))
Poje (3)
(x)))(((x)
(x(
G (x)
Equation (44) implies matching based variational formulation of V Vlog (. also known as the
bridge matching objective in data-driven SB (Shi al., 2023; Liu et al., 2023).
V log Pt arg min Eps, [l)be(X,) V., logp hace (X,(xx))1
(45)
Equation (45) recovers (13) at = 1.
A.3 Additional Related Works
In this subsection, we provide additional review existing learning-based methods for sampling Boltzmann distributions.
Learning-augmentte MCMC sampling methods-succ such as MCMC (Metropolis et al., 1953; HASTINGS, 1970), Sequential Monte Carlo (SMC; Del Moral et al., 2006) and Annealed Importance Sampling (AIS; Neal, (00l)-where
This class of methods can be thought of as extension of classical
24

---

traditional proposal distributions are replaced with modern machine learning models. For instance, Arbel et al. (2021) and Gabrié et (2022) use normalizing flows (Chen al., 2018) as learned proposal distributions, whereas Matthews et al. (2022) employ stochastic normalizing flow (Wu et al., 2020). More recently, Chen et (2025) have explored the use diffusion models (Song et al., 2021; Ho et al., 2020). However, training these models typically requires computing importance weights, which necessitates a large number energy evaluations.
MCCMC-augmenter Diffuusion Samplers Alternatively, methods of this class adopt modern generative models to sampling Boltzmann distributions and incorporate MCMC techniques to mitigate the lack of explicit target samples. For example, Phillips et al. (2024), (De Bortoli et al., 2024) and (Akhound-Sadeg)
al., 2024) employ score matching objective from score-based diffusion models (Song et al., 2021; Ho et al., 2020). In contrast, Albergo and Vanden-Eijnden (2024) base their method on action matching objectives (Neklyudov et al., 2023). However, estimating target samples requires computing importance weights, which makes these methods computationally expensive in terms of energy function evaluations.
B Proofs
B.1
Preliminary and Additional Theoretical Results
Lemma B.1 (Itō lemma (Itô, 1951)), Let Xt be the solution to the Itô SDE:
dX; fe(X,)dt + o,dW,
Then, the stochastic process (x)),,, where E ( 1], R"), ) is also an Itô process:
1 (x,( [axx((X)) Vu,(X)) 1 (o dt or (X,))) dW.
(46)
Lemma B.2 (Laplacian trick). For any tuice- function such that π (2) # 0, it holds that
1 AT( | [Vlo Alg (x))
(47)
x(x)
Proof.
AT(x) = v VT(I)
log
VT(x) V log (x) + ((x) log)(z) xxx)) log 7 (2) + Alog (x))
Theorem B.3 (SB characteristics of SOC). The optimal distribution px of the SOC problem in (20) is also the solution to the following SB problem:
arg min { (pllp base) Po = p.. P1 = =
(48)
Proof. We aim to show that there exist transform such that the SOC's optimality equation (21) can be reinterpreted as the ones for SB (40). To this end, consider
(x) = e Vi(a))
(x(x) = c Vi(x) pl (x).
(49)
One can verify that the value function Va( defined (21) can be rewritten as
(x) = hase
Pile (1)=))11() dy)
25

---

On the other hand, we can expand (x)) by
(x(x)) e p p' = ax)XX
ev(c)) i (x =)) dase (x XoJe (1(X))+V6(o)) dX,dXo
by (25)
eVilt) ( p)ocoXx = 2, xooe Vo66))
by (21)
/ Juse
(X: (X; x(=oxxo)((())) VolXo) dNo
buse (xly)) Pejo
by (49)
Combined, the optimality equation (21) for the SOC problem can be rewritten equivalently as
fı(x) = / PiR (ulz) (o(l))u(() = =m(2),
(x) oz V (r(),, where
(r(x)) =
P:lo (xlw) (1()))((() (i)
hose
We conclude that p" indeed solves (48).
Corollary B.4 (Reciprocal process of the SOC problem). The optimal distribution p' of the SOC problem in (20) is reciprocal process, i.e.,
p' (X) (X) = p (X Xo, X1)p" X1),
(51)
B.2 Missing Proofs in Main Paper
Proof of Theorem 3.1
Comparing (8a) to (21), reinterpret (ar) as value function Vz())
by reinterpreting
(1(
(x) == - log (z), g(x) == log you (x)
(8h)
log
v(x)
That is, the kinetic-optimal drift of SB solves SOC problem (4) with terminal cost g(x) := (x)
Proof of Theorem 4.1 For notational simplicity, will denote g q²h (k-1) throughout the proof. We first rewrite the backward SDE (17) in the forward direction (Nelson, 2020):
dX,, = [/2 ofV log de + V log 9i] d' + a,dWe, X1 ~
where we rewrite (xx) w.r.t. the forward time coordiante:
(x(x) / Prio (- (o)))y
(1 1000-((k-1) (1(-1)
(52)
Note that (52) admits equivalent PDE form by invoking Feynman- Kac formula (Le Gall, 2016):
0x)((x) = V (fio) + of Aoi(z), (1 = F (k--1) (z).
(53)
2
On the other hand, the dynamics of 0,y follows the Fokker Plank equation (Øksendal, 2003):
0gg. = - (ff+ of log d. + of log 9t) + 2 og
1 07 Agu.
V (ooVloog fi)
and straightforward calculation yields
1 o0? / A log qt
o, log q: = of A log - f (o di ft) V log qt 3 [ log qe
(54)
26

---

where we apply the Laplacian trick (47) to ; A | log qel2 2 + A log q-- Now, recall that is the path distribution induced by the following SDE:
d.X. = (fr(X,) + (x)) + o,dWe.
Xo ~ μ.
(55)
Invoke Ito Lemma (46) log (X,), where X, follows (55):
1 Øg + Vlog qt (fr + 0z44) + a qt
dlog q:
+ ae V log qi dW.
(54)
07 A log 0e
ft + V V Vloga
2 2
1
V log 4:l2 + Vlogae qi (orw)) dt
o. log St dW,
(56)
Likewise, invoke Ito Lemma (46) to log ti (X.), where X₂ follows (55);
d log de
Oglog Φe + Vlog Φe (fe + ),u)) + of - log oe dt + or V do dW,
o
Adi
1 20 A log de dt a2 a. VlVlog Φb dW,
V. ft
Vog 0: (oru)) +
2 Φi
2
d
1 (I log l1l + bi) + v log (o
7 o
V-ft+
+
A log Φ5 dt + V log dwW
2
V f
V log o. 21
V log (0,4) + o oi di or V dW, (57)
Subtracting (57) from (56) leads to
+ +o log o, o, V log 1|2 - 1 dt + V.V log &t
d doog 6: - d log q
l
dw, (58)
Finally, we are ready to compute the variational objective in (16):
DKL. (pll
Exap"
oo (X.) o, v log 4 (x
E Ex"
(2 (2llu((x))] d de (X,) d log de (x)) dt
d1 (X1)
log 2v (Xo)
= =xxap"
1 2 [X.)) + log
(59)
91 (X1)
20 (Xo)
0
2 (x,) log J/²
7 (8
1) (X1)
X Ex
(60)
v(11)
That is, we have shown that the variational objective DKL(pple (p is equivalent (up to an additive constant) to an SOC problem (60). Applying Reciprocal Adjoint Matching (Havens et al., 2025) with the reciprocal process from Corollary B.4 conclude that DKL (pllg
(0k-1) is minimized by p"
Proof of Theorem 4.2
For notational simplicity, we will denote p(() = p" (*(*)) throughout the proof.
Let q be the path distribution induced by backward SDE, propagating along the time coordinate 8 ::= 4:
dY, = (Y.)( a.v. (Y.) ds a.W..,
Yo
Next, rewrite the forward SDE (A) in the backward direction:
dY, = [-f or (b) o² log p ds + a,dW,, Yo pt t=1.
By Theorem B.3, we know that (+) is the SB solution, thereby satisfying
(x(x)) / bame phr (vlm)) (o(x))p((1) m(x) (61a)
u (x) o, Vog (x)), where
(x(z) = ʃ
P:o (alv)) (z(2)(((z) = (()(22) -a))))pp((((
base
(61b)
27

---

Since we are working with the backward time coordinate it is convenience to define o, == 91- and rewrite (61b) by
ex(y) p (1-ss((ll2)dd()))d eu(y)
pl (v) 491
(62)
Now, expanding the variational objective with Girsanov Theorem yields (Särkkä and Solin, 2019)
DKL (1g) By pp(h)) L. b1l V Pl + az V log (V,) es dY.))
(63)
which is minimized point-wise at
(k)
(v) (22)
Pe
(v) = V. log
Os V log (v)).
(1)
In other words, the backward SDE that minimizes (63) must obey dY, (Y,) + Vog
@o (Y.)) ds + o,dW,, Yo ~~ v,
with o, defined in (62). That is, we have concluded so far that
arg min DEL (p (K) (1q) Q1 = v}.
(64)
(x()) of the CM objective at stage k equals 2
Hence, remains to shown that the minimizer
This is indeed the case since (()) is the SB solution:
v log (R) (15) =
arg min
E [[)((X)) Vay log (X, V log 1
V log
h Po.1
C Practical Implementation of ASBS
Algorithm summarizes the practical implementation of ASBS, where we expand the adjoint and corrector matching steps (i.e., lines 3 and 4 in Algorithm 1) full details. Table 5 provides the hyper-parameters for each task. We break down each component as follows:
Harmonic prior /hharmonic
Recall the harmonic prior in (19):
Aharmonic (x) x exp(- E.,; ||,; - xz 12)
(65)
In practice, we set 1 and implement (65) anisotropic Gaussian. For instance, for 2-particle system 3D, i.e., x [ x2] R2, we can rewrite (65) as a quadratic potential,
0
0
1
0
0
exp(-bllæ1 x2ll²) exp(x Rr). where
0
(66)
0
0
0
0
0
0
0
1
2
and then sample xr from the Gaussian (x; 0, (R + cI) where we set 10-4
Noise schedule a; We consider two types of noise schedule.
The geometric noise schedule (Song al., 2021: Karras al., 2022) monotonically decays from t = to 1 according to some prescribed 3min and 3maxi
goometric
8min
Beez. Fmm
1-6
V
2
log
Aux Buts
(67)
28

---

Algorithm 2 Adjoint Schrödinger Bridge Sampler (ASBS)
Require:
Sample-able source Xo μe, differentiable energy E(x)) parametrized drift uo(t, x) and corrector hig (2), replay buffers Budj and Ben, number of stages K, numbers of AM and CM epochs Madj and Mat- number of resamples N, number of gradient steps L, time scaling X maximum energy gradient norm mmax-
1: Initialize h. := 0
> IPF initialization
2: for stage k in 1, 2. K do
3: for epoch in 2. ..... Maj do
adjoint matching
4: Sample from model {(X)) X1 (1) ) path) where ū(A) = stopgrad(u))
(4) :- stopgrad CLip(VE(X)) Okmax) + l1p (x,1)1)) (x{))) 5: Compute adjoint target or
6: Update replay buffer Budj + Bayj ((x!) X az )
7: Take gradient steps VoLAM w.r.t. the AM objective:
(0) == (x..
(XoxX)) [eellub (t, + t
8:
end for
9:
for epoch in 1,2,... Man do
corrector matching
(6) J Sample from model {(x,) X1
p where a(k) stopgrad (k)
10:
Update replay buffer Ben <- Ben U ((x)) )(
11:
12: Take L gradient steps VaCcM w.r.t. the CM objective:
(CM(Ø) -(xx.xi))BBB
[ ( (X1) Vay log peepeeeee (Xi
13:
end for end for return Kinetic-optimal drift u² ut()
15:
It is convenience to further define
Kaja = fooddr geometric
PPaxx . ((
Puis Plane
Puti Preas
82 := max
Kgjo
7/t ==
32
(68)
With them, the conditional base distribution when f O can be represented compactly by
P base (x, = ,VXX;;;; k)
(69a)
buse Xo, X1) = NV(,;; 1 No n , -
(69b)
The constant noise schedule simply sets
constant
O.
(70)
When f = O, the base SDE is effectively a standard Brownian motion whose conditional distributions obey
bose (X,(xo) = oo. 02t)) XX, X1) = NNX, t)Xo + tX, o -2)
(71a)
p base
(71b)
Replay buffers Badj and Ben Similar to many previous diffusion samplers (Havens et al., 2025; Akhound-Sadegh al., 2024; Chen et al., 2025), we employ replay buffers B in computation of both adjoint (14) and corrector (15) matching objectives. Specifically, we rebase the expectation over model samples ( onto replay buffer B, which stores the most latest [B] samples. We update the buffer with new samples every gradient steps. Note that the use of replay buffers effectively render ASBS hybrid method between on-policy and off off-policy.
29

---

Parametrization of ug and he For each energy function, we parametrize the drift ut a)) and the corrector hx(x) with two neural networks, 2o(t, a) and v I), of the same architecture.
Specifically, we parametrize the drift as no no(t, 2r) == ty (4, x), which effectively eliminates the noise schedule "a," " in matching target (see (14)), making it time-invariant for each sampled trajectory. The only exception is the conformer generation task, where we keep the original parametrization ug(t, := ut æ), which empirically yields better results. On the other hand, since (2( is independent of time, we simply set a fixed time input = 1, ie., (x) == up (1,
The specific parametrization u(t, z)) employed for each task are detailed below.
MW-5: We consider v(t, x) a standard fully-connected network with 4 layers with 64 hidden features of the following form:
output = layer_no
layer__ (x_embed (x)
t_embed
DW-4, LJ-1, LJ-55: We consider vtt, x) Equivariant Graph Neural Network (EGNN; Satorras al., 2021) 1 with layers and 128 hidden features. The architecture of EGNN is aligned with prior methods (Akhound-Sadegh) et al., 2024; Havens et al., 2025).
Alanine dipeptide: We use the same architecture as in MW-5, except with 8 layers with 256 hidden features.
Conformer generation: We consider x) a similar EGNN used in Adjoint Sampling (Havens al., 2025), except with 20 layers. Ablation study on the same EGNN architecture can be found in Appendix D.4.
Clipping 0max
We clip the energy gradient to prevent its maximum norm from exceeding Omaax
Time scaling A Following standard practices for AM objective, we employ a time scaling to improve numerical stability. Note that this does not affect the minimizer of the AM objective. We set A, = for all tasks.
Translation invariance For DW-4, LJ-13, LJ-55, and conformer generation tasks, we follow prior methods (Akhound-Sadeg) et al., 2024; Havens al., 2025) by restricting the state space zero center-of-mass (ZCOM) subspace and thereby enforcing translation invariance.
For n-particle k- d-dimensional system, i.e., [æ1:-.. ................................ : where E R the ZCOM subspace is defined as 2 2CCM = fu E RRn Σ Te = 0}. Practically, this is achieved by projecting the initial sample Xo ~ the SDE's noise dW, and the energy gradient VE((-) onto 2CCCMM Note that the output of EGNN is by construction ZCOM.
Formally, the adaption is equivalent to augmenting the SDE with projection matrix A E Buknkk
dX,; = (u,Au((,,)) AdW, Xo = AY, Yo-M, A In - 1n 1 X Ik, (72)
71
where & is the Kronecker product, I, e R²xn n an identity matrix, and 1,, E R" is a vector of ones.
Initialization and alternate procedure As ASBS is an instantiation of the IPF algorithm (see Theorem 3.2), must adhere to the IPF initialization protocol to ensure theoretical convergence to the global solution. Specifically, the IPF initialization can be implemented in two ways
Initialize with h 0 and run AM, CM, until convergence. We adopt this setup for all tasks.
Initialize with 2
(0) 0 and run CM, AM, until convergence. Since p²( huse in this setup,
the optimal corrector at the first CM stage is known analytically:
A (x)
Poj1
base
(y/=)) V log Pijo ( (yly)dy
base
Juee Por (y w) Pio (=ly)
base (xly)dy
P
V
Po (v)p (xly)dd
base
p (x)
V log p1 hase (z)
(73)
30

---

Table 5: Hyperparameters of ASBS for the each task.
Synthetic energy functions
Alanine dipeptide
Conformer generation
MW-5
DW-4
LJ-13
LJ-55
J
1)
Pharmonic in (19) with a-2, 2,
(0,, 0.25) 0.001
Pharmonic
Amin
0.001
0.001
0.001
0.001
Bmax
0.5
0
0.2
K
5
20
15
15
15
3
Madj
100
200
300
300
4000
2500
Men
20
20
20
20
2000
2000
1000
1000
1000
1000
1000
128
L
200
100
100
100
100
100
8
104
104
104
10a
10a
6.4 104
0max
100
100
100
100
150
1
07
o7
o7
In practice, we find that the two setups yield similar performance.
RDKit warm-start This warm-start starts the drift ua using RDKit samples. The procedure is inspired by the fact that (Shi et al., 2023; Liu et al., 2023):
2i o, V log Pt arg min Ep, (X,, or Vv. logp hase (X1 (X][3)
arg min E x X1 (X,, - as Va, log (X, x.)11
base (X,
(74)
"
where the last equality is due
Po.,, (2. y.
Pt.
(
base
Polo,1 (vl x, 2 p huse (21))
buse
by Markov property
Pejo,1 (y) or, ) (Ir, =).
base
(75)
Equation (74) can be understood as analogy of (45) for another SB potential pr. In practice, given RDKit samples X, o RDKit, we warm-start ASBS by minimizing w.r.t. the following objective:
Lwarmup (f) = -01 Nine X Xim
(ellu((X.) oz Va, log log pase (X, (X1 xx))
X1)
(690)
B1.ttfo.11 tonk (11
)ellu((,,)
0r (X, (X,-X,)1+ x,)13
(76)
Kije
where #1 is defined in (68) for the geometric noise schedule. We set the time scaling 7 v milt"
Note that, unlike AS, the minimizer of (76) does not equal u², since (Xo, X1) ~ @ 8 RDKi + Pú.1 are sampled independently.
D Experiment Details
D.1 Synthetic Energy Functions
D.1.1 Energy functions
In this section, we provide the exact setup for our synthetic energy experiments in Table 2. We consider four synthetic energy functions that have been widely used recent literature benchmark sampling and generative algorithms: MW-5, DW-4, LJ-13, and LJ-55.
31

---

MW-5 The MW-5 (Many-Wol Well in 5D) energy is 5-particle ID system adopted from Chen et al. (2025), where x -1:- :as IR' with 2, R, The energy function is defined as follows:
5
E(x) - > (= 5)2
(77)
where we set 8 4. This creates distinct modes centered at combinations of = in each of the dimensions.
DW-4 The DW-4 (Double-Wer for 4 particles in 2D) energy is physically motivated pairwise potential originally proposed in Köhler et al. (2020) and subsequently used in Athound-Sadege et al. (2024); Havens et al. (2025). It defines system of four particles, each living in R2, leading 8I)
state vector I = [11; 22; Tat ma) E IR with 2; E R The energy function reads
E(x) - exp
1 2T
(a(d;) - da) + b(djj do)² + c(di - do)))
(78)
where dij = =|,;;
2z 12 is the Fuclidean distance between particles and j. We follow the standard -, 0.9, dg and temperature 7 1.
configuration with 0,
IJ-133 and LJ-55 The Lennard-Jones (LJ) potentials are classical intermolecular potentials com- monly used in physics to model atomic interactions. These are defined for system of particles in 3D space, with = :Fi:...;:.) E g and z, E R The index following "LJ-" indicates the number of particles (e.g.. 13 or 55). The unnormalized energy function takes the form:
127
E(x) = 2r Σ
Tm
P' dij
Σ
||= C(x))
(79)
dis
2
where d, = ||, zاارæ is the pairwise distance and (ax)) denotes the center of mass of the particles. We use the parameter values T' 1, 1, 0.5, and r = following prior work. The LJ-13 and LJ-55 systems correspond to 39D and 165D, respectively.
D.1.2 Baselines
Here, we outline the procedure used to obtain the values reported in Table 2 for the baseline methods.
For PIS (Zhang and Chen, 2022), DDS (Vargas al., 2023), and LV-PIS (Richter and Berner, 2024), iDEM (AKhound-Sadegh) et al., 2024), and AS (Havens et al., 2025), we reuse the values reported in AS (Havens et al., 2025, Table for DW-4, LJ-13, and LJ-55 energy functions. As for MW-5, which is not included in AS, we run iDEM using their official implementation and the rest of baseline methods using our own implementation in PyTorch (Paszke al., 2019). We were unable obtain reportable results for LV-PIS and iDEM on this energy function.
For PDDS (Phillips et al., 2024) and SCLD (Chen et al., 2025), in JAX (Bradbury et al., 2018) using the default hyperparameter settings specified for the Log- Gaussian Cox Process experiment in their respective papers. To enhance stability and convergence on synthetic energy functions, we tune the gradient clipping parameters. For PDDS, we apply clipping the gradient of the energy function. For SCLD, clip both the energy gradient and the Langevin norm. In both cases, the clipping magnitude is selected from the set (1, 10, 100, 1000} based on the best validation performance. Training performed for 100,000 iterations across all runs. For SCLD, we use subtrajectory splitting with the default value of 4, so that it does not degenerate to CMCD (Vargas et al., 2024). In practice, we find that using subtrajectories yields better results.
their official implementations
D.1.3 Evaluation Metrics
In this subsection, we outline the evaluation criteria used to quantitatively assess the quality of samples generated from synthetic energy functions. We employ three primary metrics: Sinkhorn distance, geometric Wz, and energy Wz. each designed to capture different aspects of distributional similarity between generated and ground truth samples.
Sinkhorn distance To evaluate the similarity between the empirical distributions of generated and reference samples, compute the Sinkhom distance using the reguaay-re optimal transport
32

---

formulation (Peyré and Cuturi, 2019), following the implementation of Blessing et al. (2024) and Chen al. (2025). The Sinkhorn regularization coefficient is set 10-3 throughout, We use 2,000 samples from both the generated and ground truth distributions to compute the metric.
Geometric W2 For DW and LJ tasks, the potential energy functions-and consequently, the sample distributions-enn exhibi invariance to both particle permutations and rigid transformations such rotations and reflections. To appropriately account for these symmetries, employ the geometric W2 distance as defined by Akhound- Sadegh et al. (2024) and Havens et al. (2025). Formally, the 2- Wasserstein distance computed as:
/ D(a,,)) (x,, v) drdy, *ell(D,))
( v)
inf
(80)
where [T, v) denotes the set of joint couplings with prescribed marginals à (generated) and (ground truth), and (x,, y) a symmetry-aware distance between samples defined as:
( REO(s),
min
| (R @ P)yl12
(81)
Here, Os)) denotes the group of orthogonal transformations in s spatial dimensions (rotations and reflections), and ('n) represents the symmetric group over n particles. As exact minimization over these symmetry groups is computationally infeasible, we adopt the approximation scheme of Köhler et al. (2020). We use 2000 samples from each generated and ground truth distribution to compute the metric.
Energy W2 To evaluate fidelity with respect to the target energy landscape, we also compute the 2- 2-Wasserstein distance between the energy values of generated samples and those of ground truth samples. For each target distribution, we generate 2,000 samples from both the model and the reference, and compare their respective energy histograms. This scalar-based Wasserstein metric serves as a proxy for how well the generative model captures the energy histogram of the target distribution.
D.2 Alanine dipeptide
Benchmark description We adopt the experiment setup primarily from (Midgley al., 2023). Given configuration of alanine dipeptide, which consists of 22 particles in 3D, i.e., x [xq:-..: ; F222] R6 where 2; E RR3, we apply the same coordinate transform proposed by Midgley et al. (2023). This coordinate transform maps the Cartesian coordinates to internal coor- dinates, T(x) =: E I, which include bond lengths, bond angles, and dihedral angles (Stimper et al., 2022). This process effectively removes six degrees of freedom three for translation and three for theaeyy enforcing structural invariance. Non-angular coordinates are further normalized using samples with minimal energies. We refer readers to (Midgley et al., 2023, Appendix F.1) for further details. Note that the internal coordinate transformation is bijective. Hence, we can compute the energy via
E(x) = E2)
(82)
Evaluation and baselines For each sample x = (2) E R966, we extract five torsion angles, including the backbone angles o, 1 and methyl rotation angles Yi. 72. 7a. We report two divergence metrics with respect to the ground-truth distribution, which contains 107 samples simulated by Molecular Dynamics. We implement the baseline methods, including PIS (Zhang and Chen, 2022), DDS (Vargas et al., 2023), AS (Havens et al., 2025), using PyTorch (Paszke et al., 2019).
For the KL divergences, we adopt setup from (Wu et al., 2020) and compute the divergence of the ground-truth marginal to model marginal for each torsion angle:
P*(++e
DKL(P" (Olp" ()) Pt
6= c=10-
(83)
(m)
where Ps and pas are histograms of 107 samples, discretized between [- n] with 200 intervals.
For the Wasserstein-2 distance, we use the Geometric W2 in (80), where each sample now in 2D. (d), øb E IR2 Due the high computational cost, compute the value using subset of 10 samples from the test set ground-truth samples, which is fixed for all methods.
Finally, both Ramachandran plots in Figure 5 are generated using 107 samples.
33

---

120
200
300
103
Covergge Itccall 190)
40
40
ROCk
A635
4032 hammank
0.5
1.0
1.5
2.0
1.5
2.0
0.5
1.6
Threshold (Angstrüm)
id (Angstromm)
Threshald (Angström)
Threshold (Angstrām)
Figure 8: Ablation study on full recall coverage curves (without RDKit warm-start) using the same EGNN architecture in AS (Havens et al., 2025). Note that Table 6 reports the values at the thresholds 1.0 and 1.25A.
D.3 Amortized conformer generation
In this subsection, we provide some context for the experimental results found in Table 4 regarding the generation of conformers.
Benchmark description Conformers are atomic representations molecules in cartesian space with their constituent atoms arranged into local minima the potential energy surface. Molecules are defined to be graph of atoms (nodes) connected by bonds (edges); conformers are geometric realizations of that molecule. Torsion angles, rotatable bonds, particularly important degrees of freedom for defining conformations since bond lengths and bond angles are typically much more common to consider bond lengths and bond
stable due to a high sensitivity to perturbations. angles fixed, while the torsional degrees of freedom define the conformer.
The task in this benchmark is take representation of the molecular graph, usually SMILES string (Weininger, 1988), and comprehensively sample the conformational configuration space. In flexible molecules, there can be large number of conformers with many separated modes in 3n 6 dimensional space. (Where n represents the number of atoms and 6 comes from the irrelevance of rotation and translation of the conformer.) We quantify the notion of comprehensively sampling the space by comparing generated structures to a set of conformers sampled using expensive, standard search techniques (Pracht al., 2024) that were further relaxed using extremely precise density function aicory-based, quantum chemistry methods (Neese, 2012; Levine et al., 2025). A detailed description of this benchmark can be found in its source (Havens et al., 2025, Appendix F.).
Evaluation and baselines The method comparison between proposed structure and reference conformer is to use RDKit's (Landrum, 2006) implementation of Root Mean Squared Displacement (RMSD), a measure of distance between atomic structures that is invariant to translation and rotation. We set threshold RMSD for structures to match and computed the Recall Coverage and Recall Average Minimum RMSD (AMR). The experiment was performed with both generated structures and with generated structures after so-called relaxation, i.e. geometry optimization of energy, using eSEN (Fu et al., 2025). The equations for computing these metrics are:
COV-R(Ø) = T. (1...,
L): =kke E (I, RMSD(C&, C7) < <6H
(84)
AMR-R : L Σ min
(M)
(85)
KE
kE(l...,K)
(E(I.......)
where 0.75 À is the coverage threshold, max (I', 128), where L' is the number of reference conformers, K = 2L, and let {C7 heli,r) and {C& ]ke[I.K] be the sets of ground truth and generated conformers respectively. We capped the reference conformers per molecule 512 in COV-R.
The values for the baselines are adopted from AS (Havens et al., 2025).
D.4 Additional Experiments and Discussions
Ablation study between AS and ASBS using the same EGNN For the amortized conformer generation task in Table 4, we use an EGNN architecture with 20 layers, whereas AS employs the same architecture with 12 layers. In Table 6, report the results of ASBS using the same 12-layer
34

---

Table 6: Ablation study on amortized conformer generation using the same EGNN architecture as in AS (Havens et al., 2025). We report the recall at the thresholds 1.0A and 1.25A, where the latter was reported in AS.
without relaxation
with relaxation
SPICE
GEOM-DRUGS
SPICE
GEOM-DRUGS
Method
Coverage 1
AMR
Coverage 1
AMR
Coverage 1
AMR
Coverage
AMR
Threshold 1.0
RDKi ETKDG
2015)
56.94_1152
1.04
g0.81
1.15
70.21
0.79
62.55
0.93
(Hovens al., 3025)
0.96
36.23
1.20
82.41
g.68
64.26
0.89
ASBS w/ Gaussian prior (Ours)
68.61 453.4
46.03
1.08
84.77=2246
0.64
68.83
0.80
ASBS w/ Harmonic prior (Ours)
30
a.86
:2.19
86.79
0.61
70.08
0.80
warmup clemseri 270%
72.21 2:0
g.84
52. 19
1.02
87.84_
73.88 0.60
0.76
ASBS +RDKit warmup (Ours)
74.29 24
0.82
55.88
0.98.
87,25
0.60
74, 1 1 101
g.
Threshold 1.254
RDKi ETKDG (Reakoe Lemken, 2001)
72.74
1.04gno
63.51
1.15
g1.61
0.79
21.72
g.
AS (Havens et al., 3025)
0.96
60.93
1,20
94.10 14
0.68
79.08
g.
ASBS w/ Gaussian prior (Ours)
87.20
2.88
70.86.1144
1.08.00s
95.19
0.64_.005
84.66
ASBS Harmonic prior (Ours)
89.66..
0.86
94
1.08
96. 2016
0.61
83
0.80
AS +RDKit warmup Glloneer 2005
89:42 17
0.84
72.98 32.82
1.02 1034
96.65 67:50
0.60 (0:3
87.01
22:75
0,76 :334
ASBS +RDKit warmup (Ours)
90.85
77.86..1 2.8
0.98
97.28
0.60
87.81
0.75
Target
Langevin Baseline
ASBS (ours)
40
20
-20
-2
20
-40
Figure 9: Compared vanilla Langevin baseline, our iSBSa-nnttnttited with standard uni- variance Gaussian-is able to identify almost all modes without any prior knowledge of where the target modes were located.
EGNN as AS. Notably, our ASBS consistently outperforms AS on all metrics across all setups, except the coverage for GEÖM-DRUGS with relaxation and RDKit warm-start, where ASBS falls slightly behind AS by only 1.0%. Finally, Figure 8 reports the full recall coverage curves that reproduce Table
Ability of ASBS in finding modes We conduct additional experiments on the 40-mode GMM in 2D. Specifically, we instantiate ASBS with a uni-variance Gaussian source distribution centered at zero, effectively assuming no prior knowledge of the target modes, as the initial distribution does not coincide with any target modes. We also run vanilla Langevin baseline for 1 million steps, starting from the same source distribution.
Figure 9 represents the quantitative results. Notably, ASBS is able to identify almost all modes. In contrast, the vanilla Langevin baseline appears to suffer from a slow mixing rate, recovering less than half of the total modes after 1 million steps. We highlight this distinction an advantage of constructing diffusion samplers from the stochastic control and Schrödinger Bridge frameworks, which allows theoretical convergence to target distribution within finite horizon. Finally, we believe that with proper tuning of the ASBS noise schedule, its performance can be further enhanced.
Discussion on important weights Finally, we discuss the potential integration of ASBS with importance weights, emphasizing that our theoretical and algorithmic frameworks do not preclude the use of importance weights to further enhance performance robustness.
Formally, the importance weights over model path ~ p" admit the following representation:
wo( : dx) exp (L," (X:) - S/' u.((.)) dW, - log (Xi)) (x((
+ log ((Xx)
(X)
v(Xi)
(86)
35

---

which can be obtained from (59) by setting h := @1 so that q² p- is the optimal distribution of SB. Note that when the source distribution degenerates to the Dirac delta (xXo)) do (o)), the last term log Xo(Xa) becomes constant and-as discussed Section 3.2-Φ1 = pi thereby recovering the xxX
weights used in prior SOC-based methods (Zhang and Chen, 2022; Havens et al., 2025).
Equation (86) is also more concise representation than the one derived in (Richter and Berner, 2024), by recognizing the following relation through the application of Ito Lemma (46) to log (X,):
log (X.1) log (Xo)
!,' [x,, + (u (x,) V (X,, fc(X.)) dt+ / u2 (x(X)) dW,
(87)
where we shorthand vx() V log (a:).
Estimating the weight in (86) requires knowing the ratios (x
v(z) and (u(z)
m(x) which are not immediately
available with the current parametrization, us us(t, s o V log (x(()) and (x)x) ss V log (x)). One
accommodation is to reparametrize the functions with potential network v(t, [0, x X' + R.
ut x) == oe V (f,
ha(x) == (1g(1,x)
(88)
and then regress their gradients onto the adjoint and corrector targets. With that, the logarithmic ratios be easily estimated:
Φ (x)
log (o(r) (x
log
2s(1, + E(x),
log (p(()) (0.
(89)
A more detailed investigation of this importance sampling scheme is left for future work.
36


Reviews:
[{"review_id": "620GRNNas9", "reviewer": "Reviewer_RKyV", "summary": "The authors introduced the Adjoint Schrödinger Bridge Sampler (ASBS), which is a novel diffusion-based sampler for Boltzmann distributions that addresses general Schrödinger bridge sampling problems using only the target energy function.", "questions": "See comments on Weaknesses.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\nIt is known that the SOC setup for fine-tuning is limited to the Dirac measure [1] [2] for the initial distribution.\nThe authors extended it to handle arbitrary distributions with memoryless source distributions by leveraging Schrodinger bridge between two marginal distributions. Schrodinger bridge (SB) should be the right tool here.\n\nExtensive evaluations are conducted with promising performance.\n\n[1] Theoretical guarantees for sampling and inference in generative models with latent diffusions\n\n[2] Path Integral Sampler: a stochastic control approach for sampling\n\nWeaknesses:\n\nI understand that SB are complex to understand. Howevere, the writings are a bit sloppy and many important details are less evalborated, e.g.  \n\t1) Eq.(5) is quite important to understand why we should avoid memory setups and the authors at least should cite the adjoint matching paper. To be honest, even for the adjoint matching paper, some details are not detailed or derived properly as well, which makes it harder to understand. \n\t2) eq(6), if p^base is a distribution that takes a joint distribution, then we should not reuse it a non-joint distribution; the intuition of setting g(x):=log p_1^base / nu(x) is not elaborated. \n\t3) eq(14) eq(15) are not evalaborated clearly and it is crucial to understand the dynamics and why it works, please write it in more details.\n\nDidn't check proof.", "was_revised": true, "final_justification": "I thank the authors for their detailed and thoughtful rebuttal. They have convincingly addressed all of my initial concerns, and I appreciate the detailed clarifications provided. Based on this strong response and the authors' clear plan for revision, I have raised my score to 5.", "rebuttal": "We thank the reviewer for raising all valuable comments. We are excited that the reviewer identified the novelty of our technical contributions for overcoming Dirac limitations from prior methods, appreciated the advantages of SB-based diffusion samplers, and acknowledged our strong and extensive empirical results. We believe ASBS takes a significant step toward scalable diffusion samplers with flexible model design. We try our best to resolve all raised concerns in the responses below.\n\n---\n\n**1. Clarification on Sec 2 (Eqs 5 & 6)**\n\n- We understand that current Sec 2 may be compactly written, mainly due to the space constraints at submission. In response to the reviewer’s suggestion, we will provide further clarifications in the revised Sec 2, along with a self-contained, step-by-step derivation in the revised Appendix. We detail these clarification below.\n\n\n- The goal of Sec 2, as briefly stated in L86-88, is to formally introduce the mathematical formulation of the SOC problem (Eq 4), its optimal distribution (Eq 5), and how prior diffusion samplers have adopted it for Boltzmann sampling under the memoryless condition (Eq 6). This leads to a specialized, “_memoryless_”, SOC problem (Eq 7), which contrasts with our generalized, “_non-memoryless_”, SOC problem inspired by Schrödinger Bridge (Eq 9; also see L129-131). \n\n\n- As the reviewer noticed, Eq 5 is indeed the key to connect SOC problems with Boltzmann sampling. This equation, representing the optimal solution to the original SOC problem (Eq 4), can be derived as follows: \n  - By rewriting Eq 4 as KL-regularized optimization, $\\min_{p^u} \\mathrm{KL}(p^u || p^\\text{base}) + \\mathbb{E}_{p^u_1}[g(X_1)]$ using Girsanov theorem [1]. \n  - Due to the KL term, the aforementioned optimization is convex in $p^u$, allowing for an analytic solution. Setting the first-order variation to zero yields $p^\\star(X_1 | X_0) \\propto p^\\text{base}(X_1 | X_0) e^{-g(X_1)}$. \n  - Finally, substituting the normalization constant $\\int p^\\text{base}(X_1 | X_0) e^{-g(X_1)} \\mathrm{d} X_1 = e^{-V_0(X_0)}$ leads to Eq 5.\n\n  While Eq 5 is well-known in SOC literature [2,3], we acknowledge that it may not be immediately clear to readers without sufficient background—even with the interpretation provided right below (L93-96). In the revision, we will include these clarifications along with additional citations for Eq 5, including the original AM paper, which—we kindly note that—was already cited in the beginning of Sec 2 (L86).\n\n\n- Eq 5 implies that, for any SOC problems (Eq 4) to be effective in sampling Boltzmann distribution, the optimal distribution $p^\\star$ must satisfy the desired target distribution $\\nu$ at terminal time $t=1$. This requires appropriately selecting the base drift $f_t$, noise schedule $\\sigma_t$, source distribution $\\mu$, and terminal cost $g$, such that $\\int p^\\star(X_0, X_1) \\mathrm{d} X_0 = \\nu(X_1)$. In Adjoint Sampling [4], this is achieved by \n  - Enforcing the base process $p^\\text{base}$—uniquely defined by $(f_t, \\sigma_t, \\mu)$—to be memoryless (Eq 6). This simplifies the marginalization to $\\int p^\\star(X_0, X_1) \\mathrm{d} X_0 = C \\cdot p^\\text{base}(X_1)_1 e^{-g(X_1)}$ with some constant $C$, as detailed between L102-123.\n  - Finally, equating $p^\\text{base}_1(X_1) e^{-g(X_1)} = \\nu(X_1)$, which leads to the proposed terminal cost $g := \\log p^\\text{base}_1 - \\log \\nu$ in L102.\n\n  We will include these clarifications for how $g$ is set up in the revision, and we thank the reviewer for pointing it out.\n\n\n- Finally, for notational simplicity, we omitted the time specification of distribution—such as $p^\\text{base}(X_0)$ and $p^\\text{base}(X_1)$ in Eq 6—whenever it can be clearly inferred from the variables $X_0$ and $X_1$. As the reviewer correctly pointed, this is not mathematically rigorous, and we will correct these inconsistencies in the revision.\n\n---\n\n**2. Clarification on the Adjoint Matching (Eq 14) and Corrector Matching (Eq 15)**\n\n- Given our proposed SOC problem (Eq 9), which accommodates non-memoryless conditions and arbitrary source distributions, Sec 3.2 identifies two key mathematical objects, namely the optimal control $u^\\star$ in Eq 12 and the corrector $\\nabla \\log \\hat \\varphi_1$ in Eq 13. These two mathematical objects characterize the optimality condition of our SOC problem (Eq 9) and, equivalently, the Schrödinger Bridge problem (Eq 3). In other words, the diffusion sampler that solves Eqs 9 & 3 must satisfy the optimality conditions in Eqs 12 & 13. \n\n- Building on the reasoning above, our algorithmic goal is to design a learning method that finds a diffusion sampler satisfying Eqs 12 & 13, which correspond to two simple matching-based objectives (RHS of Eqs 12 & 13). However, these two matching objectives cannot be implemented naively due to their interdependency: Solving the matching objective in Eq 12 for the optimal control $u^\\star$ requires knowledge of the corrector $\\nabla \\log \\hat \\varphi_1$. Likewise, solving the matching objective in Eq 13 for $\\nabla \\log \\hat \\varphi_1$ requires knowledge of $u^\\star$. \n\n\n- In Sec 3.3, we address the interdependency between the control $u$ and the corrector $h$ through an alternating training scheme. Specifically, \n  - Starting from an initial guess $(u, h)$ of the control and corrector, we first update $u$ by solving Eq 12 with $h \\approx \\nabla \\log \\hat \\varphi_1$. \n  - Then, we update $h$ by solving Eq 13 with $u \\approx u^\\star$. \n  - This alternate update between $u$ and $h$ continues until $u$ converges to a fixed point (see Alg 1).\n\n  This effectively creates a sequence of update, $(u^{(0)}, h^{(0)})$ → … $(u^{(k)}, h^{(k)})$ → …, that may be thought of as running coordinate descent between the control $u$ and the corrector $h$. Intuitively, at each stage $k$, we first find the control $u^{(k)}$ that best aligns with the corrector from previous stage, $h^{(k-1)}$, then update the corrector $h^{(k)}$ accordingly to reflect the “memorylessness” of the current control $u^{(k)}$. We formalize these update rules in Eqs 14 & 15, and their dynamics are visualized on a 2D toy example in Fig 2.\n\n\n- Whether the aforementioned update sequence converges—and if so, whether it converges to the global solution—is a highly non-trivial question. Nevertheless, we provide a positive answer in Thm 3.2. The proof, detailed in Sec 4, essentially shows that the update rules in Eqs 14 & 15 are contractive maps that progressively bring $u^{(k)}$ closer to the optimal control $u^\\star$. Formally, the KL divergence $\\mathrm{KL}(p^u_1 || \\nu)$ is non-increasing as $k$ grows [5].\n\n\nOverall, we acknowledge that Sec 3.3 is compactly written and will include these discussions in the revision. We thank the reviewer again for the suggestion. \n\n\n---\n\n[1] Applied SDEs (Särkkä & Solin, 2019)  \n[2] Adjoint Matching  \n[3] Solving high-dim HJB PDEs using neural networks  \n[4] Adjoint Sampling  \n[5] Diffusion SB with App. to Score-Based Generative Modeling"}, {"review_id": "8V1fcVnB4T", "reviewer": "Reviewer_5V2q", "summary": "This work proposes a novel method for sampling from unnormalized densities. The method extends the adjoint sampler to allow for using arbitrary source distributions while retaining the scalability of matching-based objectives. The authors formalize their method using stochastic optimal control (SOC). In particular, they propose a SOC formulation of the Schroedinger Bridge problem which is key to their new methodology. However, this formulation, or more specifically, the terminal cost of the SOC problem depends on the Schroedinger potential which is not available. As such, the authors propose an optimization scheme that alternates between learning the control via SOC and updating the Schroedinger potential. They show that this scheme is an instance of the IPF algortithm and therefore converges to the Schroedinger Bridge between source and target density. They validate the effectiveness of their method on several synthetic energy functions, Alanine dipeptide, and amortized conformer generation.", "questions": "- Why are LV-PIS and LV-DIS reliant on importance weights as stated in Table 1?\n- To the best of my knowledge, IPF only converges to the Schroedinger Bridge for $u_0 = 0.$ How valid is the pre-taining scheme introduced in Appendix C then?\n- Can the log normalizer be computed? It seems like the importance weights in (78) are intractable\n- Why did the authors use such small buffer sizes?\n- What symmetries did the authors use for ALDP? I could not find anything mentioned in the appendix. Also, why use internal coordinates and not Cartesian as done for the conformer generation? I reckon the harmonic prior would also help for the ALDP task\n- The authors write that they use Gaussian priors for all other targets. Did they also use the corrector matching algorithm in this case? This would not be necessary since there is also a SOC formulation for Gaussian priors using DDS. A comparison would be very interesting here. \n- In the Appendix it says ‘For SCLD, we do not use subtrajectory splitting’. So, this would be CMCD with LV loss then?\n- How much did the harmonic prior help for the synthetic energy functions?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### **Strengths And Weaknesses**\n\nStrenghts\n\n- Elegant SOC formulation of the Schroedinger Bridge problem\n- Both SOC and corrector objectives are highly scalable\n- The method is very sample efficient, which is crucial for most real-world applications\n\nWeakesses\n\n- Performance on ALDP (Ramachandran plot) is not very convincing, especially since internal coordinates were used. There exist several works that obtain significantly better performance, see e.g. [1].\n- The paper is heavily based on adjoint sampling\n\n[1] Temperature-Annealed Boltzmann Generators", "was_revised": true, "final_justification": "The authors have addressed all my concerns.", "rebuttal": "We thank the reviewer for raising all valuable comments and questions. We are excited that the reviewer identified the novelty of technical contributions in connecting SB with SOC, appreciated the scalability of our SB-based diffusion samplers, and the efficiency evidenced by empirical results on real-world sampling problems. We believe ASBS takes a significant step toward scalable diffusion samplers with flexible model design. We try our best to resolve all raised concerns in the responses below.\n\n---\n\n**1. Additional experiments on ALDP**\n\n- We first thank the reviewer for raising the comments. The experiments for ALDP (Table 3 & Fig 5) were conducted in internal coordinates mainly to maintain consistency with prior SOC-based diffusion samplers, specifically PIS [2], to which our ASBS sampler also belongs. That being said, our ASBS can indeed be run on the full Cartesian coordinates for ALDP. \n\n\n- In the table below, we report preliminary results for ASBS (using harmonic source) trained from scratch in full Cartesian coordinates, without relying on any explicit target or MCMC samples. Similar to Table 3, we report the KL divergences for the 1D marginal across five torsion angles, $(\\phi, \\psi, \\gamma_1, \\gamma_2, \\gamma_3)$, as well as the Wasserstein-2 on joint distribution of $(\\phi, \\psi)$. The latter serves as a quantitative measure of the Ramachandran plot. We kindly note that, in accordance with this year’s NeurIPS policy, we are unable to provide any external link to images or updating the manuscript.\n\n\n  |  | $\\phi$ | $\\psi$ | $\\gamma_1$ | $\\gamma_2$ | $\\gamma_3$ | $(\\phi,\\psi)$ |\n  |---|---|---|---|---|---|---|\n  | ASBS (Cartesian) | 0.78 | 0.54 | 0.20 | 0.11 | 0.11 | 1.30 |\n\n\n- Our preliminary results imply that ASBS can generate samples with reasonable structures, with Ramachandran plots that match those of prior SOC-based diffusion samplers in internal coordinates (see Table 3). Notably, ASBS achieves this without using any explicit target samples during training, relying solely on the energy function of ALDP. This contrasts with previous attempts on ALDP in Cartesian coordinates—such as [1,3] mentioned by the reviewer and Reviewer jVAy—which often require explicit samples from or near the target distribution. We emphasize this distinction as a key advantage of adjoint-based diffusion samplers, which we further detail in the second section.\n\n\n- Finally, while we believe that the current results could be further improved with additional tuning and/or pretraining, we also highlight an intriguing application of ASBS: using model samples from [1,3] as its source distribution. This effectively transforms ASBS to a “fine-tuning” method, offering an alternative to importance sampling at inference time, which can be costly in terms of energy evaluation. We highlight this flexibility in choosing source distribution, a feature that is otherwise infeasible in prior adjoint-based methods, particularly Adjoint Sampling. \n\n---\n\n**2. Discussion on Adjoint Sampling (AS)**\n\n- While our ASBS is closely related to AS, as noted by the reviewer, we emphasize that the motivation behind ASBS differs significantly from that of AS. Practically, AS can only be used for sampling from a distribution, while ASBS can be used to transport from any (e.g. a similar-but-incorrect) distribution. Despite this additional capability, we found ASBS also outperforms AS in pure sampling due to the less restrictive requirements. Theoretically, ASBS is designed to solve the Schrödinger Bridge (SB) problem (Eq 3) whereas AS solves a specific form of SOC problem (Eq 7). The theoretical connection between these two problems only became clear with the presentation of Thm 3.1. This is also acknowledged by the reviewer, i.e., \"Elegant SOC formulation of SB problem”. \n\n\n- Given the practical success of AS in terms of both performance and energy efficiency, along with our theoretical results in Thm 3.1 that generalizes AS to a broader, non-memoryless SOC problem, it is natural to pursue a new learning method that retains the algorithmic advantage of AS while addressing additional complexity from non-memorylessness and arbitrary source distributions. As shown in Sec 3.3, this can be achieved by incorporating an additional corrector matching objective (Eq 15). Since Eq 15 employs a simple matching objective, the overall ASBS method (Alg 1) remains as scalable as AS (Fig 4). \n\n\n- Such a straightforward algorithmic modification from AS to ASBS, although theoretically complex, has led to significant performance gains, evidenced by all of our experiments (Tables 2,3,4). Furthermore, it does not impede convergence to global solution (Thm 3.2). To summarize, we believe ASBS complements AS with additional flexibility in selecting informative sources and utilizing richer model classes, leading to significant performance gains without compromising algorithmic scalability. Together, these advancements enable ASBS to achieve strong performance and open new avenues for future applications. \n\n\n---\n\n\n**3. Clarifications on questions**\n\n- **LV-PIS & LV-DDS in Table 1.** We thank the reviewer for raising this question. Our intention was to convey that LV directly minimizes the variance of the importance weights (IWs), and thus, methods in this category involve the computation of IWs. We acknowledge that they do not perform importance sampling, which may have been incorrectly inferred from the current Table 1. We will clarify this in Table 1 in the revision.\n\n\n- **Pretraining with IPF.** As the reviewer conjectured, the global convergence of IPF requires initialization from either $u=0$ or $h=0$. Initializing IPF with a pretrained $u$—which corresponds to the first stage of Iterative Markovian Fitting (IMF)—has been theoretically explored recently [4] and shown to maintain global convergence in Gaussian setups. However, in more complex scenarios, we still observe stable local convergence. Alternatively, we can initialize ASBS by constructing its source distribution using samples from pretrained models, thereby leveraging ASBS's flexibility in selecting source distributions without compromising global convergence.\n\n\n- **Log normalizer of ASBS.** The importance weights in Eq 78 can be computed by re-parametrizing $u$ and $h$ with two potential networks, specifically $u = \\nabla v$ and $h = \\nabla \\bar h$ as shown in Eq 80, and then calculating the importance weights using Eqs 78 & 81. Alternatively, we can train an additional backward drift $v_t := \\nabla \\log \\hat \\varphi_t$ with standard SB matching [5], and then use Eq 79 to compute the ratio $\\log \\frac{\\hat \\varphi_1}{\\hat \\varphi_0}$. While both methods are tractable, they introduce additional complexity. Exploring alternative approaches to compute the weights would be a promising direction for future research (L295).\n\n\n- **Buffer size in Table 5.** This’s a typo. The correct buffer size $|\\mathcal{B}|$ for conformer generation should be 64000, which is consistent with the order of magnitude used for other tasks. We thank the reviewer for the meticulous reading!\n\n\n- **Symmetry for ALDP.** In practice, we utilize the full internal coordinate transform from FAB [6], which converts Cartesian coordinates into bond lengths, bond angles, and dihedral angles, effectively removing 6 degrees of freedom (3 for translation and another 3 for rotation). Non-augular coordinates are further normalized using samples with minimal energies. We refer the reviewer to Appendix F.1 in [6] for further details and will include these discussions in the revised Appendix.\n\n\n- **Corrector matching (CM) for Gaussian prior.** We first clarify that CM is used for all ASBS models, including those with Gaussian priors. This is because the base processes $p^\\text{base}$ for all ASBS models are not memoryless—with $f :=0$ and $\\sigma_t$ set according to Table 5— hence necessitating the use of CM, as discussed in Sec 3.2. As the reviewer conjectured, there is indeed a memoryless base process (specifically, the one considered in DDS) for Gaussian prior, which eliminates the need for CM. This corresponds exactly to the leftmost plot in Fig 1, where significant noise is injected into sampling to ensure the conditional independence of the memoryless process (Eq 6). We have also reported its performance (i.e., DDS) on synthetic energies (Table 2) and ALDP (Table 3), where ASBS outperformed DDS in both cases. We will include these clarifications in the revision.\n\n\n- **SCLD.** We thank the reviewer for catching the error. We did use subtrajectory splitting (with the default value of 4) for SCLD, so it does not degenerate to CMCD with LV. In practice, we find that using subtrajectories yields better results. This will be corrected in the revision, and we appreciate the reviewer’s meticulous reading.\n\n\n- **Harmonic vs Gaussian prior in synthetic energies.** In the below table, we present a comparison of ASBS using harmonic and Gaussian priors. The results are consistent with those in Table 4, showing slight improvements with harmonic priors in both cases.\n\n  |  | DW4 (W2) | DW4 (EW2) | LJ13 (W2) | LJ13 (EW2) | LJ55 (W2) | LJ55 (EW2) |\n  |---|---|---|---|---|---|---|\n  | Harmonic | 0.38 | 0.19 | 1.59 | 1.28 | 4.00 | 27.69 |\n  | Gaussian | 0.42 | 0.20 | 1.60 | 1.82 | 4.09 | 38.70 |\n\n\n---\n\n[1] Temperature-Annealed BG  \n[2] Path Integral Sampler  \n[3] Scalable Equilibrium Sampling with Sequential BG  \n[4] Diffusion & Adversarial SB via IPMF  \n[5] Diffusion SB Matching  \n[6] Flow Annealed Importance Sampling Bootstrap"}, {"review_id": "VCdnO8ZFKP", "reviewer": "Reviewer_jVAy", "summary": "This paper attempts to tackle the problem of sampling from a Boltzmann distribution.  They do so by building on the recent Adjoint Sampling work, noting that the memoryless condition -- an issue first recognized in the Adjoint Matching paper -- may be solved by employing a stochastic optimal control approach to the Schrödinger Bridge problem.  The authors finally demonstrate ASBS's efficacy on a number of sampling tasks.", "questions": "1. How does ASBS perform compared to the improved RDKit baseline in the conformer generation task?\n\n2. Can ASBS scale up to any of the even larger scale experimental settings I described in the strengths/weaknesses section?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "I found the paper to be quite well written, that the theoretical results -- especially that the memoryless condition can be relaxed via the proposed SOC approach -- were significant, and that the experimental results showed ASBS's efficacy.  As such, I believe the paper merits an acceptance.\n\nI don't have many notes or questions regarding the theoretical section of the paper as I found the theoretical results to be satisfactory.  However, I have a few concerns regarding the paper's experimental validation.  Further, I feel that while ASBS performed well in the presented experimental settings, the experimental settings considered in the paper are not ones which significantly push the boundary on amortized samplers (e.g., ALDP in internal coordinates and the various synthetic energy functions have been solved for some time now).  I already feel the paper merits acceptance, but if the authors were to include further experiments showing that ASBS scales well enough to tackle problems like alanine tripeptide or larger molecules in, e.g., cartesian coordinates (a more challenging task due to how internal coordinates allow one to bypass problems with very sharp energy functions) I would be happy to raise my score to a strong accept.  Note that prior work (albeit it is a Boltzmann generator) in [1] have been able to successfully to these sorts of problems, so it would be interesting to see whether ASBS could scale to this.\n\nAs a side note, I believe that the RDKit baseline is not representative of the actual state-of-the-art performance of RDKit based conformer generation, see [2] which achieves superior performance to most deep learning based conformation sampling methods using some different settings of RDKit than the ETKDG baseline used in this paper. \n\nReferences: \n\n[1] Tan, Charlie B., et al. \"Scalable equilibrium sampling with sequential boltzmann generators.\" ICML (2025).\n\n[2] Zhou, Gengmo, et al. \"Do deep learning methods really perform better in molecular conformation generation?.\" arXiv (2023).", "was_revised": true, "final_justification": "I believe this paper should be accepted to NeurIPS.  The theory is interesting and experimental results good enough that I believe it merits presentation at the conference.  I recommend a 5.", "rebuttal": "We thank the reviewer for raising all valuable comments. We are excited that the reviewer identified the novelty of technical contributions in relaxing the memoryless condition, appreciated the well-written presentation, and acknowledged our strong empirical results. We believe ASBS takes a significant step toward scalable diffusion samplers with flexible model design. We try our best to resolve all raised concerns in the responses below.\n\n---\n\n**1. Additional experiments on ALDP**\n\n- We first thank the reviewer for raising the comments. The experiments for ALDP (Table 3 & Fig 5) were conducted in internal coordinates mainly to maintain consistency with prior SOC-based diffusion samplers, specifically PIS [4], to which our ASBS sampler also belongs. That being said, our ASBS can indeed be run on the full Cartesian coordinates for ALDP. \n\n\n- In the table below, we report preliminary results for ASBS (using harmonic source) trained from scratch in full Cartesian coordinates, without relying on any explicit target or MCMC samples. Similar to Table 3, we report the KL divergences for the 1D marginal across five torsion angles, $(\\phi, \\psi, \\gamma_1, \\gamma_2, \\gamma_3)$, as well as the Wasserstein-2 on joint distribution of $(\\phi, \\psi)$. The latter serves as a quantitative measure of the Ramachandran plot. We kindly note that, in accordance with this year’s NeurIPS policy, we are unable to provide any external link to images or updating the manuscript.\n\n\n  |  | $\\phi$ | $\\psi$ | $\\gamma_1$ | $\\gamma_2$ | $\\gamma_3$ | $(\\phi,\\psi)$ |\n  |---|---|---|---|---|---|---|\n  | ASBS (Cartesian) | 0.78 | 0.54 | 0.20 | 0.11 | 0.11 | 1.30 |\n\n\n- Our preliminary results imply that ASBS can generate samples with reasonable structures, with Ramachandran plots that match those of prior SOC-based diffusion samplers in internal coordinates (see Table 3). Notably, ASBS achieves this without using any explicit target samples during training, relying solely on the energy function of ALDP. This contrasts with previous attempts on ALDP in Cartesian coordinates—such as [1,3] mentioned by the reviewer and Reviewer 5V2q—which often require explicit samples from or near the target distribution. We emphasize this distinction as a key advantage of adjoint-based diffusion samplers, which we further detail in the second section.\n\n\n- We would like to highlight that the amortized conformer generation task (Table 4) already includes larger molecules, with 50-65 atoms in the test sets of SPICE and GEOM-DRUG, respectively, whereas the suggested alanine tripeptide consists of 33 atoms. Given ASBS's strong performance in amortized conformer generation and its capability to generate the Boltzmann distribution of ALDP in Cartesian spaces, we believe ASBS has great potential for tackling larger molecules. Scaling ASBS for amortized Boltzmann generators would be a promising direction worthy of its own in-depth investigation.\n\n---\n\n**2. Discussion on the new RDKit baseline**\n\n- We thank the reviewer for referring us to the new RDKit baseline [2], which post-processes RDKit samples using a K-means clustering algorithm. Following the reviewer’s suggestion, we applied their method to our GEOM-DRUG test set and found that the recall converges at the same 0.75Å threshold, as reported in Table 4, to approximately 54.4%. This performance is comparable to ASBS with a harmonic prior and relaxation post-processing.\n\n\n- We note, however, that [2] appears to generate a much larger number of conformers than the evaluation pipeline used in Table 4. That said, we believe that ASBS could similarly benefit from clustering post-processing by generating a comparable number of conformers. Exploring such post-processing techniques, which have proven effective in prior work, would be an interesting direction for future research. We thank the reviewer for highlighting this opportunity.\n\n---\n\n[1] Scalable Equilibrium Sampling with Sequential BG  \n[2] Do DL Methods Really Perform Better in Molecular Conformation Generation  \n[3] Temperature-Annealed BG  \n[4] Path Integral Sampler"}, {"review_id": "yDvEofaCET", "reviewer": "Reviewer_NUeM", "summary": "This paper considers how to modify a reference SDE in order to sample from an unnormalized density. Like prior literature, it proposes to correct the SDE with an additive drift that should be learnt (Eq 2). Overall, there are two ways of learning the drift in the literature:\n\n*(i) This drift is learnt by minimizing a trajectory-level cost (Eq 4) and a general terminal cost*\n\nComputationally, this problem can be efficiently solved using so-called adjoint sampling (Eq 14). \n \nTheoretically, the resulting SDE does *not* sample the target density unless some additional, restrictive assumptions are made (memoryless schedule and Dirac initialization). \n\n*(ii) The drift is learnt by minimizing a trajectory-level cost (Eq 3) under a terminal condition*\n\nComputationally, this classical problem called Schrodinger Bridge can be solved using an Iterative Proportional Fitting algorithm. \n\nTheoretically, the resulting SDE *does* sample the target density. \n\nThe authors identify a special terminal cost (Eq 9) so that (i) and (ii) are theoretically equivalent, and in particular the resulting SDE samples from the target density. Because the special terminal cost depends on an unknown quantity, the authors propose to learn it on-the-go (Eq 15). This way, the authors aim to use the computational efficiency of (i) to solve the problem in (ii).", "questions": "Q1. The authors propose a nested optimization problem, where the outer loop updates follow Eq 14 and the inner loop updates follow Eq 15. How brittle is the interplay between these two updates? For example, in the inner loop which approximates Eq 15, how many iterations of the optimizer are required to get stable results? \n\nQ2. Could the authors illustrate their algorithm's ability to find the modes, by illustrating it on the synthetic 40-mode Gaussian mixture in 2D from Figure 2 of [1]? Please use an initial distribution that has the same density as one of the modes of the mixture (so as to not assume that the user knows where the other modes are). \n\nThe resulting plot could be reported in the Appendix and could compare only ASBS and a vanilla Langevin to get a visual understanding of how the authors' algorithm deals with the \"search problem\" (finding the target modes) as compared to a basic local exploration algorithm (Langevin). \n\nHere is an additional discussion of why this plot would be useful. While the results on molecular tasks are impressive, they do not provide a visual understanding of which modes are found by the algorithm. For example, is it those modes that are closest to the initial distribution that are found by ASBS? Similarly, Figure 2, while illustrative, is an example where the target distribution is symmetric and the initial distribution is initialized in the barycenter of the target modes, so even a Langevin process would be an efficient sampler.", "limitations": "Yes.", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n\nThe paper is very clearly written and proposes to alternate between adjoint sampling updates (Eq 14) and corrective updates (Eq 15) to learn a corrective drift to an SDE for sampling from an unnormalized density. The alternate optimization scheme (Eqs 14-15) is connected to the classical Iterative Proportional Fitting algorithm which is appreciated. The benchmarks with other diffusion samplers is are satisfactory and the experiments on synthetic and molecular problems are appreciated. \n\n## Weaknesses\n\nOne of the main difficulties in sampling from an unnormalized density is finding the modes. It would be nice to visually inspect the algorithm's ability to find the modes as in [1]: please see Q2 for a more detailed request. \n\nIt is also unclear how the ASBS algorithm's ability to find modes scales as a function of the dimensionality and of the number of modes. \n\n[1] L. Illing Midgley, V. Stimper, G. N. C. Simm, B. Scho ̈lkopf, and J. Miguel Hernandez-Lobato. Flow annealed importance sampling bootstrap. In Interna- tional Conference on Learning Representations (ICLR), 2023.", "was_revised": true, "final_justification": "I maintain my recommendation for a strong acceptance. \n\nThe theory is well-presented and impactful. The authors satisfactorily conducted experiments that answered the questions I had in my review, on the stability of their optimization scheme and on the mode-finding ability of their algorithm.\n\nI have also read the comments by the other reviewers which conclude overall that the theoretical contributions are worthwhile for the community, while the experiments are satisfactory but could be improved, which the authors did in their rebuttal.", "rebuttal": "We thank the reviewer for raising all valuable comments and the comprehensive summary. We are excited that the reviewer identified the novelty of technical contributions and connection to the Iterative Proportional Fitting algorithm, appreciated the well-written presentation, and acknowledged our strong empirical results. We believe ASBS takes a significant step toward scalable diffusion samplers with flexible model design. We try our best to resolve all raised concerns in the responses below.\n\n\n---\n\n**1. Additional experiments on mode searching**\n\n- We thank the reviewer for raising the comments. Following the reviewer’s suggestion, we conduct additional experiments on the 40-mode GMM in 2D from [1]. Specifically, we instantiate ASBS with a uni-variance Gaussian source distribution centered at zero, effectively assuming that users have no prior knowledge of the target modes, as the initial distribution does not coincide with any target modes. We also run a vanilla Langevin baseline for 1 million steps, starting from the same source distribution.\n\n\n- The table below presents the count of identified modes out of a total of 40. Notably, ASBS is able to identify almost all modes. In contrast, the vanilla Langevin baseline appears to suffer from a slow mixing rate, recovering less than half of the total modes even after 1 million steps. We highlight this distinction as an advantage of constructing diffusion samplers from the stochastic control and Schrödinger Bridge frameworks, which allows theoretical convergence to target distribution within a finite horizon. Finally, we believe that with proper tuning of the ASBS noise schedule, its performance can be further enhanced.\n\n  |  | # of identified modes out of 40 |\n  |---|---|\n  | Vanilla Langevin | 15/40 |\n  | ASBS | 38/40 |\n\n\n- We kindly note that, in accordance with this year’s NeurIPS policy, we are unable to provide any external link to images or updating the manuscript. However, we believe these are valuable discussions and will include the figures in the revision. We thank the reviewer again for bringing up the topic.\n\n---\n\n**2. Clarification on nested optimization (Eqs 14 & 15)**\n\n- Regarding the sensitivity of the interplay between Adjoint Matching (Eq. 14) and Corrector Matching (Eq. 15), we observe in practice that both optimizations exhibit stable convergence due to their formulation as simple regression (i.e., matching) objectives. While exact convergence to the minima of Eqs. 14 and 15 is required for global convergence (see Thm 3.2), we found that early termination with a fixed number of iterations did not cause training to diverge or become unstable in any of the tasks considered.\n\n\n- In practice, the Corrector Matching (Eq 15) often requires significantly fewer iterations to converge compared to the Adjoint Matching (Eq 14). Specifically, the number of iterations for CM ($M_\\text{crt}$) are typically 2 to 10 times fewer than for AM ($M_\\text{adj}$). We refer the reviewer to Table 5 in Appendix for the exact values of $M_\\text{crt}$ and $M_\\text{adj}$ on each task.\n\n\n---\n\n[1] Flow Annealed Importance Sampling Bootstrap  \n[2] Non-Equilibrium Transport Sampler"}]

Author rebuttal:

Meta-review: This paper introduces the Adjoint Schrödinger Bridge Sampler (ASBS), a diffusion-based framework for sampling from unnormalized densities, built on a refined formulation of stochastic optimal control and the Schrödinger bridge. All reviewers agree that this work is exceptionally strong and merits oral presentation.

The contributions of this paper are groundbreaking. It advances the theoretical foundations of diffusion-based sampling with a bold relaxation of the memoryless condition in adjoint sampling, opening new directions for the field. The algorithmic formulation is not only elegant but also conceptually unifying, drawing connections to classical iterative proportional fitting in optimal transport while extending them in novel ways. On the practical side, the method achieves remarkable scalability and sample efficiency, setting a new benchmark for tackling complex multimodal distributions.

What makes this paper stand out is its rare combination of strengths across the full spectrum: rigorous theory, innovative algorithms, practical relevance, and convincing experiments. The empirical results are extensive and show the method's unique ability to navigate diverse modes, including those far from initialization, a clear indication of its power and robustness. The reviewers also highlighted the excellent clarity of exposition and the thoughtful rebuttal, which successfully addressed all initial concerns.

In conclusion, this is a groundbreaking submission that combines theoretical depth, algorithmic innovation, and practical effectiveness. It is likely to shape future research in this area, and I strongly recommend acceptance as an oral.
Decision: Accept (oral)