Adjoint Schrödinger Bridge Sampler

• ASBS is a diffusion-based sampler that formulates Boltzmann sampling as a Schrödinger Bridge problem, enabling scalable sampling without target samples.
• It employs a two-stage matching approach—Adjoint Matching and Corrector Matching—to optimize controlled diffusion and debias diverse source priors.
• Empirical evaluations show ASBS achieves high sample quality and efficiency across synthetic energy functions, molecular models, and discrete state spaces.

Adjoint Schrödinger Bridge Sampler (ASBS) is a diffusion-based sampler for approximate sampling from a Boltzmann distribution when the target law is specified only through an unnormalized energy function $E(x)$, rather than through direct target samples (Liu et al., 27 Jun 2025). In its original formulation, ASBS is designed to combine three properties that earlier diffusion samplers did not typically achieve simultaneously: scalability, matching-based training without target samples, and flexibility beyond the restrictive “memoryless” setup used by earlier adjoint-based diffusion samplers. Its central construction recasts Boltzmann sampling as a Schrödinger Bridge (SB) problem and solves that bridge through a stochastic optimal control (SOC) view, yielding a controlled diffusion that transports a source distribution $\mu$ to the target $\nu$ while minimizing kinetic control energy. The name ASBS has subsequently also appeared in a sampler-oriented framing of Adjoint Schrödinger Bridge Matching for image generation, and the framework has been extended to discrete state spaces (Shin et al., 17 Feb 2026, Guo et al., 9 Feb 2026).

1. Problem setting and defining objective

The original ASBS paper considers sampling from a Boltzmann distribution of the form

$$\nu(x)=\frac{e^{-E(x)}}{Z}, \qquad Z=\int e^{-E(x)}\,dx,$$

where the energy $E(x)$ is known but the normalizing constant $Z$ is intractable (Liu et al., 27 Jun 2025). The practical problem is to generate approximate samples from $\nu$ without relying on expensive MCMC chains and without access to target samples. This is the setting in which many diffusion-based Boltzmann samplers become difficult to scale, because they often require importance-weighted estimation or other approximations of unavailable target samples.

ASBS addresses this by learning a controlled stochastic process whose terminal law approximates $\nu$. The controlled dynamics are written as

$$dX_t = \big[f_t(X_t) + \sigma_t u_t^\theta(X_t)\big]\,dt + \sigma_t\,dW_t,\qquad X_0\sim \mu,$$

where $f_t$ is the base drift, $\mu$0 is the noise schedule, $\mu$1 is the learned control, and $\mu$2 is the source distribution. The objective is to choose $\mu$3 so that $\mu$4.

A distinguishing feature of ASBS is that $\mu$5 is not restricted to a degenerate source. The method is explicitly formulated for arbitrary source distributions, including Gaussian priors and domain-specific priors such as a harmonic oscillator prior for molecular systems. This shift is central to the method’s scope, because it removes a prior design constraint that had limited earlier adjoint-based samplers.

2. Schrödinger Bridge and stochastic optimal control formulation

ASBS is grounded in the Schrödinger Bridge problem, posed as a path-space KL minimization relative to an uncontrolled reference process (Liu et al., 27 Jun 2025). In the paper’s formulation,

$\mu$6

subject to

$\mu$7

Here $\mu$8 denotes the path measure induced by the controlled process and $\mu$9 the uncontrolled base process. The paper interprets this as a kinetic-optimal transport problem: among all controlled paths that match the source and target marginals, the SB selects the one with minimal control energy.

The theoretical move that organizes the rest of ASBS is the SOC reinterpretation of that bridge. The paper states an SOC problem with terminal cost

$\nu$0

where $\nu$1 is the SB terminal potential. The optimal SB control then has the form

$\nu$2

with Schrödinger potentials satisfying forward and backward integral equations. The corresponding path-density factorization

$\nu$3

is the mechanism by which the source distribution is “debias[ed]” at the endpoint. In the original presentation, this factorization is not merely formal: it is what permits learning from model trajectories while still enforcing the target marginal through the terminal potential.

This formulation places ASBS within the intersection of SB theory, diffusion generative modeling, and SOC. In that sense, ASBS is not a generic score-based sampler; it is a controlled transport construction whose control law is tied to SB potentials rather than directly to target-score supervision.

3. Adjoint Matching, Corrector Matching, and the learned debiasing mechanism

The core algorithm alternates between two matching objectives, Adjoint Matching (AM) and Corrector Matching (CM), both of which depend only on samples from the current controlled process rather than on target samples (Liu et al., 27 Jun 2025). This is the point at which the SOC perspective becomes operational.

With a current corrector estimate $\nu$4, the control is updated by minimizing the AM objective

$\nu$5

This stage trains the drift to match an adjoint signal formed from the energy gradient and the current corrector.

The second stage updates the corrector. In the paper’s description, the corrector estimates the SB terminal potential gradient $\nu$6, and its role is essential when the source prior is nontrivial. The corrector is characterized variationally through a regression involving $\nu$7, and the paper emphasizes that this identity reduces to the Adjoint Sampling objective when the source degenerates to a Dirac delta. The practical significance is explicit: for arbitrary priors, the corrector replaces the earlier memoryless workaround with a learned debiasing mechanism.

The alternating AM/CM procedure is initialized with $\nu$8 and then repeats stagewise updates of the control and corrector. In implementation, the paper reports the use of replay buffers, gradient clipping on the energy gradient, and time-scaling factors; the control and corrector are represented by neural networks, often EGNNs in molecular settings.

The main convergence theorem identifies AM and CM with the two half-bridge steps of Iterative Proportional Fitting (IPF): AM solves a forward half bridge and CM a backward half bridge. Under the assumption that each matching stage reaches its critical point, the alternating procedure converges to the SB solution,

$\nu$9

This global convergence guarantee is one of the main theoretical differentiators of ASBS within matching-based Boltzmann samplers.

4. Removal of the memoryless restriction

ASBS is best understood in relation to the earlier Adjoint Sampling construction and to the broader critique of memoryless diffusion processes (Liu et al., 27 Jun 2025). Adjoint Sampling used a special SOC problem with a Dirac source $$\nu(x)=\frac{e^{-E(x)}}{Z}, \qquad Z=\int e^{-E(x)}\,dx,$$0, zero base drift, and a terminal cost involving $$\nu(x)=\frac{e^{-E(x)}}{Z}, \qquad Z=\int e^{-E(x)}\,dx,$$1. That setup relied on the so-called memoryless condition, written in the original paper as

$$\nu(x)=\frac{e^{-E(x)}}{Z}, \qquad Z=\int e^{-E(x)}\,dx,$$2

meaning that the source and terminal states are independent under the base process. Under that condition, the initial-value bias disappears and the target marginal becomes easy to realize.

ASBS removes this restriction entirely. The paper’s key claim is that the memoryless condition is highly restrictive, especially when useful source priors are nontrivial. In molecular applications, for example, a harmonic prior can be substantially more appropriate than a simple Gaussian, yet such a prior is excluded by the earlier memoryless setup. The corrector is introduced precisely because non-memoryless processes induce an initial-value bias through the initial-value function $$\nu(x)=\frac{e^{-E(x)}}{Z}, \qquad Z=\int e^{-E(x)}\,dx,$$3.

A later paper on “Efficient Generative Modeling beyond Memoryless Diffusion via Adjoint Schrödinger Bridge Matching” develops this critique in a broader generative-modeling setting and, in its appendix, refers to the sampler-oriented framing as ASBS (Shin et al., 17 Feb 2026). There the memoryless condition is formalized as

$$\nu(x)=\frac{e^{-E(x)}}{Z}, \qquad Z=\int e^{-E(x)}\,dx,$$4

and standard diffusion is described as an SB special case under such a base process. The paper argues that memorylessness leads to independent data-noise pairing, noisy training targets, and geometrically curved reverse trajectories. Its non-memoryless SB construction is organized in two stages: first learning a forward coupling as a data-to-energy transport problem, then training the backward generative dynamics under that induced coupling. The paper reports lower trajectory straightness $$\nu(x)=\frac{e^{-E(x)}}{Z}, \qquad Z=\int e^{-E(x)}\,dx,$$5, lower trajectory variance, improved forward-backward consistency, and image-generation results including FID $$\nu(x)=\frac{e^{-E(x)}}{Z}, \qquad Z=\int e^{-E(x)}\,dx,$$6 on CIFAR-10 and strong low-NFE performance on latent FFHQ.

Taken together, these papers locate ASBS within a broader research program: replacing memoryless endpoint independence with informative coupling structure, while retaining simple matching objectives.

5. Empirical behavior in Boltzmann sampling and molecular modeling

The original ASBS paper evaluates the method on three families of tasks: classical synthetic energy functions, alanine dipeptide Boltzmann sampling, and amortized conformer generation (Liu et al., 27 Jun 2025). Across these settings, the empirical emphasis is not only on sample quality but also on efficiency under the constraint that only the energy function is available.

On classical synthetic energy functions, the benchmarks include MW-5, DW-4, LJ-13, and LJ-55. The paper reports that ASBS achieves the best or near-best performance across all metrics, including Sinkhorn distance and geometric Wasserstein distances, and that generated energy histograms closely match the ground truth. The LJ-55 result is highlighted as a case in which ASBS scales to a high-dimensional setting where other methods struggle. A practical claim made in this section is that ASBS attains strong sample quality at lower energy-evaluation cost than most competing diffusion samplers; the only overhead relative to Adjoint Sampling is the additional corrector network.

For alanine dipeptide, evaluation focuses on torsion-angle marginals and Ramachandran plots. The paper states that ASBS achieves the lowest KL divergence across all five torsion angles and that the joint $$\nu(x)=\frac{e^{-E(x)}}{Z}, \qquad Z=\int e^{-E(x)}\,dx,$$7 distribution is very close to the ground truth. At the same time, the method is described as somewhat mode-seeking, which the authors identify as a known behavior of SOC-based diffusion samplers. Low-density modes may therefore be missed, and the paper notes that importance weighting could mitigate this in future work.

In amortized conformer generation, ASBS is tested with both a Gaussian prior and a harmonic prior. The harmonic prior is described as particularly useful because it encodes a physically meaningful geometry bias. The paper reports that ASBS with harmonic prior often matches or outperforms Adjoint Sampling even without RDKit pretraining, and that with pretraining it achieves the best results on most metrics. The stated conformer-generation advantages include improved coverage and lower AMR both before and after relaxation. These experiments are the paper’s clearest empirical demonstration of why arbitrary, non-memoryless priors matter in practice.

6. Extensions, discrete generalization, and terminological scope

The name ASBS now denotes a small family of closely related constructions rather than a single fixed algorithmic object. In the 2026 discrete extension, “Discrete Adjoint Schrödinger Bridge Sampler” develops a version of ASBS for discrete state spaces and continuous-time Markov chains (CTMCs) (Guo et al., 9 Feb 2026). The target remains an unnormalized energy-based law,

$$\nu(x)=\frac{e^{-E(x)}}{Z}, \qquad Z=\int e^{-E(x)}\,dx,$$8

but the state space is discrete, $$\nu(x)=\frac{e^{-E(x)}}{Z}, \qquad Z=\int e^{-E(x)}\,dx,$$9, and the dynamics are given by controlled jump rates rather than an SDE. The paper’s main theoretical claim is that the core mechanism of adjoint matching is $E(x)$0, provided the reference kernel has an additive structure.

That extension requires a specific structural assumption: the state space is taken to be the cyclic group

$E(x)$1

This group structure is described as necessary for the AM derivation, because it supplies a discrete analogue of additive noise. The reference process is a uniform additive CTMC with single-coordinate jumps, and the paper derives a discrete SB problem, a discrete SOC formulation with terminal cost $E(x)$2, and a discrete analogue of the optimal SB control

$E(x)$3

Training alternates between controller and corrector objectives formulated through Bregman-divergence regression targets, and generation uses $E(x)$4-leaping.

Empirically, the discrete paper evaluates DASBS on Ising and Potts models, against LEAPS, DFNS, UDNS, MDNS, and Metropolis-Hastings. It reports competitive or best-in-class sample quality among uniform-based discrete neural samplers, together with substantial gains in efficiency and scalability. For Ising high temperature, the table reports DASBS at about $E(x)$5 steps and $E(x)$6 runtime, compared with LEAPS at $E(x)$7 steps and $E(x)$8. The same paper also states several limitations: experiments are restricted to synthetic benchmarks, the method requires discrete score-like quantities such as $E(x)$9, importance weighting can reduce effective sample size, and the cyclic-group assumption limits universality.

The broader generative-modeling paper on ASBM adds another layer of terminological scope (Shin et al., 17 Feb 2026). There, ASBS appears as a sampler-oriented interpretation of a two-stage non-memoryless SB framework for high-dimensional data, including image generation and one-step distillation. That usage does not replace the original 2025 ASBS definition, but it does indicate that the adjoint Schrödinger-bridge viewpoint has become a general design principle: first recover a meaningful endpoint coupling under a non-memoryless bridge, then train generative dynamics by matching objectives under that coupling.

Across these variants, the common core remains stable. ASBS denotes an adjoint, SB-based, matching-trained sampler whose defining technical move is to replace memoryless endpoint independence with a learned debiasing or coupling mechanism, while avoiding reliance on target samples during training.