DriftLite: Lightweight Drift Control for Inference-Time Scaling of Diffusion Models
Authors: Yinuo Ren
Wenhao Gao
Lexing Ying
Grant M. Rotskoff
Jiequn Han
Venue: ICLR 2026 Poster (2026)
Keywords: Diffusion Model
Inference-Time Scaling
Variance Reduction
Sequential Monte Carlo
Guidance
Primary area: generative models
TL;DR: We introduce DriftLite, a lightweight method that reduces variance in particle dynamics, enabling scalable inference-time adaptation of diffusion models.

Abstract: We study inference-time scaling for diffusion models, where the goal is to adapt a pre-trained model to new target distributions without retraining. Existing guidance-based methods are simple but introduce bias, while particle-based corrections suffer from weight degeneracy and high computational cost. We introduce *DriftLite*, a lightweight, training-free particle-based approach that steers the inference dynamics on-the-fly with provably optimal stability control. DriftLite exploits a fundamental degree of freedom in the Fokker-Planck equation between the drift and particle potential, and yields two practical instantiations: *Variance- and Energy-Controlling Guidance (VCG/ECG)* for approximating the optimal drift with modest and scalable overhead.  Across Gaussian mixture models, particle systems, and large-scale protein-ligand co-folding problems, DriftLite consistently reduces variance and improves sample quality over pure guidance and sequential Monte Carlo baselines. These results highlight a principled, efficient route toward scalable inference-time adaptation of diffusion models. Our source code is publicly available at https://github.com/yinuoren/DriftLite.

Published as a conference paper at ICLR 2026
DRIFTLITE: LIGHTWEIGHT DRIFT CONTROL FOR INFERENCE-TIME SCALING OF DIFFUSION MODELS
Yinuo Ren', Wenhao Gaoi, Lexing Ying"!, Grant M. Rotskoff-1, , Han
I Institute for Computational and Mathematical Engineering. Stanford University
2 Department of Chemistry, Stanford University
Department of Mathematics, Stanford University
4 Center for Computational Mathematics, Flatiron Institute
[y ren, whgao, lex lexing, ig, rottsoof.
* *Correspondence to: Jiequn Han <flaadffattrroonsiiitte....
ABSTRACT
We study inference-time scaling for diffusion models, where the goal is to adapt pre-trained model new target distributions without retraining. Existing guidance-based methods are simple but introduce bias, while particle-based cor- rections suffer from weight degeneracy and high computational cost. We intro- duce DriftLite, lightweight, training- free particle-basee approach that steers the inference dynamies on the fly with provably optimal stability control. DriftLite ex- ploits fundamental degree of freedom in the Fokker-Planck equation between the drift and particle potential, and yields two practical instantiations: Variance- and Energy-Conirolle Controlling Guidance (VCG/ECG) for approximating the optimal drift with modest and scalable overhead. Across Gaussian mixture models, particle sys- tems, and large-scale protein- ligand co-folding problems, DriftLite consistently reduces variance and improves sample quality over pure guidance and sequential Monte Carlo baselines. These results highlight principled, efficient route to- ward scalable inference-time adaptation of diffusion models. Our source code is publicly available at https:: /ttps://qithub. com http:///at..tccccoooooooo/////// www.oooooeee//iittteeee
INTRODUCTION
Diffusion- DSckseDickstein et all 2015: Ho et all 2020 Song & Ermon, 2019 Song et all 2020)
and flow-based (Zhang et anl 2018 Lipman et al, 2022 Albergo & Vanden-Eyynden, 2022: Lm
etall 2022; Ren all 20255) models have revolutionized generative modeling, achieving state-of-
the-art performance
domains ranging from creative media synthesis (Rombach et all, 2022: La
etal 2023 all 2022: Austin atal (22T fundamental scientific discovery (Xu 2022
Watson al. 2023% Duan et all 2023; etal 2024: Zhu all 2024: Zemeet 2025: Duan et all. 2025). They typically rely neural network approximate time-dependent vector field, which guides a stochastic process from noises to a complex target. However, training is resource-intensive making impractical retrain from scratch for every new setting. This renders lightweight adaptation of pre-trained models to target distributions that is both compelling and essential.
To this end, spectrum of adaptation methods has emerged. At one end are guidance-based tech- niques, the most popular and straightforward inference-time techniques, which inject new informa- tion into the drift term, such as classifier (Dhariwal Nichol, 2021) or classifier-fres guidance (Ha & Salmans, 2022) and its many variants (Chung et all 2022: Trppe et all, 2022: Bansal et all, 2023; Song et all, 2023at He all, 2023; Guo all, 2024 Zheng all, 2024: Rojas all, 2025).. While simple and effective for many tasks, these methods are often heuristic and introduce uncontrolled bias (Chrdambaram el all 2024; Wiletal ()225)) a significant trawback for scientific applications where sampling accuracy is paramount. On the opposite are methods that resort to extra train- ing, such as fine-tuning (Fan & Led 2023: Fan et all 2023 Black el all 2023; Clark all 2023 Wallace et all (2024) in the LLM context (Ouyang all 2022: Ratalovv all 2023). learning within a stochastic control framework (Dommgo-Ennee Enri-E et all, 20Z4R; Uehara ef all 2024: Thome ton all, 2025), similar to leariin-bbase samplers (Zhang & Chen. 2021; Vargas et all, 2/125

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Domingo-Ente Ennchh et al. 2022h; Richter & Bernen 2023: Zhu et al 2025), adding additional train- ing objectives (Venkatramen et all 2024: Thormton et all 2025), but this shifts the computational burden back to retraining, forfeiting the efficiency of inference-time approaches.
Between these ends lies a middle ground training-free free but more sophisticated inference- ap- proaches. A promising direction formulates the problem in Bayesian and Monte Carlo sampling frameworks (Du el all 2023; Xu & Chi, 2024: Wi et all 20248; Coeurdoux et all 2024: [Bruna &
Han. M022) Zheng et all (025).. In particular, Sequential Monte Carlo (SMC) methods (D=l Mora et all. 2006; Doucet et all, 20000 have been recently introduced to correct for the bias of guidance
by simulating the target dynamics with weighted particles (Wi ef al.
2023 Cardoso ctall 2023;
Skreta all 2025: Chen et all, 2102) Singhal et all 2025: Lec all
2026) Despite their strong
theoretical grounding and asymptotic guarantees, these particle-based methods face a critical prac- tical bottleneck: weight degeneracy. As the simulation progresses, the weights of a few particles grow exponentially while the rest decay, causing the effective sample size to collapse. To mitigate this, one may increase the number of particles, raising computational cost, or use fewer particles, resulting in instability and degraded sample quality.
Our work introduces DrifiLite, lightweight, training-free inference-time scaling approach that re- solves the inherent instability of particle-base basee methods without sacrificing mathematical rigor. By exploiting fundamental degree of freedom in the fokker-Planck equation, we actively control particle drift on the fly. This proactive steering mechanism absorbs sources of weight variation, pre- venting the weight collapse common in passive reweighting schemes and dramatically improving stability. The method's modest and scalable computational overhead requiring only the solution of small linear system per step, makes it fundamentally lightweight, especailly compared with meth- ods that require retraining or extra training of additional control networks. Unlike learning-base control (Vargas al. 20235: Richter Bernen, 2023) or heuristic control frameworks (He all, 2025), DriftLite is a training-free solution derived directly from the principle of variance reduction. Comparing with computationally intensive PDE solvers (Albergo & Vanden- Eijndenn, 2024) and tra- jectory balance-based GFlowNet methods (Bengio et all, 2023: Sendera et all, 2024; Venkatraman cLall 2022), our method is applied on the fly without any parameter updating or backpropagating through the diffusion model. It is designed to scalably match an entire target distribution in high- dimensional, continuous systems, more rigorous than targeting sample-focuse metrics eM all 2025) or solving problems in simpler discrete settings (Chertkov et all 2025).
Our Contributions. Building on this insight, our work makes the following contributions:
We formulate and exploit a fundamental degree of freedom in the Feynman- Fey-taa-Kac-type Fokker- Planck equation (2.5), establishing principled trade-off between the particle drift and the reweighting potential, and show that it can be directly exploited to actively minimize particle weight variance.
We introduce Driftl.ite, lightweight and training-free framework that computes a control drift on-the-fly to stabilize the sampling dynamics. We derive two practical instantiations, Variance- Controlling Guidance (VCG) and Energy-Coontoollng Guidance (ECG), which are computation- ally efficient and require solving only a small additional linear system at each time step.
We conduct extensive experiments on challenging benchmarks, including high-dimensional Gaus- sian mixture models, molecular particle systems, and large-scale protein-ligand co-folding. Our results demonstrate that DriftLite substantially reduces weight variance, stabilizes the Effective Sample Size (ESS), and improves final sample quality over current baselines.
2
PRELIMINARIES
In this section, we establish the problem setting, including the fundamentals of diffusion models and the inference-time scaling tasks central to our study.
2.1
DIFFUSION MODELS
We begin with pre-trained diffusion or flow-matching model, to which we refer the base model. This model defines both a forward process (xx, (t,)s<(0.))) governed by the following stochastic differ- ential equation (SDE) and Pokker-Planck (FP) equation:
da., u,, )ds + U,dw, (SDE), (xpa(x) -V. (x)( + U2 (p) (FP), (2.1)
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where ut is the forward drift, Ps is the marginal distribution at time s, and (w,),20 is Wiener process. Po represents the data distribution, and pr is simple prior, typically a standard Gaussian.
Generative modeling is performed using the backward process. Letting T - be the reverse time and denote *,- = *T--t- the backward process (ix, )eeo,r) is then described by:
dž, vy (. )d V,dw, (SDE), (a)) V [T)) Y/Z (a) (a)) (FP),
where 8t is the backward drift. The process starts from the noise distribution po pr and recovers the data distribution PT = Po. In traditional diffusion models, the backward drift v, is related to the forward drift u. (xx))) Fa via the score function V log Pr:
Ú. V? Vlog)((>)).
(x)x)) (x)) +
(2.2)
The word "pre-trained" signifies that we have access to the forward drift us and a reliable NN approximation of the score V log iog which in turn defines the backward drift iy.
2.2 INFERENCE-TIME SCALING
Our goal is to adapt the generative process of pre- pre-trained model new, related tasks at inference time. This approach avoids the significant computational cost and data requirements retraining from scratch, making it desirable to leverage existing models. We focus on two primary scenarios:
Annealing: Given a factor ", the goal to sample from @r X P This is common in physics for generating low-tem -ewtemaertture samples concentrated around primary modes of a distribution (Karz Czewski et all 2022),, using a model trained on easier-to-obtain high-temperature data.
Neward-Tilting Given a reward function r(z), the goal is to sample from X Po exp(r). This can be interpreted as posterior sampling with Po being the prior and the reward r being the pos- terior likelihood. It is widely used in applications, such as inverse design (Chung et all 2022), where the reward function encodes the desired properties of the generated samples.
In this work, we mainly focus on the settings where the reward is wiiee-iifereetiiabl.. In scien- tific applications,
typically a physics-based energy with analytic gradients (e.g., Passaro et all (2025)), which guarantees cheap access to derivatives at inference time. For black-box rewards, one could in principle approximate derivatives using stochastic estimators.
Distribution Path Selection. We can unify both scenarios by defining the target compactly as
(T()) x exp(r(x)) = po())) exp(r(x)).
To sample from ęT, we define a modified backward process that evolves along a path of distributions (q) (4))ee
[0. (E(OT
that smoothly connects from initial noise
target Φr- Following recent works (Skreta
atl 2025: Chen et all 2025), we adopt a both conceptually and computationally simple path:
(x())) x Pr (x=) exp (r)())),
where the reward ri interpolates from an initial state ro chosen such that go is easy to sample from, to the final reward rr r. While complex paths can be learned via optimal control (L. all 2025), we focus on such pre-defined paths to maintain a training-free framework.
Guidance-Bace Based Dynamics. A common and intuitive approach, to which we refer as pure guid- ance (Nichol all, 202T; Ho & Salımans, 2022), is inject the new information directly into the drift term by replacing the original score V log Pa with a heuristic score V log 4 corresponding to the marginal qu. leading to the following Fokker-Planck equation:
V2
0.q. (x: = V (x))
+ 2 (q)
(2.3)
where the modified drift 3, is defined below (cf., Eqn. )):
(x()) - (x: + 0 + V2 2
log + (r))
(2.4)
However, this method is known to be intrinsically biased because it fails to account for the changing normalization constant of 9, over time Chidembarmm et all 2024). To correct this bias, the true dynamics must include self- self-normalizing reweighting term, formalized below.
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Proposition 2.1 (Guidance-Based Dynamics). The exact time evolution of the density
(o) the following Baynman-Kac-tyae Fokker-Planck equation:
V?
0eq((z) V [+)()))((<)))
Aq((=) + (z())
(2.5)
2
where Vz is the same drift in pure guidance (2.4), and the reweighling potential go 9x(x))
Gi(z) Egg [G((.)] is given by:
U?
[-ū ( log
UP Vra 2
G. -.=h- -
(Ar. 7(1 >) log ll)
2
We refer readers to App. A for the proof. The PDE describes dynamics that diffuse with the guidance drift 8t while densities continuously reweight according to the centered potential Gt-
Weighted Particle Method. The corrected PDE
(2.5) be simulated using Sequential Monte
Carlo (SMC) (Doucet et all, 200R Del Moral et all 2016), where the density is approximated by
empirical distribution formed by ensemble of N weighted particles { w he[M]:
(6) (0)
(4) dt Vidw, (1) dzz = (,) d log w (x ) = ) - E. wo G(( (%),
i [M]
(2.6)
E [N].
We refer to this baseline as Guidance-SMo method is provably convergent, with the KL divergence to the target scaling as O in the diffusion context (Andrieu atall 2018; Huggins & Roy, 2019 Domingo-Enrich et, 2020 Cardoso etall 2023: Chen 2025). A brief justification of this method is given in App. A.2
(G-SMC) (Skreta al. 2025; Chen al 2025). This
METHOD: LIGHTWEIGHT DRIFT CONTROL
While the principled dynamics outlined in Prop. I offer path to unbiased sampling, their reliance on weighted particles introduces the critical vulnerability of weight degeneracy. As the simulation progresses, the exponential dependency of the weights w the potential ge leads rapid weight degeneracy and collapse of the effective sample size. This instability makes the standard Guidance- SMC approach computationally inefficient, especially with a limited number of particles.
This section introduces our solution: DrifiLite, a lightweight, training-free free framework that actively controls the drift stabilize the weights. We develop three steps: (1) we formulate fundamental degree of freedom in the governing Fokker-Planck equation (2.5), (2) we exploit this freedom to formulate an objective for minimizing the variance of the reweighting potential gt. and (3) we derive two practical, computationally efficient algorithms (VCG and ECG) for achieving this control.
3.1 DEGREE OF FREEDOM IN THE POAKEK EQUATION
Our key insight is that we can dynamically modify the particle SDE to counteract the sources of weight variance. Instead of passively reweighting particles, we can proactively steer them by "of- floading" the problematic parts of the potential ge into new, corrective drift term. This is enabled by a degree of freedom within the Fokker- Planck equation, which we formalize below.
Proposition 3.1 (Degree of Freedom). For any control drift b,(x), the Feynman Saynman-Kac-type Fokker-Planck equation (2.5) is equivalent
V?2
(x) = v [ (x) + bx(x)) q (z)] + where the residual potential is (b((x) (x)) + he(;;; by (xx) with control potential h, being:
2 (x) + ())))((())
(3.1)
(æ b.) = (> log (x) (r)(w)) (x( V b.())..
Proof Sketch. The core of the proof detailed in App. Briefly, we have -V (b, (bx()))((())) + g.(x) hu(x)he(z (x; b,) 0, since (x; b.) is constructed using V log q: ? log p. + Vr. An important property is that the correction term has zero expectation under qt, Le.. E, (:) = 0.
This proposition provides a powerful tool: we can introduce any control drift b, to alter the dynam- ics, as long as it is compensated by an extra control potential (e(;; bt). Since large variance in
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the potential 9t is the direct cause of weight degeneracy, goal is to choose i, strategically to minimize the variance of the new residual potential Φv. An ideal control would make Φi constant, completely stabilizing the particle weights. In fact, perfect, variance-eliminate control always exists for any base potential gt. as shown in the following proposition:
Proposition 3.2 (Optimal Control, Informal Version). There exists a unique curl-free control (i - (a) such that of (a) = (x(x) + (,(;; by) 0 for all æ, where the optimal scalar potential A (x) the solution to the following Poisson equation:
VA( ( (x).
(3.2)
The proof and further discussion provided in App. that the control potential satisfies gi (z)//(>;; b.) = (9))))b((z) from its definition.
A Intuitively, Eqn. (. follows by noticing
This mechanism is closely related to twisted proposals in SMC (Briers et aD. 2010 Whiteley & Lec.
20 Heng cf all (020)) and its recent applications (Lawson et all 2022: Zhao et all 2024 Lu &
Wang 20224) and Prop. B_] can be viewed as diffusion formulation of this degree of freedom.
3.2 IN SEARCH OF OPTIMAL CONTROL
While Prop. guarantees a perfect solution, solving the high-dimensione PDE in (3 at every time step is computationally intractable. In contrast to learning- caaning-based methods that uses neural net- works approximate the optimal control via backpropagation (Albergo Vanden-Eyjnden, 2024; Vargas et all (0233)) we propose two training-free, practical methods that approximate this optimal control by balancing effectiveness with efficiency. Both methods share a core strategy: restricting the search for the control drift b, to finit-ddimensional subspace. This simplification is key, as it transforms the complex problem of minimizing the residual potential di into solving a small linear system. This reduction from intractable PDE tractable linear solve makes the control truly lightweight, hence the name DrifiLite.
Controlling Guidance (VCG). The most direct approach to find control b, that explicitly minimizes the variance of the residual potential:
min Varzenq (cb [z)] = Vara-g, (9x (z) + h4 b.)]
(3.3)
Instead of parameterizing b, with a neural network (Albergo Vanden- Eignen, 2024), we seek lightweight solution by approximating it as linear combination of basis functions.
Ansatz 3.3 (Linear Control Drift). The optimal control drift bj (x) is approximated as (x) =
Ei= (z); where (s())))((())) are pre-defined vector bases and 9, (0) ..... 8%)
the coefficients to be found.
Under this ansatz, the residual potential becomes (x(x) = (z) + En= (2), where hz =
ha( st). The objective (6.3) corresponds to standard least-square problem, whose solution is obtained by solving an n linear system A,0; C₁, where Aij Egg [hih?]] and c; = EE, [g.
Mnergy-Controlling Guidance (ECG). An alternative approach directly targets the curl-free op- timal control by in Prop. 3 by variationally solving the Poisson equation (6.2). As shown by Yu (2013, this equation is the Euler- Lagrange equation for the following energy functional:
/
min EsAr A
q, (=)))
(z)) dæ.
(3.4)
We can efficiently find an approximate minimizer using the Ritz method for the scalar potential A.,
Ansatz 3.4 (Linear Control Potential). The optimal scalar potential (T) is approximated as (x Ei_ (z), where (s) are scalar bases. The control drift is then given by b,(x)) = V (.) = Σi_ (.))
Substituting into the energy functional (3.4) again yields a linear system of equations A,O; Ct, where Aú F.g. [vst"" Vs!] and e; [9:st]-
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3.3 PRACTICAL IMPLEMENTATION
Choice of Bases. The effectiveness of VCG and ECG depends on the choice of suitable basis functions. While the formal solution for the optimal control bū is intractable (cf., App. A44,, its structure reveals that the ideal control is a function of temporally locally available quantities like the score log Pt. the reward gradient Vrt. and the potential 9e (containing the forward drift it and higher-order terms). This suggests that natural low-rank approximation is obtained by projecting the intractable optimal control onto the span of locally available vector fields, as defined in the following.
Variance-Comin Controllin Guidance (VCG): We use the following vector basis functions:
81 (2) = (r), 82(Ø) - V log (ar), (a( = (xr)).
Note that using 82 requires computing the Laplacian A log Pe (2)). which can be approximated efficiently with Hutchinson's trace estimator (Hutchinson, (1999) in high dimensions.
Energy-Controlling Guidance (ECG): We use the corresponding scalar potentials:
(x) = rr(2), 52(x:) = log (xr): 83(2)) Ut(z);
where Ü. is a potential such that VÜ, = it, This method is especially convenient when the log- likelihood log is readily available from upstream training tasks Sykhound-Sadepe et af, 2025: Gl 2025: : Thortton al 2025).. If not, approximations or alternative bases may be used, such as the norm V log p. or random projections of the score V log Pt & for random &.
For annealing tasks, reward- based bases ($1 and $1) are automatically dismissed.
Weighted Particle Simulation. As discussed in Sec. 2.2 we simulate the Deynman-Kac-tyan Fokker-Planck equation (E-II) using the SMC/weighted particle method detailed in Alg. The key difference from G-SMC Eqn. (2.6) is the use of the controlled drift ह. + b, and the residual potential Φe = 9t + he(-; by). To prevent weight collapse, particles are resampled when the Effec- tive Sample Size (ESS) drops below threshold These principled versions with resampling denoted VCG-SMCJECG-SMC. For high-dimensional problems where resampling introduces ad- ditional stochastic instability, we also consider simpler variants, denoted VCG/ECG, which the low-variance residual potential di, retain continuous path-level weights, but omit resampling steps.
Our method adds modest computational overhead. The linear solve itself is negligible, while the dominant additional cost comes from evaluating basis functions and building small n X n linear system each time step. where is the number of bases, typically n in our experiments. The components of this system (A, and c;) are computed as expectations over the current weighted par- ticles, reusing terms like the score V log Pe and the reward gradient Vr, that are already computed for the base guidance drift. While accurate evaluation of the score Laplacian A log Pt can improve control quality, efficiency is preserved with stochastic approximations, and thus per-step over- head remains constant in dimension and fully parallelizable across particles, resulting in moderate runtime increase compared to the pure guidance baseline (cf., empirical results Tabs. E and M
4 EXPERIMENTS
In this section, we empirically test the performance of DriftLite by designing a series of challenging annealing and reward-tille tilting tasks, comparing our DriftLite methods (VCG and ECG with and with- out SMC) against two key baselines: Pure Guidance (PG) (2,33 (Ho & Salimans, 2022), Guidance- SMC (G-SMC) (2.5)) (Skreta alll 2025; Chen all 2025). Our implementation uses JAX (Brade bury et all 2018 to ensure efficient, parallelizadddeeee computation on GPUs. Our source code is publicly
available at bttps://qithin http:///.mmmmmmooee////rereeeeee
4.1 GAUSSIAN MIXTURE MODEL.
We begin with a 30-dimensional Gaussian Mixture Model (GMM) (ef., App. B._ for detailed set- tings), controlled environment where the exact score V log and the potential log Pt are known analytically, allowing us to isolate and evaluate the performance of the sampling algorithms them- selves, free from any confounding errors of a learned score network. We evaluate the methods with multiple metrics, including the Negative Log-Likelihood difference (ANLL), Maximum Mean Discrepancy (MMD), and Sliced Wasserstein Distance (SWD) (ef,, App. B.

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Algorithm 1: DriffLite-VCG/EEGG-SS Implementation
Input:
Original drift path vt, original potential path It, time steps (fx 1 k=n- reward r(æ))
schedule Bt, basis functions, number of particles N, ESS threshold т.
(8) Initialize particles (6) ~ Po and weights w/ < for L,...,
for to M - do
Form weighted estimates of Ad, and C22 using {<x,2 w/, BEINj Solve Ar, 0t. Ct. to obtain the control drift by, (-);
(.)(+) f Uea (.) (.) 1).9. 9t. (+) t (-) + he, (: bi
log logw(y) w log Win (6) + +9t. ( (tx+1 tx), W & softmax (wi.): al1 P Fith
(4) + -vi (1 (+x-1 (x) + Ven Vla+1 txz where 2(1) N/(0 I);
ESS((WIH <T or periodically then
(c) Resample (x JIE[N] according to (win lIE[N] and reset W224 for all i;
Output: Final samples (2 WT JEENN] from the last completed pass.
PC MMN O (46)
G-SMC (MBD 0.248)
VCG-SAIC (MND D.CES)
ECG-SMC (ANG 0.62))
-2
-20
-00
20
Figure 1: Qualitative comparison of sampling methods the GMM annealing task (1 2.5).
Annealing. We first test the ability to sharpen the GMM's modes by annealing, which tests each method's ability to maintain the correct relative mode weights. As shown in Fig. MI the pure guid- ance (PG) method produces visibly biased samples, while G-SMC suffers from mode collapse, direct consequence of the weight degeneracy that our work aims to solve. In contrast, our meth- ods (VCG and ECG) accurately sample from the correct modes, also corroborated with quantitative comparisons in Tab. EL A closer look at the ESS and potential variance evolution during the inference dynamics Fig. R reveals why DriftLite succeeds. Our control mechanism reduces the variance of the reweighting potential by several orders of magnitude compared to G-SMC. This directly prevents weight degeneracy, leading to a stable Effective Sample Size (ESS) throughout the simulation and superior final sample quality. Notably, ECG, while not directly minimizing variance, achieves sim- ilar stabilizing effect, validating the energy-based control perspective. Fig. B shows the performance of all methods the number of particles varies. It indicates that methods not only outperform the baselines but also converge more efficiently, achieving better results with fewer particles.
The results of reward-tilting task where the distribution shifted towards region defined by a quadratic reward (Figs. M. A and III and Tab. ! confirm our findings from the annealing task. We refer to App. CJ for further experimental results.
Reward-Tilting
Learning-Based Baseline. To contrast DriftLite's training free inference-time control with amor- tized drift-learning, we additionally implement training-based baseline, Neural Controlling Guid- ance (NCG), which parameterizes the control drift by neural network and optimizes the variance objective (B*) via backpropagation. See App. for full setup details and quantitative results.
Iterative Refinement. Furthermore, we introduce an iterative refinement procedure, where the leamned control drift ö, + b, and potential d, 9t + he (:; b.) from one full pass are used as the base dynamics for the next. As further discussed in App. C this process progressively reduces variance and stabilizes ESS over multiple rounds (cf., Figs. T and [X), further enhancing sample quality (cf., Tabs. [T and (6). The monotonic decrease in variance across rounds also acts as a proxy for the reduction of approximation error in the linear control ansatz (Thms. 3% and B.4)).

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PG
VDG
VEG-SMC:
FDR-AMC
ESS
MMD
SWD
100
Variance
10'
Moan
Cov Frob.
107
10"4
107
10
200
400 500
800
1000
Step
Figure 2: Evolution of ESS and potential vari- ance during inference on the GMM annealing task (7 = 2.2). Our methods (VCG/ECG) sub- stantially reduce variance and stabilize ESS.
Figure 3: Performance metrics versus number of particles for the GMM annealing task (7 = 2.0). Our methods consistently outperform baselines and show strong scaling.
Referonce
(MMD 0.007
VOG SMC (MND 0.021)
(MND 0.022)
Figure 4: Qualitative comparison of sampling methods on the GMM reward-tilting task (a 200.0).
4.2 PARTICLE SYSTEMS
Next, move more realistic scientific benchmarks where the score is approximated by NN
trained on finite data. We evaluate on two standard systems with complex, multimodal energy landscapes: a 2D 4-particle Bouble-Well (DW-4) and a 3D 13-particle Leonard-Jones system (LJ-
13), both widely used as benchmarks (Klein et all, 2023; AKhound-Sadegh et alj 2024 2025: Liu
all 2025: Skreta 2025) Zhang all 2025).
The score is obtained by training an E(n)- Graph Neural Network (EGNN) (Satoras
atall 20216) (cf., App. B.2). For particle systems, ground-truth reference obtained from under- damped Langevin dynamics simulations with BAOAB splitting scheme (Leimkuhler & Matthews.
2013) (cf., App. (. The EDM framework (Karras et anl 2022) is adopted for both training and
inference (cf., App. Hrd) We measure performance using additional metrics that capture physical correctness, including the Radial Distribution Function (RDF) for structure and the energy distri- bution for thermodynamics (cf., App. (B.)). Based the GMM results showing VCG's superior performance over ECG and the lack of pre-trained log-likelihoon we proceed with only the VCG variants of DriftLite in the following experiments.
Double- Double-Well-4 Well-4 (DW-4). We first consider the DW-4 system (cf., App. B This system features two energy minima separated by barrier. The annealing task requires the sampler correctly populate both modes, even when they are sharpened at low temperatures. As shown in Fig. B3. VCG-SMC achieves nearly perfect match with the ground-truth RDF and energy distribution. This demonstrates that by using variance reduction to maintain an ensemble of high-quality particles, Die effectively leverages global information to navigate challenging energy landscapes where baselines fail to do so.
Motivated by Schebek et all (2024), we consider applying an additional harmonic potential as reward, and the reward-tilted distribution corresponds to another DW-4 system with different con- figuration. The quantitative results Tab. Il confirm that our methods consistently outperform base- lines by large margin across all metrics. The ESS/potential variance plot in Fig. I confirms the stabilizing effect of our method on ESS. An ablation study in Fig. [ demonstrates that our method converges the number of particles increases across metrics.
Lennard-Jones-1 (LJ-13). We conclude with a highly challenging annealing task on the LJ-13 system (ef., App. B.D), complex benchmark known for its rugged energy landscape and singular
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Table 1: Performance comparison on Particle Systems (DW-4 and LJ-13). Results are =ean over 5 runs. Best results per column are in bold.
DW-4, Amscaling (2' 2.0,7
DW-4, Reward Tilting (T'
=20,, 0.5)
Method
ANLL
MMD
SWD
WTDF
W
ANLL
MMD
SWD
WT
W
PG
0.159-1212
0,400
1.088
0.208-aa0e
0.551
1000
0,867
0.771 2ooss
4,714
0.627
1.837 2000
G-SMC
0.038 go
1.365
1.012
0.208
101
g.190 0.329 poms
0.087 gome
0.194 gooe
0.118ga00 0.330
VCG
+.043++ao2
0.014+0000
0.037-0006
0.043
8.000
0.663
+aoss
0.699
+1.05 0.614+0.339
+0438
0.161 +o.ss
0.461 +0.44
VCG-SMC
0.032
0.014
0.035 8.002 0.060_0a0m
0.031
0.296
0.021 10000 0.048 2882
0.107 0.296.gams
Method
LJ-13, Annealing (T
2.0,
2.5)
LJ-13, Reward- Tilting (T
2.0. 0.8)
ANLL
MMD
SWD
Wine
W
ANLL
MMD
SWD
W1
W
PG
13.5 13.58
0.598_oom 1,603 _oa0
11
49975
0.719 0.797 100%
0.070 5.430
G-SMC
13 13,64_591
0.616gam2 ao0o0
0.040_0.00
13. 69
1.783
0.081
1o0e
0.084_ao03
g.023
1.784
VCG
gamm
0.144
0.069 2.am 22.86.a177
0.695
2o.22
0.702
0.036
2.790
VCG-SMC
-0,699
iiw 0.102 1ones
0.098 2/14 0,002 1.28
1.734
0.015 1.00
0.015
0.022 1.735 po10s
Analytical Reference
Analytical Reference
PG (94) 0.037) 0-SMC (W; = 0.717)
0.08
(My -1 31.559) G G-SMC (V2 5,3122)
3.0
-.008)
VCD 22.475)
2.5
VDG-SWYC (W, 0.001)
0.05
VOG-SMC (W) 0.370
RDF 9in
Denatyy PL
2.0
3.8
1.5
3.5
0.04
3.4
0.07
0.30
1.00
100
11
0.75
1.00
1.25
1.50
1.75
2.00
2:25
2.50
0.00
-200
-280
-190
-185
-150
-175
-170
-100
Painwise Dissance
Energy
(a) Radial Distribution Function.
(b) Energy Distribution.
Figure 5: Comparison of generated distributions for the LJ-13 annealing task (7 2.5). VCG-SMC the only method that successfully recovers all three peaks in the (a) RDF and closely matches the (b) Energy Distribution. Insets provide a zoomed-in view.
behaviors at short distances. Fig. B presents the result of demanding inference-time annealing task from T' = 1.0 to 0.4 with = 2.5. The target distribution exhibits third peak in its RDF corresponding to a structural feature almost absent at the initial temperature (Fig. M. In a powerful demonstration of its capabilities, VCG-SMC is the only method that successfully discovers and samples from all modes, matching both the RDF and energy distribution with high precision. Metrics in Tab. 0 further confirm significant performance gap over the basclines in this complex setting.
We refer readers to App. t for additional experimental results and visualization on DW-4 and LJ- 13 systems, with results with varying base temperatures T, annealing factor 7, constraint strength X' and number of particles N.
4.3 PROTEIN-LIGAND CO-FOLDING
Lastly, we apply DriftLite to the protein-ligand co-folding problem (Abramson et all 2024: Bryani atan 202e), a central task of structural biology and drug discovery. The goal generate 3D pro- tein structures and their binding partners (ligands, particularly small molecules) simultaneously and in mutually dependent manner, given the protein sequence and the ligand identity. This problem extends the classical protein folding problem (Jumper et all 2021; Back et all 202T) and is crucial for elucidating protein-ligand interactions. Despite the recent progress achieved by diffusion mod- els, notably AlphaFold3 (Abramson et all 21124), Protenix (Ceameetal) 20255,, and Boltz-2 (Passara alall 2025), a persistent challenge is that purely data-driven generative approaches tend to overem- phasize global structural similarity while often producing conformations that violate basic physical constraints (Bnttenschoen et all 2024; Masters et all, 2022). Recent studies demonstrated that in- corporating physics-basee based steering potentials can help mitigate this limitation (Passaro et all 2025).
We adopt and follow the experimental setup of Boltz-2 (Passaro et all 2025), an open weight diffu- sion model, the base model, and apply VCG-SMC steer the generation of protein-ligand struc-

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Published as a conference paper at ICLR 2026
Table 2: Performance comparison on steering the physical validity of protein-liganm co-folding.
Results are mean Esld over 3 runs. Best results per column are in bold.
Mothod
Valid Fraction t Clash Free Fraction
Bond Length
Bond Angle 4. Internal Clash 4 Chiral Atom
Chain Clasbes
Base
0.00
0.490+accon
55 4::0
133.00
138.67 =4.6
118.33
398 44.1
FKS
0.379 2001s
52.67 1::00
127.33
140
126,332151
377.00
G-SMC
0.838
0.945 hocss
42.33 13,06
98.00
31.33 4-497
2.33
31.67
VCG-SMC
0.8564acce
0.950 acets
24.33 29929
19.00
32.33
1.00
-0.00-..0
Reference
Base
G-SMC
VCG-SMC
Figure 6: The reference and predicted complex structure of Hst2 bound to 2' -O-benzoyl ADP ribose. The reference corresponds to the experimentally determined crystal structure (PDB ID: 7F51). The unsteered base prediction inverted a chiral center in the ligand (highlighted with a red box). G-SMC failed to correct this issue and even broke the bonding, whereas VCG-SMC successfully guided the generation toward the correct chirality and preserved a chemically meaningful structure.
tures toward physically valid conformations using a physics-based potential as reward. We compare our method with two additional baselines: the unsteered model (Base) and Feynman-Kat Steering (FKS) (Singhal et al, 2025).. We assess physical validity using the widely adopted PoseBuster V2 benchmark (Buttenschoen et all, (024)). Results are summarized Tab. 2 In this task, evaluating the physics-based reward and its gradient adds relatively little overhead. VCG-SMC exhibits the strongest performance with fewer or without rule violations, improving the quality of partially valid structures, and increasing the proportion of fully valid ones. This underscores its effectiveness in complex real-world setting. An example highlighting these improvements is shown Fig. H Im- plementation details provided in App. R Additional experimental results the ESS evolution and runtime comparisons are provided in App.
CONCLUSION
We introduce DriftLite, lightweight, training-free free framework that resolves critical trade-off in the inference-time scaling of pre-trained diffusion models. By identifying and exploiting a funda- mental degree of freedom in the Fokker-Planck equation, DriftLite actively controls the sampling drift, thereby mitigating the weight degeneracy that plagues previous particle-based methods. Our practical instantiations, VCG and ECG, impose modest and scalable overhead while dramatically improving the stability and accuracy of inference-tine scaling. Experiments further confirm their effectiveness and strong scaling with the number of particles, and we observe that the VCG variant is generally more robust, while the ECG holds promise in several specific scenarios. Across par- ticle and protein systems, our approach consistently produces higher-quality samples and handles complex distributions more robustly compared to existing inference-time scaling baselines.
While DriftLite proves effective, its reliance on fixed set of linear basis functions presents a poten- tial limitation. Future work could explore more expressive yet still efficient representations for the control drift, such as compact neural networks or adaptive basis sets, including those involving the posterior mean (Chung al, 2022). Our current instantiations also assume wwice-differentiabiliy of the reward r and extending DriftLite to less smooth rewards is an open direction. We anticipate that future work will extend DriftLite more complex physical systems, including LJ-55, ALDP, by pairing our inference-time scaling framework with continued advances in generative modeling, Furthermore, we have focused on annealing and reward-tilting tasks with non-heuristic targets and accuracy demands, but the DriftLite framework is broadly applicable beyond these tasks. Extend- ing it to other generative problems, such as podduct-of-eppertt models or conditional generation, a promising direction for future research. It is also interesting to study the inference- time scaling problem in the discrete settings under both the uniform (Campbell et all, 2022; Ren et all, 2024; 20252) and the absorbing (Lou et all, 2023: Shi et all 2024) frameworks.
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ACKNOWLEDGMENTS
This work was conducted during Yinuo Ren's internship and visit at Center for Computational Math- ematics, Flatiron Institute, an internal research division of the Simons Foundation. We thank the Scientific Computing Core, Flatiron Institute, for the computing facilities and support. This material is based upon work supported by the National Science Foundation under Award No. CHE-2441297 Grant Rotskoff. Lexing Ying acknowledges support by the National Science Foundation under Award No. DMS-2208166
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A PROOFS
In this section, we present the omitted proofs of several propositions and additional discussions of the theoretical claims in the main content.
A.1 PROOF OF PROPOSITION 2
To aid reading, we reiterate Prop. below.
Proposition A.1 (Guidance-Barl Based Dynamics), The exact time evolution of the density (q,) follows the following Fokker-Planck equation:
V? deq((x)) = [x)) + sq((=) + (x))
2
where in is given by
Ú2 V2
vy (x) = -xx))) +
V log]n(x) Vrx()))
ú, + V (x)
(1-1)) + (v)(
and the reweighting potential gx(x)) = Gx(x)) Ee [Gr(.)] is given by:
G. = - (1-7)V- ã, + t?
(Dr 7(1 -1)1 log Pe 11
2
t² Vrr)
Vr. ū, n 0. log +
2
Proof. We begin with the definition of the target density qr (x)):
log (=()))) log (.()) + rx( Zz
where we define the log peg--pattiion function
Z, log / pily)7 exp((:())))
and taking the time derivative gives:
o log (z) yu log (og)((=)) + +x()) o,Zr
(A.1)
Recall that the Fokker-Planck equation for the backward process marginals p. as follows:
V22
(x) = V (x) + Ap)(z),
which can be expressed in terms of the log-density as:
O (x)) = -p, (x) (v.()))
(x) +
2 Pr (x)
(A.2)
V/2
=- (.) 2. ( V (x)
lo
|| lz)))
2
We posit that the time evolution of di is governed by the fokker-Plande equation with an additional reweighting term:
V 2 (x) + (x))
(z)
(x))
and dividing both sides by (z.)), we can solve for the potential (z):
V22
go (x) = 0 log (x) + q. (x)V [x()))
2 gi (q)
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Since have
d / (0
/ (:)))d 0, di
and
/( (-v. + V2 Aq)(4)) da = 2
by applying the divergence theorem and assuming suitable boundary conditions, the reweighting term must have zero expectation, i.e.
Ex [9(())] =
(2) (
Thus, write (x) Gz(a:) Egg, (G.)), where we define Ge(x) by absorbing the spatially independent term Zz
Gx(x) = g:(x) ()) + 0,4
V?
0 ( + 0,4, 0 (z) (+()))
2
V22
=((()) + 90, log - (x) (x)))
2 9: (z)
V22
=x(x) + 7 - (x) (x) v (z)
(A log + A log (2))))
2
V22
(x)) + (x) (o))
A log de (a)
log (x)))
2
= =x(x)= ~, ( V (x)
(x) V Vlog (x) + (x)
V log (x)
(1)
(II)
VV2
V? (A (z) + V log de (a:) 1?),
(A (o: + VV (ogT
2
(mI)
where the second equality is due to Eqn. II, and the second-to -last is due to Eqn. (A.I.
We now substitute the expressions for θ log Pt and log q and simplify term by term.
(I) Divergence Terms:
(1) = --- (x( V Ūz(x) V (or((x - (v)()))
=V. (-11 ((1)-))ū((00) 07
V22
8? + V?
(r(( =-(1 -) --)
2 (n)
(II) Inner Product Terms:
(II) = - loog +) (Vl(l==>)oooggee
Vlogg(z)
Voogg()) (x) (1 Vog (> Vr(())
=(5x(x:) - vx))) (V (x) Vr((x)
((1 - )x) + Ú + V?
(r)
Ú + V2
+ ((1 - -x)((x))(+++ + (x) +
Vri(z))
Vr((x)
2
Ú² + V22
07+ + V22
-vr - + +7034 log (x))
Vr((at)
71
V (x)
2
2
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(III) Laplacian Terms:
?V?
V22
(11I) ( lolog + V (ogg)(
(A log + l
2
V22
(7A log (x) (()o0=)))(10000))))))))))))>)))1
V log |
1 logg Vr.(z)]
2
V
Ar,(z) - V22
2 (7 - log Lo (1) + V log P. (z) Vr,(a) + [v,()))]3)
2
Combining all the simplified terms, we arrive at the expression for G, stated in the proposition.
Remark A.2. This proof similar to the proof in related works (Skreta et al 2025: Chen et al 2025), where Fokker-Planck equations are derived for each task lask-specific setting. While being more general, our approach also omits the computation of the time derivative of the log-partition function Ü,Zú since we notice that it cancels out the final expression for Gt. This simplification makes the proof more straightforward and concise.
A.2 JUSTIFICATION OF WEIGHTED PARTICLE SYSTEM
Proposition A.3 (Weighted Particle Simulation). Let qi Rdd + Rzo solve
V2 (a) (c()))g(x)).
(.ge(x)) = - VV [()())) +
Then this PDE can be simulated in the mean-field limit system
- oo by the weighted particle
(d ( ह. (z) (2) dt + Vidw d log we De (m) )dl,
E [N].
i E [N],
where the weights satisfy the normalization Σ of gr:
(2) E
w!
1, and the empirical centered version
N
(x(x) (G()) - Σ
(
(z( (8)
w
==1
ensures preservation.
The proof of this argument is standard (Moral 2004 Doucet.ctal 20XE Andrieu 2010) under typical assumptions, including local Lipschitz continuity and linear growth of the drift öt bound- edness of the diffusion coefficient Vr, moderate growth of Gr. sufficient regularity of qe justify integration by parts, and either fast decay at infinity or no-flux boundary conditions to eliminate boundary terms. We provide proof sketch below for the reader's convenience. We also point that similar arguments also apply to the weighted particle simulation for the controlled dynamics in Prop. B.I. which we omit for simplicity.
Proof Sketch. The main steps follows:
Step (Empirical measure). Define the weighted empirical distribution
5
m
Step (Test function evolution). For y E C (Rd), consider
N (p, = Σ
w
(2x)
t=1
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under the SDE and combine with the weight dynamics via the product rule. This yields
Apply Itö's lemma (((a)") (0)
V? Ao + D. u4 di + dMM d(p, = U1 V +
2
where M is martingale term.
Step 3 (Limit N + oo). By law of large numbers and propagation of chaos (Sznimma,, 1991 Lacker 2018), we have the weak convergence:
(( (x, as oo,
while the martingale term vanishes. Passing to the limit gives the weak form of the PDE:
d ʃ ( at
a
V2 / (.(()) ( + Ap(x) + (z)))(())) (a()dd.
Step 4 (Integration by parts). Using divergence theorem identities, we have
ʃ (2()) / ((x))
[ (o.()))((z))
V22 dq(()) (x()))(()) dæ, 2
for all test functions g.
Hence, we have
V2 0.g. (x) - ().()) + Ag)(a)) + ())))))()),
and the proof complete.
A.3 PROOF OF PROPOSITION 1
The following proposition is the same Prop. B.1l but with a more detailed proof.
Proposition A.4 (Degree of Freedom). For any control drift be(x), the Fokker-Planck equa-
tion
V2 (q((z) + (x))
0eq((z) = - [=))))
2
is equivalent to the following one with additional control drift term bx()):
V2 Aq. + (x))
- - [() (((((()))) ()))(()))))))))))))))))))))))))) q.
where the residual potential is ()(x)) = (x) + (x;; by) with the control potential he(;; b.) defined as:
h:(w; b.) = V> log (w) (x) by (x) be(x).
Proof. The terms added to the right-hand side of Eqn. (2.5)) to obtain Eqn. (B.l) are:
-V. (b(((()))())) (ax E.g. (xx: b.))).
First, we prove that Egg (2; b.)l 0:
E2 (<: bx)] = / (x) Vogg::())) bx(x)) + V (x(x) dæ
(=))) 0,
where the last equality follows from the divergence theorem, assuming appropriate boundary condi- tions (e.g., be vanishes at infinity).
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Then, show that the remaining added terms cancel each other out:
V (b)()))(())) + (m)), b.)
(b,(z) + (og (x) b,(2)) + V bx(x)]
= (b.()) + V ((( = O.
Since the added terms sum to zero (given Eg (x; b.)] 0), the two PDEs are equivalent.
A.4 FORMAL SOLUTION FOR THE OPTIMAL CONTROL DRIFT
In Prop. B.2, we claim that there exists uniform optimal control drift as we rewrite its mathematical rigorous version in the following proposition.
Proposition A.5 (Optimal Control), Let C Idd be a bounded Lipschitz domain. Assume that 9: Ω R is measurable and uniformly elliptic: there exist constants O < X S A < such that
X (z))) A, for a.e. E Ω
and suppose 9t E (Ω). Then there exists a unique potential A (Q) solving
V (((()))VAA())) (((x))((x) in H (9)).
Defining the control b, (z) (x) E (2(22; one has
1 V (9((z)b((z)) = in (2).
(1(x) = g((x) + (a b,) = (z) + (((x)
In particular, b is the unique curl-free control that drives o, to zero.
Proof. This follows directly from the Lax Max-Miagram theorem. The bilinear form
alu, v) = / q: Vu Veda,
u, E Hy
is bounded and coercive, while the linear functional
L(u)
y.gevdx
continuous on H) (9t). By the Lax- Milgram theorem, there exists a unique A; E (2) satisfy- ing (B.2). The existence and uniqueness of weak solutions to such elliptic PDEs are standard results; see, for example, Exans (2022, Chapter 6).
While the existence and the uniqueness of the solution the Poisson equation (3.2) are shown above, we present a formal solution for the control drift by in the following, leading to our choice of basis functions in Sec. 3.
Let fz( = ge (a (z)) By the Helmholtz-Hove Hodge theorem, any sufficiently smooth vector field fi can be decomposed into curl-free component (the gradient of scalar potential A) and divergence-fret free component u.i. so that
f((x) V A(xx) + +ui (x),
where Tn 0.
Substituting this decomposition into the equation gives:
(vA(x) + (x)) AA(x) = =::0===4(0pp000 (x)).
This is a standard Poisson equation for the scalar potential A. The solution for can be expressed formally using the Green's function for the Laplacian in d dimensions, Ga y):
A(x) Ga(æ ()
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The desired control is then given by br(x) (((x)) (A(z)). Taking the gradient of A(x) with respect to a and using integration by parts with the property VaGa( y) -VyGGa - y), we get the following formal solution:
/ VoGala - ()
VA(x) =
/ Golæ ,)V,
/ G.(z - y) y) (g. ( + (v)( gly)dy.
where the term Vge introduces higher-order derivatives of the reward function:
Vg.(2) = 3, V'r(x) ( -- -
- vū((x))V((a)) - (VV Vr()).
This formal solution is computationally infeasible weighted by the unknown density (3)) that we are trying to simulate. However, its structure mo- tivates choice of local basis functions: the reward gradient Vr,(z), the score V Vogg (x:). and the forward drift ū. (z), as we discard higher-order derivatives.
it requires integrating over all space y,
B ADDITIONAL IMPLEMENTATION DETAILS
In this section, we provide additional implementation details for the experiments, including the prob- lem setup with parameters, a detailed description of the network architecture, training, inference, and evaluation procedures.
B. PROBLEM SETTINGS
Gaussian Mixture Model (GMM). In the Gaussian Mixture Model (GMM) task, the data distri- bution is set as
40
1
N (æ; p4; 501),
40
where each component μn is sampled from Unif([-400 40]).
For the annealing task where the target distribution is qr (T(x) x po(x)), the reference samples are obtained by rejection sampling with the following proposal distribution:
proposal x
N
x: pi:
50
1).
40
For the reward-tilting task, the quadratic reward function given by
r(x) - xI- -4) 14),
where the target mean is sampled from A O, 100I) and the covariance matrix Σ = al with being a parameter controlling the spread of the reward. The posterior distribution corresponds to another Gaussian mixture model, in which each component acquires an updated mean and weight after incorporating the quadratic reward:
(x
Σ
W.N a:: (a: E.)
with posterior covariance
5 (Σ + (50I) (501)-1)-1
posterior means
;. ((00)) - + 1
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and mixture weights reweighted according to the evidence.
2/2 (M: p1) (E 501) - (mu: -4)
i/ w, exp
For all GMM experiments presented in Sec. I we set the number of particles N 213 and perform resampling whenever ESS drops below 0.9. All plots in the GMM experiments are plotted by projecting onto the first two dimensions.
Bouble-Well-4e (DW-4). For both DW-4 and LJ-13 systems, the target Boltzmann distribution of the following form:
po( x exp(-E(x)) = exp (- H(x) ( A n
Σ 1r: - F122) +7(²)
(B.1)
with the potential H(x) system-specific, and a harmonic potential of strength À imposed as a phys-
ical constraint.
In the reward-tilting task, we consider the quadratic reward:
X r r(x)
rr T11
2
and thus the reward-tilted distribution is
Po(x) exp
- T
1 H(x)
A 12
Σ
Ir, F|(
X n
Σ 2
rr - (T1)
1 (- [1(x)+ A XT ||: F(3
exp
2
i.., another Boltzmann distribution of the same temperature but with a different constraint strength *+ATT. This reward thus corresponds physically to strengthening harmonic confinement around the center of mass, i.e., sampling the same particle system at the same temperature but with an increased trap stiffness.
First introduced by (Köhller all 2020), the double-well potential is defined on system of particles the two-dimensional space (x E R-1x2). The potential energy function given as:
How(w)
[a(de da) b(dis do)² + cddij do)
where is the coordinate of particle and dı = ||:: r denotes the distance between particles and j. We use standard parameters: 0.0, b 4400 0.9, and dg 4.0. This would yield double-well potential with two minima di 4 V5. For the harmonic potential constraint in the Boltzmann distribution Eqn. (B.,), we set the constraint strength as X = 0.05. For all DW-4 we set the number of particles 2 and perform resampling
experiments presented in Sec. 4.2 every K 100 steps.
Lennard-Jones-13 (LJJ-33). The Lennard-Jones potential is a widely used model for simulating interatomic interactions with both repulsive and attractive components. our case, is defined on a system of 13 particles in three-dimensional space (x E [R 13333 The potential energy function is given as:
Σ
dij )
Hu((:)
2
2T d
We use the following parameters: 5 = 1.0 and = 2.0. For the harmonic potential constraint in the Boltzmann distribution Eqn. (B.,), set the constraint strength as À 1.0. For all LJ-13 experiments presented in Sec. 4.2 we set the number of particles = =213 and perform resampling every K 50 steps.
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Anelysical de. 4.0
Analytica
0.5
RDE gtr
0.4
0.15
Denciyy
0.10
0.1
0.0
0.00
-17.5
-15.0
-12.5
-10.0
-7.5
-3.0
-2.5
Distance
Energy 2
(a) Radial Distribution Function.
(b) Energy Distribution.
Figure 7: Reference distributions for the DW-4 system at the base temperature T 1.5: (a) Radial Distribution Function (RDF) and (b) Energy Distribution.
14
Anelyrical Reference
0.00
Analytica Reference
1.2
0.05
RDR gif
Denoty
0.0
0.03
0.6
0.02
0.4
-70
-00
-20
Pairwise Distance
Energy e
(a) Radial Distribution Function.
(b) Energy Distribution.
Figure 8: Reference distributions for the LJ-13 system at the base temperature 1.0: (a) Radial
Distribution Function (RDF) and (b) Energy Distribution.
Protein-ligand Co-folding. The protein-ligand co-folding problem is an extension of the classi- cal protein folding problem: rather than predicting only the structure of the protein itself, the task is to simultaneously predict the structure of its interacting counterpart, a ligand that is typically small molecule. This problem lies at the core of structural biology and is essential for understanding how proteins and ligands interact, which in turn underpins the elucidation of biological pathways and the design of new drug molecules to modulate biological activity. this work, we focus on addressing the issue of physical validity in protein-ligand co-folding. Diffusion-based models often overemphasize global structural similarity while occasionally generating unphysical conformations. To mitigate this issue, we incorporate physics-based potentials to steer the generative process, effec- tively modifying the data distribution (x) with a physics-based potential function:
Pranger x po (xx exp (r(z)).
Following Passaro all (2025),, we use a physics reward that penalizes local constraint violations defined in the PoseBuster V2 benchmark. Let æ denote all atomic coordinates, and let Ephys (x) be weighted sum of potentials:
(xh (x)
albond Uboed (x) Gangls (x) +achir Ucbir (x) ( Uint-clash (x) Orebain Uchain-clash (x),
with nonnegative weights a. The steering reward is then defined as
r(x) (2)).
For each covalent bond, bond angle, and chiral center, we apply a flat-bottom potential function that penalizes deviations from the corresponding physical rules while permitting small fluctuations within an acceptable tolerance. This ensures that generated structures remain physically plausible without being overly constrained. Further implementation details of these potentials are provided in
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Passaro etal (20))) Since also adopt the pre-trained model weights from Passaro et (2IZ2), we do not describe the model architecture or training process in details here for which we refer the reward gradient Vr(z) single basis,
readers to the original work. In this experiment, and its Laplacian Ar(x) is obtained through auto-differentiation.
B.2 NETWORK ARCHITECTURES
The score function V log Pr in both particle experiments (DW-4 and LJ-13) is approximated by E(n) Equivariant Graph Neural Network (EGNN) (Satorras et all 202Tbaa; Köhler et all 2020 Klem 2023: Midgley all (023).. The network maps a time scalar and particle coordinates = (ri..... TnJ to an updated set of coordinates representing the score. All MLPs use Swish activations.
Initialization. Input coordinates are first centered. The scalar time t is encoded using sinusoidal positional embeddings di. The resulting vector is then passed through an MLP create the initial node features h which are identical for all nodes.
1 T = T;
(0)
Tj
h
MLP (t).
π j=1
Equivariant Layers. The network consists of L sequential Equivariant Graph Convolutional Lay- (EGCL). For each layer I {0,.... L 1}, initial message mij computed for each edge
(1)
(i,j) using an MLP PE (0)
m (1)
5h ([h Ilr - -0)
where [...] denotes concatenation. An attention mechanism then refines each message by multiply- ing it with learned gating coefficient:
(1) m mij (Qull (m
where (4) is a single-layer MIP and a is the sigmoid function. These refined messages are used to produce equivariant updates for the coordinates and invariant updates for the node features via MLPs oz and of Both updates employ residual connections.
(1+1)
(m (1)
=r
h
(4+1
= h + 1h Ent
m
where C a normalization constant for the coordinate update.
Final Output. The final output of the network the set of coordinates from the last layer, (L)) which is re-centered to guarantee overall translation invariance. For each particle, the output is given by centering the coordinates:
7
For the DW-4 experiment, we use L layers with a hidden dimension 64, and for the LJ-13 experiment, use L layers with hidden dimension of 128. The normalization constant C is set to 1.0 for both experiments.
B.3 SAMPLING DETAILS
To generate configuration samples, simulate the underdamped Langevin dynamics for system of particles D dimensions. The dynamics target canonical Boltzmann distribution:
7 (x v) x exp (
1
U(())
|001
2
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where (() is the target potential energy (U) H(x) Σ lr: -
2 in case),
the particle velocities, is the temperature, and we assume unit mass. The corresponding stochastic differential equations (SDEs) are:
dæ; v,dt
d = F ( - yo, + /2,Tdwwr
where F(Ø) (x)) is the force, is the friction coefficient, and (wr)220 represents a standard Wiener process. For both experiments, we use y 0.5.
Initialization. Particle positions 2o are initialized on perturbed lattice separated by the charac- teristic length of each system, and the center of mass is removed. Initial velocities vo are drawn from the equilibrium Haxwell-Bolzzmann distribution, with each component sampled independently from (O,
Numerical Integration. The Langevin SDEs are numerically integrated using the BAOAB split- ting scheme (Leimkuhler & Matthews, 20E3, which is a highly accurate and stable method for thermostatted systems. For discrete time step At. the update from state (æn, vn)) to (In+1 Bx+1) performed in five sequential steps:
At
B: Un+1/2 Un +
F(x.),
A
A: #n+1/2 = Fn +
12/2/2
2
O: 0p+1/2 = - Un+ - + T - e(1--e2))
At
A: En+1 In+1/2 +
n+1/2
At F F(xn+1),
B: Un+1 = 8 1/21/2
2
where En is a vector of independent standard normal random variables, i.e., &n ~ ~N
Sampling Protocol. The simulation begins with a burn-in phase, during which the system is evolved for Nbun steps to allow it to equilibrate with the stationary distribution. Following this, the sampling phase begins. To ensure that the collected samples are approximately independent, the system evolves for Ninterval steps between each saved configuration. This process is repeated until the desired number of samples is obtained.
B.4 TRAINING AND INFERENCE DETAILS
We train score-network using the Elucidating the Design Space of Diffusion Models (EDM) framework (Karras et all 2022). This approach frames the learning problem as a denoising task, where single neural network is trained to remove noise from corrupted data across continuous range of noise levels.
Network Preconditioning. The component neural network F(;; or) that is wrapped with a set of preconditioning functions dependent on the noise level a. This design ensures numerical stability and improves performance across all noise scales. The final denoiser. (xnoisy o)). which predicts a clean sample from a noisy one, is defined as:
(Tnoisy ; o) Cskip mnossy + Cour(a) (oin(r))oooss (noise
(B.2)
The functions Cakip Cin- and Cour provide scaling and skip connection, while Cuoise (a) creates time-like embedding from the noise level.
We adopt the preconditioning functions follows:
2 2² + 2
Cakip (a)
Cin Cin(a) Va data
Cour (o)
Va² + o Cnoise (a) log a,
data
where datta is byperparameter varying with the task. We set d = 1.8 for the DW-4 experiment and ddata 0.68 for the IJ--3 experiment.
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Training Objective. The network is trained by corrupting clean data samples Fetean with Gaus- sian noise of given standard deviation 0, creating noisy samples Eoois Telean + de, where (0, I). The training objective is to minimize the weighted mean squared error between the denoiser's prediction and the original clean data. The loss function is given by:
CEDM Po wu(o)) D (anossy σ)
Felean [1]]
where the distribution of a and the weighting function w((r) will be specified later.
Following the EDM framework, the noise level is sampled from log-normal distribution:
log NNP mean= Pad);
with P mem 1.2 and Pud 1.2. To avoid numerical instability, we clip the noise level to be within the range minn max] with min 0.002 and max 80. The weighting function is set
0² data
w(o)
o²o
Inference Process. While the model is trained as denoiser, sampling requires the score function, Ve log Faossy: o). The trained denoiser is converted the score during inference using the following exact relation:
V a log p σ) D (uosssy) σ))
Tooisy
x²
The noise level a coincides with the forward time in our setting in Equ. (2.1) and we have mmax- The time discretization is chosen
MM = { (tx [t
(omiie
(6/2/0
11pp) ))
M
M
k=0
where we choose smaller range of noise levels for the parameters o'nn 0.005 and o 50 to avoid the boundary part of the noise level, which may be less accurate due to the lack of training and singularity. We p experiments and M = 1000 time steps for particle system experiments.
control the spacing of the discretization. We use M 500 for GMM
Training Settings. For the neural network F(::o) in the denoiser (B we adopt the EGNN architecture described in App. B. For the training process, we use the Adam optimizer (Kinmma & Ba, 2014) with learning rate of 5 x 10 and a batch size of 1024. We train the network for 106 steps, where we sample fresh dataset of 106h data points with the same protocol as described in App. BH for every 10 steps. The warm-up period is only performed once before the first batch. All training is performed on single NVIDIA A100 GPU. The training code is based on both the JAX library (Bradbury et all, 2018) and the Optax gradient processing and optimization library (DeepMind al), (020)).
B.5 EVALUATION METRICS
To quantitatively assess the quality the samples generated by each method against ground-truth reference distribution, employ suite of five complementary metrics. Let the generated samples be a weighted set (t) (+)) HE[N and the reference samples be { [x w ]j([Na))
Negative Log-Likelihood Difference (ANLL). The Negative Log-Likelihood (NLL) measures how well set of samples fits the target probability distribution p(æ). ). For weighted set of samples, it is estimated the weighted average of the negative pogpbbbbabilitie::
NLL =
Σ
(4) w
log ()))
1=1
We report ANLL, which is the difference between the NLL of the samples generated by method and the NLL of the reference samples: ANLL = NLLmethod - NLL A lower absolute value indicates better fit to the target distribution.
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Maximum Mean Diserepancy (MMD). Maximum Mean Discrepancy (MMD) is integral probability metric defined in a Reproducing Kernel Hilbert Space (RKHS) with characteris- tic kernel k(.,). Any probability distribution P admits unique mean embedding pp e H. The squared MMD between two distributions and Q is the squared RKHS distance between their embeddings:
MMD -|upp - || (k(a,, -)] Ey-ookyy -)1
Using the kernel trick, this definition can be expressed entirely in terms of kernel evaluations, avoid- ing explicit computation of the embeddings. For two weighted sample sets, the unbiased empirical estimator is
2MD Re (k:(,,,)) :]] Ey -)llx E (k( + Ey (ky, y/)l
2 P. ry (kk(((r y)l
Nior
N Niar
w 1 OZ x00) +
w ret w hi (6) (5)
ret ref 28 rel
w
w ref
20 ( 1ef)
,,==1
i=1
We use the RBF kernel
| y11
k< y) exp
2
with σ 20 for the GMM experiment and 0x 5 for the DW-4 and LJ-13 experiments.
Direct evaluation of MMD requires O(N²) kernel evaluations, which becomes expensive for large sample sets. To scale up computation, we use Random Fourier Features (RFF) approximate the RBF kernel with an explicit oow-iimesioona feature map zxx) E RR, reducing the complexity to O(NN)).
Bochner's 's theorem states that a shift- invariant kernel can be expressed as the expectation of a product of complex Fourier features with respect its spectral density p(w):
k(x, y) = Euupp(w) [eww"
x =e- w: y
For the RBF kernel, p(w) is Gaussian, 0x II) The RFF approximation replaces the expec- tation with Monte Carlo average over f/2 sampled frequencies we N 2)) and random phases by Unif(O, 2x)) yielding the explicit feature map
2 2(x) / cos(ww] b1), cos(w) b5/2), sin(w] x + b1), ..... bi)....,,si(([T] + by/2))
This ensures that squared Euclidean distance between the mean feature vectors:
s k(x y). The MMD can then be efficiently approximated as the
Niot w ret 2
MMD² || [z(=)
Ey E
Σ w 2(x0
1 (cf
In our experiments, we use 2048 random features (ʃ 2048) for experiments.
Sliced-Waid Wasserstein Distance (SWD). The Wasserstein distance measures the minimum cost to transport mass from one distribution to another. As computing exact 2- Wassersstein distance between high-ddmensional empirical distributions with up to 2 particles in our experiments is computa- tionally prohibitive, we report Sliced-Wasserss Wasserstein Distance (SWD) as a computationally efficient approximation that involves projecting the sigh-dimensional distributions onto series of random 1D lines, calculating the ID Wasserstein-2 distance for each projection, and averaging the results:
(WD)
no (Q))do ~
W3 (To. (P). no, (Q)),
where Tp denotes the projection onto a line with direction -D p the number of projections, and 0, Un-1() In experiments, use p 10 for all experiments.
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Radial Distribution Function Wasserstein-1 Distance (WIDF). The Radial Distribution Func- tion (RDF), (r), fundamental descriptor of the spatial arrangement of particles in a system, measuring the probability of finding a particle at a distance from another. For simplicity and fol- lowing the literature (AKhound-Sadegh et all, 2024; St 2025), we define it as the ensemble- averaged interatomic distance distribution:
g(r)
2 N(N -1)
(8(r - ||: rr 1)); Σ K
1)
where (-) denotes the ensemble average over configurations. With slight abuse of notation, we still refer to g(r) as the RDF without the shell factor and density normalization.
To assess the structural accuracy of our samples, we compute the RDF for both the generated and reference configurations, yielding two ID distributions, Imethod (r) and (ra(r). The WRRDF metric is the I-Wasserstein distance between these two distributions, W1 (9method Iwt), which quantifies the difference in the learned physical structure.
Energy Wasserstein-1 Distance (Wf). The energy of configuration is its negative log- probability, E(x) llog (x). Due the unknown normalization constant for the reward-tilted and annealed distribution ((x) x p" (()) we report the energy up to constant, i.e.,
((x) -y log r(z).
By evaluating the energy for every sample, obtain two 1D distributions of energies, Pmmethod and pret The W metric is the 1-Wasserstein distance, W1(PP (Pamethod PPet), between these energy distributions. This metric evaluates how well a method captures the correct energy landscape and the relative probabilities of different configurations.
Physical Validity. To evaluate the physical plausibility of generated protein-ligand structures, we adopt the validity metrics from PoseBusters (Buttenschoen 2024), which assess adherence to fundamental physical and chemical rules.
The fraction of generated structures that satisfy all validity checks simultaneously. serving as an overall measure of correctness.
Valid Fraction:
Clash-Free Fraction: The proportion of structures without severe steric clashes, i.e., unphysical overlaps between atoms.
Bond Length and Bond Angle Violations: Counts deviations from standard covalent geometry. short too long, while bond angle violations
Bond length violations occur when bonds correspond to unrealistic angular geometries.
Internal Clashes: The number of steric overlaps within the same ligand molecule, reflecting poor internal consistency.
Chiral Atom Errors:
The number of stereocenters incorrectly assigned, such inversions of chi-
rality.
Chain Clashes: The number steric overlaps between ligand atoms and protein atoms, indicating violations of intermolecular packing constraints.
C ADDITIONAL EXPERIMENTAL RESULTS
This appendix provides supplementary results that further explore the performance of methods. DriftLite- VCG(-SMC) and Mixture Model (GMM) and the particle-based systems (DW-4 and LJ-13).
under various conditions for both the Gaussian
C.I ADDITIONAL EXPERIMENTAL RESULTS OF GMM
We first present more extensive ablation study the GMM. Tab. B details the performance of all methods across a wider range of annealing factors, y E {1.5, 2.0, 2.5, 3.0}. The higher the annealing factor becomes, the more challenging the inference-time scaling task as it accentuates
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the modes of the distribution and increases the energy barriers between them. The results reinforce the conclusions from the main text: as ry grows, the performance gap between our Dri methods and the baselines widens considerably. VCG-SMC, in particular, consistently achieves the best or near-best performance across all metrics, showcasing its robustness.
Similarly, Tab. _ extends the reward-tilting experiments to different reward strengths (a {100, 200, 300, 400)). For the definition of the parameter Ø, we refer to the problem settings in App. I. Again, our methods demonstrate superior stability and accuracy compared to Pure Guid- ance (PG) and G-SMC, which degrade significantly as the reward becomes more peaked (smaller reward covariance scale r).
Table 3:
Performance ablation for the GMM annealing task with varying annealing factor y. Results are +td over 5 configurations. Best results per column (within each y block) are in bold.
Annealing (y 1.5)
Anncaling (7 2200
Method
ALL
MMD x-0-
SWD
Mean L/2
Cov X10
ANLL
MMD
SWD
Mean Lg
Cov Xx30
PG
-1,19640s21
4,491
11.90=mm0
58.98
5,274
-3. -3.0744mm4
6.576-22m6
18.0646.04
80. 80.10
9.347=2240
G-SMC
0.834gomo
1.74
3.097
-0.527
2.057
11.44
58.71
2.097go.9
ECG
0.287-0093
0,185+0.03
0.622
-009
2.982+a331
0,350+am)
0.324+au4
0.415
1.209 +a.16
5,323+0a58
0.750-0009
ECG-SMC
0.219 0.162:2004
0.605
20159
2.667
0.335
2ams
0.198-gams
0.185
2000
0,779
3.782
0.446-0004
0.222_oows
0.166
0.590_eam
2.661
0.338
0.204ga00
0.188
0.672.imm
2.97 17
0.379_oom
VCG-SMC
0.203
0.162:00.02
0.638
2005 2.8522000
0,34610m1
0.192::0m0
0.166::000s
0.606 2.866
0.344 8.827
Annealing
2.5)
Annealing (y 3.0)
Method
ANLL
MMD
SWD
Mean L2
Cov '10
ANLL
MMD
SWD
Moan L2 Cov F xie
PG
-5.016g120
0. 20.27
92,04
-4.95 2,
-4.9
0.758
19.55_409
93.39.2..44
1.654
G-SMC
-0.801
13.88_1m0
18.11_18
1101
-0.69220414
0.493_om
18.31_1m
102.719%8
4.973
ECG
-0.427
0,248
6,443
30.08
3.135
22:00
-1.201
0.353
8.967
39.24
4.858
ECG-SMC
0.169
0.021
1.00220018.
4,886_
0.564
0.184_o00
0.031
1.672
1.
0.850
VCG
0.204
0.023
0.717
3.351
-0722
g.420
0.209_0.000
0.029
0.859_0347
29:0
0.505
VCG-SMC
0. 0.179=0ass 0.018
0.613
2.867
0.380:=a05t
0.174
(..19
0.691 =0.10
3.319
0.415
Table 4: Performance ablation for the GMM reward-tilting task with varying reward strength o.
Results are mean over 5 runs. Best results per column (within each a block) are in bold.
Method
Reward Reward-Tilting Tilting
100)
Reward Tilting (a 200)
ANLL
SWD
Mean Lg
Cov x100
ANLL
MMD
SWD
La
x10
PG
21
0.00
13.57
85.34
8.159
5.454
0.825
20o4
12.77
23.51
5.845
G-SMC
1.43
0.24
2.62540070
15.42+1.39
0.683+0630
0.422
0.086+0.225
1.072+0.00
5.735+2680
0,463+0225
ECG
+.854
+.119+014
1.120+144
4.496+5557
0.160+0.22
0.777
0.115
+
1.306+aass
5.613+1330
0,287+0.13
ECG-SMC
0.309
0.020
0.234
0.996
0.065
2000
0.304
0.025
00:00
0,360
1.795
0.115
VCG
0.262+
0.032+0mm
0.284
1.118+a2m6
0.059
-eats
0.256
8.899
0
0.394
1.601
0.100+000
VCG-SMC
0.338ga1s3
0.020gamm
0.236ga.12
0.931
0.061
0.348
20004
0.020
0.342
1.6362066
0.113
Method
Annealing (o- 300)
Annealing (o 400)
ANLL
MMD
SWD
Mcan
Cov F
x108
ANLL
MMD
SWD
Mean L2
Cov x103
PG
1.888
0.660
11.06gsas: 5,238 28.23
0.618
0.497
45 45.60 9.1
3,952
G-SMC
0.207 1a10s
0.047
0.664gom
3.447
0.284_a:5
0.253
0.036gma0
0.814gmmm
4.917
9ss
0.377
ECG
0.664
0.132
1.57240413
7.141
0.434
0.465
0.122
1.61
7,590
0.535
ECG-SMC
0.661 1933
0.048g00m
0.740ga335
3.965
0.258
gooos
0.327
0.039
0.8881o1
4.433goms
0,394
VCG
0.034
0.498
1
0.157
0.290_01121
n.033
0.550
2.238
0,203
VCG-SMC
0.315
0.022:-am)
0.355
1.544
0.150==0024
0.292
0.022
0.454
2.072 0.1844
Fig. 5 illustrates the evolution of ESS and variance for a milder reward-tilting task (a = 50.0), showing that even in less challenging scenarios, control mechanism actively stabilizes the particle weights. Fig. IC shows the per- formance of all methods as a function of the number of particles for a strong reward-tilting task (a = 400.0). It clearly indicates that our VCG and ECG methods not only outperform the baselines but also converge more efficiently, achieving better results with fewer particles.
Finally, Fig. E and Fig. III complement the figures in Sec.
As shown in Tab. B. for the 30 30-dimensional GMM problem, the relative runtime for our VCG-SMC and ECG-SMC methods is only around 5 to 6 times that of the standard Guidance-SMe (G-SMC) baseline. This modest increase in computation time is largely due the effective parallelization of
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PC
VCI
VEG--MMC
EDG-CMC
VDG
VEG--MIC
FORHAMC
1.0
10
MMD
SWD
101
10
100
0.0
Variance
10
10P
Moan
Cov Frob.
10
100
200
500
500
1000
Step
Particiss
Figure 9: Evolution of ESS and potential vari- ance for the GMM reward-tilting task (a 50.0).
Figure 10: Performance metrics versus num-
ber of particles for the GMM reward-tilting task (a = 400.0).
Table 5: Elapsed time comparison for different sampling methods. Results are mcan= over runs. The relative runtime, including propagated standard deviation, is computed with respect to the G-SMC method.
Anncaling (? = 2.5)
Reward- -illtnn (a 100)
Method
Runtime (s)
Relative Runtime
Runtime (8)
Relative Runtime
G-SMC
6.44 13.28
1.00
6.90
1.00
VCG-SMC 39.7 39.784099
6. 6.18433.15 X
40.22
5.83
ECG-SMC
39.1 I +0.50
6.07 #3.09 X
39.33
5. 5.70+2.49
the core algorithms. These results underscore the controlled additional cost required to implement the variance reduction technique, making them commonly practical. A more optimized implemen- tation could reduce this runtime gap even further by exploiting more GPU resources.
We further conducted GMM experiments on the annealing task (7 = 2.5) and the quadratic reward- tilting task (a 100) under matched wall-clock budget (Tab. ( In these runs, the baselines (PG, G-SMC) used 8192 particles and 2000 diffusion steps, while DriftLite variants (VCG/ECG, with or without SMC) used 1024 particles and 1000 diffusion steps. Under this configuration, the wall- clock times of DiiftLite methods are comparable to, and in some cases slightly lower than, those of PG and G-SMC, despite using substantially fewer particles and steps. At the same time, DriftLite variants, especially VCG-SMC and ECG-SMC, achieve markedly lower MMD, SWD, and mean- errors, while maintaining ANLI close to zero. These results indicate that the added control drift substantially improves sample quality at essentially fixed computational cost, mitigating the variance and weight degeneracy that limit the uncontrolled SMC baseline.
Comparison with a training-based neural drift controller (NCG). contextualize the trade- off between training-free inference-time control and amortized drift-learning approaches, we im- plemented a prototype training- based baseline, which we refer to as Neural Controlling Guidance (NCG).
In NCG, the control drift parameterized by neural network (x,) taking time and state as inputs, in contrast to DriftLite which solves for the control coefficients via small linear systems under a linear ansatz (cf. Eq. (3.3) and Ansatz 3.3). Concretely, we train b by minimizing a regularized version of the same variance objective:
(0) Var [9)(x) h4 + Al
with À = 0.1. We use U-Net architecture with encoder feature dimensions [128, 256, 512] and decoder feature dimensions [256, 128). For training, we maintain an ensemble of N 8192 particles (matching the inference-only baselines). and after each time step, we backpropagate (oL(()) and update parameters using learning rate 5 x 10 14- We repeat this training inference multiple refinement rounds, carrying the updated dynamics from the previous round as the base for the next.
iiiningginferenee procedure
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Table 6: Performance with matched wall-clock budgets for the GMM tasks. Results are s over 5 runs. Best results per column are in bold.
Amnealing
Reward- Tilting
Method
ANLL
MMD
SWD
L.
Runtime
ANLL
SWD
Mcan
Runtime (8)
PG
1..0
0.73
18.52+300
90.58_22.59
17.28
37.92
0.96
49.20
112.27
15.98=051
G-SMC
-0,56-ass
0.46+0m
18.87+am
102.17+30.0
18.45
1,23+
0.25+am
2.81
17.13
16,74-aa1
VCG
4.m 0.061am
1885
1.76
15.12
0.36
4.09
0.83
3.19
14.96.001
VCG-SMC
0.04
0.08
1.75+a041
7,33++161
15.77 +0.13
15.77
0.25
0.005
0.59
2,57-+050
15.55-0.10
ECG
-0.75-11: 0.40
10.22
50.45 =9.10
13.92 =
0.39
0.23
1.80
8.49
14.73=aw
ECG-SMC 0,08100s
0.07
2.69
14.29
14,83 1
40.01
0.061am
1.03
5.93
15.30gas
Table 7: Comparison with training-based Neural Controlling Guidance (NCG) on the 30D GMM. NCG-SMC-r denotes the r-th refinemen//training round. Inference Only evaluates the learned drift from round 20 without backpropagation. Lower is better for all metrics. 'Best-performing NCG round among those reported.
Method
MMD
SWD
Mean Lę
ANLL
Runtime
PG
0.825
21.86
66.09
-0.65
4.59
G-SMC
0.143
6.45
24.94 94
0.04
5.53
VCG-SMC
0.017
0.52
1.23
0.14
7.83
ECG-SMC
0.019
0.68
1.83
0.14
6.99
NCG-SMC-1
0,437
12.29
51.44
0.39
70.79
NCG-SMC-S
0.135
3.53
13.67
0.34
307.34
NCG-SMC-9-
0.091
2.45
9.11
0.05
543.77
NCG-SMC-12
0.223
13.35
37.96
0.01
721.23
NCG-SMC-16
0.143
7.61
21.73
0.16
957.72
NCG-SMC-20
0.199
9.36
24.02
0.08
1194.31
NCG-SMC-20 (Inference Only)
0.190
8.91
29.04
0.26
240.88
Overall, NCG improves substantially during early refinement rounds but exhibits non-monn monotonic be- havior as training continues, highlighting the optimization sensitivity of variance-basse drift learn- ing. Moreover, even without training (Inference Only), evaluating a high-capacins neural controller incurs significant inference-time overhead relative to DriftLite's lightweight linear solve.
C.2 ADDITIONAL. EXPERIMENTAL RESULTS OF PARTICLE SYSTEMS
Here, we provide additional ablation studies for the DW-4 and LJ-13 particle systems, demonstrating the robustness of our findings.
ESS and Potential Variance Evolution. In Fig. 12 present similar visualization in the GMM example (Figs. and H) where similar trends are observed: pure guidance leads to a much rapid drop in ESS during the initial stage of the inference process, while the aariance-controllin guidance postpones the drop when the resampling kicks in. We also observe that the time when the curve of the potential variance Var,, [9t] of all methods meets coincides with the drop ESS of the poriance-controlin guidance, which may correspond to the splitting of modes, and resampling may be crucial to handle.
Elapsed Time Comparison. The results for the DW4 system show that the performance scaling of the advanced methods is even more favorable than what was observed for the GMM task (Tab. D From Tab. &. we can see that the relative runtime of the VCG-SMC methods is only about twice that of the G-SMC baseline. This remarkable efficiency stems from the nature of the problem itself; the total computational cost dominated by repeated, computationally heavy score evaluations. Consequently, the fixed algorithmic overhead from our DriftLite variance reduction methods be- comes negligible as a fraction of the total runtime, underscoring their practicality and efficiency for computationally intensive systems.
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G-SMC
VCG-SMC
PG
VOG
VCG-MMC
10a
ESS
MMD
SWD
10
10-4
0.25
10t1
10'
W!
10-3
102
10
400
215
212
Step
Figure 11: Evolution of ESS and potential vari- ance for the DW-4 annealing task (7 2.0).
Figure 12: Performance metrics versus number of
particles for the DW-4 annealing task (? = 2.0).
Table 8: Elapsed time comparison for different sampling methods the DW4 System. Results iea over runs. The relative runtime, including propagated standard deviation, is computed with respect to the G-SMC method.
Annealing (7 2.0)
Reward-Tilting (X 0.5)
Method
Runtime (s)
Relative Runtime
Runtime (s)
Relative Runtime
G-SMC
281.68
1.00 x
124.12
1.00
VCG-SMC 674.3216.89
2.39 X
674.12
2.37 20.770.07 X
Ablation on Number of Particles N. Next, we present the ablation study on the number of par- ticles on the DW-4 system as in the GMM example (Fig. (2). Despite a more complex problem nature and high-level evaluation metrics (RDF and energy distribution), our methods still present robust scaling in MMD and SWD, and show promising decay in W and Wf, highlighting the effectiveness and robustness of our methods.
Ablation on Base Model Temperature T. In Tab. we investigate the effect of the base model's training temperature (T' E {1.0, 1.5)) on an annealing task for the DW-4 system in addition to the base temperature = 2.0 as used Tab. M The results show that our methods, particularly VCG- SMC, are effective regardless of the starting temperature. They successfully anneal the system to the target low-temperature state, consistently outperforming the baselines and confirming that the control mechanism adapts well different initial dynamics.
Table 9: Ablation results on Annealing with y = 2.0 from Base Temperature T E {1.0, 1.5).
Results meangsid over 5 runs. Best results per column (within each T block) are in bold.
Method
T 1.0
T = 1.5
ANLL
MMD
SWD
WRRD
w
ANLL
MMD
SWD
W
0.219
0.343 10.1s
0.833_00
0.194_000.
0.054iamm
0.546
0.259 Loos
0.690
0.217 0.320
G-SMC
2.571 0.359
0.973
0.147
0.201
1.645
0.639+o.185
1.681 +a527
0.1 16
0.325
VCG
-0.066+000
0.026
0.076-6011
0.045 samn
0.127
+aais
0.083++.12
0.032+0015
0.091
0.03444000
0,558+oom:
-0.044 0.019
0.048_9913
0.053_0011
0.053
0.077
0.015
0.035
18011
0.011 iatos 0.078..ame
Ablation on Annealing Factor 7 We further test the methods with varying annealing strengths (Y (1.5,22 2.0, 2.5)) for both the DW-4 and LJ-13 systems. For the DW-4 system (Figs. I and [4] and Tab. лI)) the visual and quantitative results confirm that VCG and VCG-SMC maintain high accuracy even as y increases. For the more complex LJ-13 system (Fig. IN and Tab. D.. the chal- lenge is greater. While all methods struggle with the most challenging annealing tasks, VCG-SMC consistently provides the most physically plausible results, capturing the structural features (RDF) and energy distributions far more accurately than competing methods. This highlights its superior performance in complex, high-dimensional energy landscapes.
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0.30
0.25
PG (MW, 0.329)
0.76
0.20
VCG 0.577)
G-SMC (W/; 0.30.)
VOG SMG (14
RDF ofr
2.35
230
245
200
245
Denstyy
0.15
0.4
Araayiia Refermoo
0.10
02
VCG (P; 6,025)
G-SMI: (W) 0.1855
0.05
0,150
203
(W)
0.00
-225
-00.0
-2.5
-25
-22
-20.0
-175
Paindssa Dispance
Energy
(a) Radial Distribution Function.
(b) Energy Distribution.
Figure 13: Comparison of generated distributions for the DW-4 annealing task (7 = 2.0): (a) Radial Distribution Function and (b) Energy Distribution.
0.22
Areeytical (Wy 0.548)
0.8
0.24
0.5s
0.25
G-SMC (W/ 0.202)
VDG 0.924)
0.5s
0.20
VCD SMC (W)) 0.055)
RDF gift
2.35
250
246
2800
Denetty P2
0.15
044
(W)
0.23
02
VCG IW:- 0,070)
G SMC (W; 0.387)
0.05
0.00
-32.5
-20.0
-37.5
-25.0
-22
-20.5
-17.5
-150
Paindse Distance
Energy E
(a) Radial Distribution Function.
(b) Energy Distribution.
Figure 14: Comparison of generated distributions for the DW-4 annealing task (7 2.5): (a) Radial Distribution Function and (b) Energy Distribution.
Table 10: Performance ablation for the DW-4 annealing task (T = 2.0) with varying annealing factor y. Results are tean over 5 runs. Best results per column (within each y block) are in bold.
Annealing
1.5)
Annealing (y 2.5)
Method
ANLL
MMD
SWD
W.O0P
WF
ANLL
MMD
SWD
W
W
PG
14
0,198 0.527-6356
0.132 oac
0. +.179
0.199 1,332
0.400-c.1s
1.888
0.208
0.551
G-SMC
+n0s:
0.003
0.143==am1
0.038 -oas
0.026:amu
0.038-6118
0,365=oas
1.012-220
0,190+
VCG
0.010+000 0.024=c005
0.041
2000
0.307
40.065
0.024=
0.061
20013
+0
VCG-SMC
-0.013+a077 0.009 0.022 9,002 0.037
0.020 -0.060
0.023
0.056
0.119.0.00
0.059
Table 11: Performance ablation for the LJ-13 annealing task (T = 1.0) with varying annealing factor
7. Results are mea over 5 runs. Best results per column (within each y block) are in bold.
Anncaling (7 1.5)
Anncaling ( = 2.0)
Mothod
ANL.
MMD
SWD
W-RDF
W.
ANLI
MMD
SWD
WRRDF
W
PG
3.424
0.718
0.806 =008
0.026
1.855
0.622
+ 12.73+0.00
G-SMC
3,289
0.220
goom
0.022
3.517
0.927
0.416
0.017 4
4.25
VCG
1.64210ss us
0.027 Lonor
0.029gnoom
gooms
0.058
gnom
8.848
1.191
1
0. no02
0.135
0.088
21 gm
VOG-SMC
2.221 40353 0.024:aam4
0.024
0.018::am:
a.
0.734::..41 0.091
0.092-4aut1
0.007 +u00 1.958
Ablation on Reward Strength X' Finally, we perform ablation on the reward strength (X' E (0.2, 0.5. 0.83) for the reward-tilting tilting task on both particle systems. The results, shown Tab. I for DW-4 and Tab. I3 for LJ-13, are consistent with previous findings. As the reward strength increases, making the target distribution more distinct from the base distribution, the performance of the baseline methods deteriorates rapidly. In contrast, VCG-SMC maintains excellent performance,
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2
(W) 0.054)
0.7
PGG (Ml, 12.780
G-SMC (W; 0.033)
G-SMC (V/s = =0.331)
(W; (.000))
VDG 21.357)
(W) +0.000)
VOB (Wy 1.577)
RDF 6
Denetty P.
3.2
0.04
1.5
0.03
1.0
0.02
0.91
0.99
3.00
1.00
05
1.0
1.5
2.0
30
0.00
-150
-150
-140
-130
-120
-110
Paindsa Distance
Energy
(a) Radial Distribution Function.
(b) Energy Distribution.
Figure 15: Comparison of generated distributions for the IJ-13 annealing task (7 2.0): (a) Radial
Distribution Function and (b) Energy Distribution.
demonstrating its capability to accurately steer the particle distribution toward sharply defined target region. This confirms the effectiveness and robustness of our drift control mechanism across a wide range of inference-time scaling challenges.
Table 12: Performance ablation for the DW-4 reward-tilting task (T = 2.0) with varying reward strength X. Results are msa over runs. Best results per column (within each X block) are in bold.
Reward Bowtad-Tilting 0.2)
Reward-Tilting Tilting (X'
Method
ANLL
MMD
SWD
W-RDF
W
ANLL
MMD
SWD
W
PG
0.182 gass
0,768
1.808
0.305_as0 0.3684amm 2.480.2211
0.775
1.677
0.852
3.867
G-SMC
0.094 Laass
1000:
0.046
0.060_ooo
g.094g
goms
0.079 gome
2o04s
g.150
0,536goms
0,386=1000
0.:03+
1.122=0359
-.005
0,138
1,050+286
0.671
tons:
1.768
+0302
0.282++m.21
1.162++0s0
VCG-SMC
0.016
0,0141aam
0.032 8,004
0,059
0.093
0.518 aunz
0.028
0,065
0.149
0.520
Table 13: Performance ablation for the LJ-13 reward-tilting task (T' 1.0) with varying reward strength X. Results are mean over 5 runs. Best results per column (within each X block) are in bold.
Reward-Tilting (A' = 0.2)
Reward-Tilting (X' = 0.5)
Method
ANLL
MMD
SWD
W-RDF
W.
ANLL
MMD
SWD
W.Rne
W
44812
0.718
0.806 0.026
40.00
3.253
45013
0.718 =0000
+0000
0.050 40705
G-SMC
0.83++am2
0,031
+omm
0.030+oo1s
0.010
0.834=0um
1,277
0.055
0.049
+0.00
0.016:amm
1.278
VCG
0.106
0.559
0.588
0.024
1.869
1.470
0.669
20.32
0.675
0.024
1.869
VCG-SMC
0.765
+amr 0.012+
0.015 0002
0.016+aco1
1.219 +.00
1.218 +
e.01
0.015
0.016
4.219
C.3 ADDITIONAL EXPERIMENTAL RESULTS OF PROTEIN LIGAND CO-FOLDING
In Fig. IG. we show representative trajectories of the normalized Effective Sample Size (ESS) for three -igtein-ligand complexes (PDB IDs 7n7h, 7zu2, and 8d39). Even with relatively small particle budget, our method (VCG-SMC) maintains substantially higher ESS than the baseline G- SMC throughout the middle and late stages of the trajectory, indicating that the controlled drift keeps much larger fraction of particles effective. We expect this advantage to become even more pronounced as the number of particles increases.
Table shows that both particle-based methods are about 3.2x slower than the unsteered base sampler, reflecting the cost of propagating and reweighting small ensemble of particles. Impor- tantly, VCG-SMC incurs essentially no additional wall-clock overhead compared with the G-SMC baseline (31.78s 31.67s per complex), while delivering consistently higher ESS and improved physical validity on PoseBusters. Thus, the stability gains from DriftLite come at negligible extra cost beyond standard SMC correction.
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Published as a conference paper at ICLR 2026
GM5MC
VCB SMC
G SMC
VCG SMC
0 SMC
VOG SMC
1.0
7n7h
72u2
8039
0.6
0.0
38 0.6
0.
(a) PDB ID 7n7h.
(b) PDB ID 7zu2.
(c) PDB IID 8d39.
Figure 16: Evolution ESS during inference on the protein ligand co-folding task. The curves are smoothed using& Gaussian kernel over time, with the unsmoothed traces shown in the background for reference.
protein-ligand
Table 14: Elapsed time per generated protein-ligand complex for different sampling methods, av- eraged over a random subset of 20 structures. The relative runtime is computed with respect to the G-SMC method.
Method
Runtime (s)
Relative Runtime
Base
10.05
0.32
G-SMC
31.67
1.00
VCG-SMC
31.78
1.00
C.4 ITERATIVE REFINEMENT
While a single pass of inference-time scaling effective, corrective power is finite; once the simulation reaches the terminal time, any residual mismatch between the particle and target distri- butions cannot be further addressed. The key insight behind our iterative approach that each pass produces improved sampling dynamic. The updated drift and potential:
tht (x) (x) (xx),
gtth (x) 9x(x) hz(x;b))
encode richer information about the target distribution. Our iterative refinement procedure leverages this accumulated knowledge rather than discarding it. By iterating the procedure, each round builds upon the refined dynamics of the previous one, creating virtuous cycle that progressively sharpens the sampling path.
This iterative process can also be viewed practical method for approaching the optimal control drift in Prop. 3.21 Since the control b, computed each step under a linear ansatz (Thms. G and (.4), it provides an approximation of the true optimal control. This approximation becomes increasingly accurate as the underlying dynamics and particle distribution are improved in each round, allowing the linear model operate better-conditioned problem.
Algorithm. The iterative refinement algorithm, detailed in Alg. M transforms the single-pass method (Alg- M1) into multi-stage process of progressive improvement. The fundamental differ- ence lies in the cumulative application of control. Whereas the base algorithm applies calculated control drift just once, the iterative method repeats the entire simulation K times.
The central mechanism is the permanent absorption of the learned control into the system's 's dynam- ics. After each full trajectory simulation "round), the control terms are folded into an "effective" drift of and potential g.ft. While particle positions are reset to the initial noise distribution at the start of new round, these effective dynamics are preserved and carried forward. The refined dy- namics from round j thus serve as the improved baseline for round + 1. This process iteratively sharpens the sampling path, guiding particles more efficiently in subsequent rounds without any external training or global optimization.
Results. Our empirical results decisively validate the iterative refinement strategy on a Gaussian
Mixture Model (GMM) target across both annealing and reward-tilting tasks.
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Published as a conference paper at ICLR 2026
Algorithm 2: Iterative Refinement for Inference-Time Scaling (cumulative drift/potential updates)
Input: Original drift path vt, original potential path 9t. time steps {tx M reward r(æ), schedule Br, basis functions, number of refinement rounds K, number of particles N,
ESS threshold т.
Initialize effective drift ofth (.) +- vv(()) and potential of ge(-);
for jorj+ 1 do
Initialize particles Lig po and !!! + for I,. for k + to M - do
(1)
Form weighted estimates of A., and Ct2 using {(x)) wir ( hEI:: Solve As Ou Cu. obtain the control drift (.)
woff() (.) f vi (-) + (t). goff() + 9iz (ff(() (-) + he, bi:)
log Whit1 -log wit + Dixe (x) (1x+11111 - 11), Wit+1 t- softmax(wr
(8)
(6) Wog
Ftath Tie + vie (xx")) (tx+1 - -x)++ + Vu vx++1 (1) where 2(0) NO
10
ESS( .) < or periodically then
Resample (ar ]iE[N] according to (w/ HEE[N];
Reset Wia N for all
Output: Final samples [(x7 (1) MEEN from the K-th refinement round.
The evolution of the sampler's internal state, shown in Figs. IV and I provides clear evidence of the algorithm's 's success. With each successive refinement round (progressing from blue to red), the variance of the control estimates exhibits a striking, monotonic decrease. This variance reduction directly leads dramatically more stable ESS, which is maintained at near-optimal levels through- out the later stages the simulation. This provides direct confirmation that the refined dynamics act substantially more efficient proposal distribution, trend that holds robustly across both experimental setups.
The final sample quality metries, reported in Tabs. I and IG complete the picture. While the improvement in downstream metrics, such as MMD and SWD, is not always strictly monotonic with every iteration, the refinement process consistently uncovers solutions that superior those from the initial pass. The best-performing configurations (highlighted bold) are frequently found in later iterations. This confirms that multiple refinement passes are invaluable for navigating the optimization landscape to find higher-quality final samples.
1,0
1.00
0.8
10th
Variance
Varier
10'7
10
40
100
200
300
430
500
100
200
300
400
500
Step
Step
(a) ECG.
(b) VCG.
Figure 17: Evolution of ESS and potential variance across multiple refinement rounds for the GMM annealing task (y = 2.5). Later rounds (red) show monotonically decreasing variance and more stable ESS.
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Iterative refinement performance on the GMM annealing task (7 3.0). Results are tsa over runs. Best result per column is in bold.
Table 15:
VCG-SMC
ECG-SMC
Itor:
ANLL
MMD
SWD
Meun Lg
Cov F x103
ANLL
MMD
SWD
Mean La
Cov F x10
2.01
0.691 20:49
3.319 =oaes
0.415 g00e
0.184_oav
20000
1.672
1.795
0.850
0.159
0:019
20797
3.906go.r
0.450
0.178
0.178_oom
0.028go00
1,105 10
0.787 1.748
0.176:2002
0.019
0.599
20.42
3,083
-4794
0.406
0.166-0015
0.027
1.201 2026
5.990
0.762
0.152
0.01
0.720
3.346_0.4s
0.168 0.168_00rr
0.029
1.696
0.81620199 1.
5
0.165 goods
0.018
0.694
3.124_0705
3.124
0,386
0,158
0.025
1.09610.06
4.854
0.693
0.019+a.00
4074
3.378
0.412+
0.164-000
0.031 +0,007
1.494
7.1384270
0.908
0.161
0.019.a.01
0.736
3.343
0,408
0.160
0.031
1.477
7.010
0.874
g.0s
0.019
20:70
3.407
20:09
g.420
0.181 0005
0.02910000
1.406
6.748.11:9
0.806
0.18420000
0.019in00t
0.670
3.238
0.395 10000
0.187
0.187_0066
0.027
1.137
5.66911519
0.739
1.00
ESS
1.00
0.95
0.00
0.00
10'
Variance
10a
Vaniance
10"2
10nd
100
200
300
400
500
100
230
300
400
500
Step
Step
(a) ECG.
(b) VCG.
Figure 18: Evolution ESS and potential variance across multiple refinement rounds for the GMM reward-tilting task (a 100.0). Later rounds (red) show monotonically decreasing variance.
Table 16: Iterative refinement performance on the GMM reward-tilting task (o 100.0). Results are sta over 5 runs. Best result per column is in bold.
VCG-SMC
ECG-SMC
ANLL
MMD
SWD
Mean La
Cov F x103
ANLL
MMD
SWD
Mean La
Cov F. x
g.020
0.236
0.931 2030
0.061 iane
0.309
0.309_aa0 _0007
0.0201000
0.234
0.996
0.065
0.294
0.020+4a.0
0.24
1.240
0.058 +aret
0.284-6051
0.851 4
0.054++0025
0.244
0.019:4a00
0.225 200m
0,983-6310
0.053 2004
0.273
0.020+0000
0.256
1.310
0.062
0.
n.020
0.268
1.38--012
0.065
0.288
-007
0.020
0.267
+oms
1.2764487
0.063
0.280
0.019.4amm
0.259
0.259.4a.12
1.087
1,087
0.057 0.057+aame
0.278
0.020+0002
0.263
1.36440065
0.055
4.319
g.020-gaom
20013
0.862
26622
0.053 1ome
0.333-o08
-004
0.020
0.255 20me
1.2922nma
0.069
0.311
0.020.noee
0.270
1.404
0.058
0.322
0.020
0,285
1.504g9913
0.060
0.365 1012
n.020
0.263
1.277
0.057 4are0
0.338 0099
0.020
0,21640m54
0.97644477
0.049
L0025
0.
0.019:4a.0:
8-861
1.321-6099
0.218++am
a.007++a.m
8430
39


Reviews:
[{"review_id": "IL1USe7Tcb", "reviewer": "Reviewer_12SX", "summary": "This paper introduces DriftLite, a training-free method for steering diffusion models at inference time using an improved sequential Monte Carlo (SMC). To mitigate the path degeneracy problem common in SMC, the authors derive an optimal, variance-minimizing drift for the governing Feynman-Kac PDE and propose two practical approximations: Variance-Controlling Guidance (VCG) and Energy-Controlling Guidance (ECG). These approximations can be obtained by solving a linear system at each sampling step, adding only a small computational overhead. The methods are analyzed on Gaussian mixture models, where they are empirically shown to reduce variance and improve sample quality. They are also tested on practical benchmarks, molecular systems and protein-ligand co-folding, demonstrating improved inference-time steering capability over the standard SMC baseline.", "questions": "1. I have little expertise in functional analysis and found the formal proof of Proposition A.5 difficult to follow. Could you provide a more intuitive explanation for why solving the Poisson equation (Eq. 3.2) yields the optimal control?\n2. Line 269, \"... where reweighting can be unstable\": Aren't VCG and ECG without resampling still weighted with path-level weights? Those weights should still have high variance. I don't understand the rationale behind this (if you want to mitigate the path degeneracy from resampling, then you can consider using tempered weights for resampling, e.g., [1]).\n3. Why were different $\\gamma$ values used for each figure 1, 2, and 3?\n4. What is the main computational bottleneck of the algorithm in practice? (Hutchinson's estimator? or solving the linear equation?)\n5. If $r$ is given by a large neural network, can this method still be used (in terms of memory and time complexity)?\n6. Why didn't you consider the ALDP experiment (which is more multi-modal compared to LJ-13, to my knowledge)?\n7. (suggestion) Line 417, \"from $T=1.0$ to $0.4$\": it might be better to say \"with $\\gamma = 2.5$\" to avoid any confusion.\n\n---\n\nReferences  \n[1] Choi, Sanghyeok, et al. \"Reinforced sequential Monte Carlo for amortised sampling.\" arXiv preprint arXiv:2510.11711 (2025).  \n\n---\n\nLLM usage disclosure: I used LLM to check the grammar and make each sentence clearer.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. The paper is clearly written, and the core mathematical arguments are generally well-explained.\n2. The work tackles a significant but often-neglected challenge of the weight degeneracy (or path degeneracy) in SMC-based diffusion steering.\n3. The paper translates a mathematical insight into novel and practical (training-free) algorithms.\n4. Their experiments are well-designed to support their arguments. The proposed methods clearly outperform the SMC baseline on real-world benchmarks like molecular systems and protein-ligand co-folding.", "weaknesses": "1. The method's practical limitations could be discussed in more detail. It introduces a significant computational overhead, with experiments showing up to a 6x increased runtime over the SMC baseline. Furthermore, the reward-tilting framework fundamentally relies on access to the gradient of the reward, which is inaccessible in many black-box applications, limiting its practical scope.\n2. The paper would be strengthened by adding empirical analysis of the approximation error from VCG and ECG.\n3. No source code is provided.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6el88e5Ogc", "reviewer": "Reviewer_ncQu", "summary": "The authors propose a method for learning to sample from a combination of tempered and/or tilted distribution, where the exponent of a reward function describes the tilting factor. Importantly, they note that the Feynman Kac PDE can be manipulated by adding an additional drift term which is compensated in the re-weighting factor, reducing the problem of sampling from the new target density to finding an optimal drift. It is an important direction of research since re-weighting typically relies on self-normalized importance sampling which can lead to importance weights blowing up. Thus, finding a good additional drift function can reduce this problem by essentially reducing the variance of the importance weights. The authors further demonstrate the improved performance of their framework, which relies on efficiently finding the optimal control through solving of a linear system of equations.", "questions": "- How do the authors evaluate their method on LJ-13 and the other benchmarks? How are the ground-truth samples obtained to compute metrics such as MMD?\n- It would be good to provide some of the standard metrics like 2-Wasserstein distance.\n- What is the reward-tilting used in LJ-13, and what is the motivation behind the specific choice of tilting applied?\n- Why do the authors not compare against reward-based diffusion finetuning tasks and methods, e.g. [4-6]?\n\n[4] Venkatraman, Siddarth, et al. \"Amortizing intractable inference in diffusion models for vision, language, and control.\" Advances in neural information processing systems 37 (2024): 76080-76114.\n\n[5] Domingo-Enrich, Carles, et al. \"Adjoint matching: Fine-tuning flow and diffusion generative models with memoryless stochastic optimal control.\" arXiv preprint arXiv:2409.08861 (2024).\n\n[6] Fan, Ying, et al. \"Dpok: Reinforcement learning for fine-tuning text-to-image diffusion models.\" Advances in Neural Information Processing Systems 36 (2023): 79858-79885.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The authors provide a rigorous and correct assessment of the setting, where existing methods rely on self-normalized importance sampling which suffer from the problem of large variance of importance weights and the sensitivity to the number of particles used for integration.\n- The experiments indicate that the proposed method leads to improved performance across a wide array of settings, substantiating the generalizability of the proposed method.", "weaknesses": "- A lot of the theory proposed in this work is actually already well known and not new, and the authors should have introduced it as such. In particular, Prop 3.1 is the basis of [1-3] which highlights that the Feynman Kac PDE can be manipulated with additional drifts as long as they are similarly compensated for in re-weighting.\n- Is there a reason why the authors did not consider the more difficult setting of LJ-55 to evaluate their framework? \n- I strongly urge the authors to provide comparisons to [2-3] which follows the same idea, learning a neural network model as an additional drift to reduce the variance of importance weights. A clear comparative analysis between modeling the drift as a linear coefficient of basis functions or a neural network would make this manuscript a lot better.\n- In continuation of the previous point, it is further important to highlight the training and inference costs of the proposed method against [2,3]. While I agree that the proposed method only requires solving a system of linear equations, it does require this at all time-points every time during inference. On the other hand, learning a time-conditioned neural network requires an upfront cost of training but then at inference only requires an additional forward pass at each step. Is solving this system of linear equations at every step cheaper than a forward pass at each step? Such an analysis has been missing from this work.\n\n[1] Skreta, Marta, et al. \"Feynman-kac correctors in diffusion: Annealing, guidance, and product of experts.\" arXiv preprint arXiv:2503.02819 (2025).\n\n[2] Albergo, Michael S., and Eric Vanden-Eijnden. \"Nets: A non-equilibrium transport sampler.\" arXiv preprint arXiv:2410.02711 (2024).\n\n[3] Vargas, Francisco, et al. \"Transport meets variational inference: Controlled monte carlo diffusions.\" arXiv preprint arXiv:2307.01050 (2023).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "EKzpDbUbjR", "reviewer": "Reviewer_hiDh", "summary": "The paper proposes DriftLite, a lightweight alternative to reward-driven particle dynamics derived from the Feynman–Kac formulation of the Fokker–Planck equation.\nWhile the principled dynamics (Prop. 2.1) yield unbiased sampling by reweighting trajectories according to a reward function, they suffer from severe weight degeneracy and are impractical for high-dimensional problems.\nDriftLite replaces explicit weighting with a Variational Coefficient Generator (VCG) that learns a low-rank, three-basis decomposition of the drift field, achieving a balance between theoretical faithfulness and numerical stability.\nThe method demonstrates stable performance on challenging high-dimensional tasks, including protein–ligand systems.", "questions": "Could the authors clarify whether the choice of three basis functions in the VCG has a theoretical grounding?\nIn particular, does this low-rank representation guarantee that the dominant drift modes of the reward-driven dynamics are captured, or is it mainly an empirical observation?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "## Strong theoretical grounding.\nProposition 2.1 provides a clean derivation from the Feynman–Kac representation, clarifying why naive guidance or reward-based drift correction leads to unnormalized density propagation. In addition, Proposition 3.1 theoretically provides design space within the Fokker-Plank equation.\n\n## Elegant practical solution.\nThe transition from weighted to unweighted dynamics via the VCG formulation is conceptually neat and computationally efficient.\nRepresenting drift using three physically motivated basis components—potential, diffusive, and reward—yields a low-rank parameterization.\nDirectly minimizing the variance of the potential term under a simple parameterization reduces the problem to a linear system, making the overall procedure remarkably simple.\n\n## Empirical validation on high-D systems.\nAlthough the approach projects the control drift $b$ onto a three-dimensional subspace, it scales to complex molecular and protein–ligand environments without losing stability, indicating that this design effectively captures the dominant drift modes.", "weaknesses": "## Partial theoretical exposition.\nWhile Prop. 2.1 and 3.1 are elegant, the paper omits intermediate derivations linking the reward term to the weighted dynamics; some readers may struggle to follow the jump from theory to implementation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RZ7tetTX6A", "reviewer": "Reviewer_cAK7", "summary": "The paper extends previous work on adapting diffusion models at inference-time to new tasks using Sequential Monte Carlo (SMC) methods, focusing on reward-tilting and sampling from an annealed version of the base model distribution. In particular, they target the problem of sampling from the sequence of distributions $q_t(x) = p_t(x)^\\gamma \\exp(r_t(x))$, where $p_t(x)$ is the original diffusion model distribution and $r_t(x)$ is some time-varying reward function, such that at $t=0$ we approach some predefined clean data distribution $p_0(x)^\\gamma \\exp(r(x))$. Following previous work, $N$ particles are sampled concurrently for each $t$, and the diffusion model is used to move these particles towards lower $t$ in the $q_t(x)$ chain. The resulting proposal particles are reweighted using Feynman-Kac-type equations and periodically resampled, enabling the particle ensemble to approximate the sequence of target distributions. \n\nThe new idea in the paper to analyze the Feynman-Kac-type equations and notice that there is additional freedom in choosing the proposal distributions, such that they can add an additional term to the drift in the Feynman-Kac equations if they correspondingly change the time evolution in the weights. The authors then show that there exists an optimal added drift such that it is the gradient of a scalar potential that follows a particular PDE. With this added design space, they propose two choices for the added drift: 1) Minimizing the variance in the change of the weights 2) minimizing a Lagrangian of the aforementioned PDE such that at the minimum, the optimal drift is obtained. In practice, they parameterize the added drift as a linear combination of $\\nabla r_t(x), \\nabla \\log p_t(x)$, and the original diffusion drift $u_t(x)$ for the variance-minimizing version, and a similar choice for the PDE-optimizing version. This results in a simple $3x3$ systems of equations per particle per step, resulting in a relatively lightweight correction, although evaluating the required matrices requires backpropagation through the diffusion denoiser. \n\nThe authors evaluate the method on a 30-dimensional Gaussian Mixture Model, two toy particle systems, and on a protein-ligand co-folding task. They notice that the method consistently improves effective sample size and reduces weight variance, outperforming the naive SMC baseline. On the protein-ligand task, they demonstrate that using physical energies as the rewards, the method can improve the physical validity of the generated samples.", "questions": "- Although not considered in previous literature on SMC, it seems that another simple baseline would be to not use SMC and instead interleave Langevin dynamics steps after each generative step, similarly to [1], such that the wall-clock time is matched with the proposed method. I would be curious to see how well this could work on some task, although this is not a priority. \n\nOverall, I will start out with a marginal reject due to the concerns raised in the weaknesses, but am open to changing the score with the rebuttal. I think the core idea in the paper is interesting and mostly seems to work well and potentially could be extended further. As such I am not initially strongly against accepting the paper with the assumption that the weaknesses are cleared out or I am incorrect about them. \n\nReferences \n\n[1] Du, Yilun, et al. \"Reduce, reuse, recycle: Compositional generation with energy-based diffusion models and mcmc.\"", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- The core idea of defining a correction to the proposal distribution that directly optimizes the variance or consistency with the target distributions, is principled and seems to be novel at least in the diffusion context. It seems to be quite simple to implement, does not cause a huge overhead, and seems to legitimately improve the performance of the SMC method considerably on the tasks chosen. \n- The experiments on the GMM, DW-4 and LJ13 are quite thorough and well-presented, and the protein-ligand task is an example of a realistic problem that the method could help with and the method clearly works better than more naive SMC approaches. \n- Overall, the paper is well written and not too difficult to follow, with the caveat that the mathematical exposition may be difficult to follow to readers that are not familiar with this way of describing SMC methods in advance.", "weaknesses": "Disclaimer: I am not an expert on SMC or SMC-based methods for diffusion models, and as such it is possible that I am missing some details, wider context or some previous literature. As such, I am open to being corrected. \n\n- The method incurs significant cost on the wall-clock time compared to the baselines, 2.3x-6x in the experiments for which the wall-clock time was reported. This is not mentioned in the limitations or in the main paper, however, and the conclusion instead states that the method causes minimal computational overhead, potentially misleading the reader or a reviewer who does not have the time to read through the Appendix. Backpropagation through the denoiser at each step requires much more memory than the forward pass and, and my assumption is that it may limit the batch size in practice. \n- As such, it might be more fair to compare to the baselines by adjusting the inference hyperparameters such that the wall-clock time is equal. E.g., I would expect that the G-SMC method improves with more particles, and potentially with more diffusion steps. \n- Following that, I would be interested in seeing the ESS and wall-clock time comparisons for the protein-ligand task as well, especially since this seems to be an example of a higher-dimensional case. As an alternative, it would also be interesting to see how the method scales, e.g., to the LJ55 task or some other task that is more than about 30-dimensional. \n- Do the authors agree that the method could be described as a \"twisted\" proposal for SMC? And further, even the initial choice of using the modified diffusion reverse process would be also simply another choice of a twisted proposal. My understanding is that this is a standard idea in the Sequential Monte Carlo literature, although the basis function scheme and the variance-minimization is novel at least in the diffusion context to my knowledge. The problem is that this seems to set the first contribution of 'identifying a fundamental degree of freedom in the Feynman-Kac type FP equation' to be less novel than it sounds, but I am open to being corrected on this. In any case, I think that positioning the method in its proper wider SMC context would be helpful for correctly contextualizing the paper. \n- In the protein-ligand, task, would another simple baseline be to take the base model samples, and optimize them directly using the gradient of the energy function? Does the method outperform this?\n- The paper [1] seems to be a relevant prior work on annealed distributions with diffusion models with SMC (and is the first paper to do this?). It may be that the proposed method is better than this, but I think it should be compared to at least in some context to clearly show this. \n- Further, the paper [2] does reward-tilted sampling with SMC by using a different twisted proposal by using the $p(y|x_0(x_t))$ gradient. It would be good to compare to that as well in the reward-guided experiments. \n\nReferences \n\n[1] Thornton, James, et al. \"Composition and Control with Distilled Energy Diffusion Models and Sequential Monte Carlo.\" The 28th International Conference on Artificial Intelligence and Statistics (2025).\n\n[2] Wu, Luhuan, et al., \"Practical and Asymptotically Exact Conditional Sampling in Diffusion Models\", 37th Conference on Neural Information Processing Systems (NeurIPS 2023).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)