Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

MPC-Flow recasts inference-time guidance of flow-based generative models as a sequence of short-horizon optimal-control sub-problems, making it scalable to massive architectures without backpropagation through the full trajectory.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 06:08 UTC pith:YXVGFMA6

load-bearing objection Useful MPC framing with a real 32B scaling demo, but the Theorem 3.2 guarantee has a factor-of-λ slip and the fast K=1 variant fails on null-space problems—worth refereeing after a fix. the 3 major comments →

arxiv 2601.23231 v2 pith:YXVGFMA6 submitted 2026-01-30 eess.IV cs.LG

Solving Inverse Problems with Flow-based Models via Model Predictive Control

George Webber , Alexander Denker , Riccardo Barbano , Andrew J Reader This is my paper
classification eess.IV cs.LG
keywords flow-based generative modelsflow matchingmodel predictive controlinverse problemstraining-free guidanceimage restorationoptimal controlreceding horizon
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Flow-based generative models provide strong priors for inverse problems, but steering their sampling dynamics to satisfy measurements is expensive when posed as one trajectory-wide optimal-control problem. MPC-Flow argues that the full-horizon optimization can be replaced by a receding-horizon sequence of local control problems: at each time the controller solves a short-horizon problem from the current state, applies only the first control, and re-plans. The paper proves that in the idealized case where each local sub-problem is solved exactly, the applied controls coincide with the global optimal control, and it shows how different horizon choices produce a spectrum of guidance algorithms, including a single-step variant that avoids backpropagation through the flow model. This makes training-free guidance practical at scale: the authors demonstrate it on a 32B-parameter flow model (FLUX.2) in 4-bit quantisation running on a consumer GPU. Across denoising, deblurring, super-resolution, inpainting, and non-linear deblurring, the framework matches or beats existing optimal-control-based solvers while using far less memory and runtime.

Core claim

The paper establishes that conditional generation with pre-trained flow models, normally formulated as a single optimal-control problem over the whole generation interval, can be decomposed into a sequence of receding-horizon sub-problems without sacrificing optimality in the ideal case. Theorem 3.1 states that if every sub-problem, minimizing control energy plus terminal fidelity over the remaining time, is solved to optimality, the applied controls coincide with the global optimal control. Theorem 3.2 states that a Δt-horizon policy is globally optimal when its terminal cost is the true value function. In practice the paper approximates the value function with a single Euler step, V(t,x)≈Φ

What carries the argument

The central object is model predictive control: a receding-horizon strategy that, at every time t, solves a finite-horizon optimal-control problem from the current state, applies only the first control increment, and then re-plans. The paper instantiates it with two horizon choices: receding-horizon control over the full remaining interval (H=1−t) and Δt-horizon control over a single step (H=Δt). The workhorse of the fast variants is the one-step Euler approximation of the value function, which predicts the rest of the trajectory as a straight line and therefore avoids differentiating through the flow network. The control-energy term plays the role of path regularisation, keeping the guided

Load-bearing premise

The fast variants assume that what happens for the rest of the trajectory is well predicted by a single straight step from the current point; when this is false — strongly curved flow, or measurements that leave some image details completely unobserved — the control cannot steer into those unobserved directions, and the method's practical advantage collapses.

What would settle it

Run the K=1 single-step variant on an inpainting problem with a large masked region: the gradient of its objective with respect to the control has no component in the masked pixels, so those pixels should remain essentially untouched. A run that nonetheless fills in coherent texture in the masked region would contradict the paper's mechanism; alternatively, on a deliberately curved flow, the single-step value approximation should visibly mis-estimate the true cost-to-go, which would show the assumption is load-bearing.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Because controls are computed from local sub-problems rather than the full trajectory, memory no longer scales with the number of integration steps, which is what makes a 32B-parameter model guidable on a 24GB consumer GPU.
  • The single-step (K=1) variant removes backpropagation through the flow model entirely, so it can be applied to quantised or otherwise non-differentiable large models.
  • The framework applies to non-linear forward operators such as non-linear blur, where the paper reports the best SSIM among the compared methods.
  • Re-planning from the current state corrects errors accumulated at earlier time steps; the paper finds this yields better reconstructions than a single global solve with comparable compute.
  • Choosing K controls a spectrum from global optimal control to greedy one-step control, giving practitioners a direct trade-off between reconstruction quality and runtime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • An extension the paper leaves implicit: because the terminal cost is arbitrary and differentiable, MPC-Flow is not limited to inverse problems — any reward function (style, alignment, safety) can be plugged in, and the control-regularisation term gives a principled knob for how far the guided sample may depart from the prior.
  • The single-Euler-step value approximation is closely related to the Tweedie estimator used in diffusion posterior sampling; improving it, for example by learning the value function, could give the fast variant both speed and the ability to fill null-space directions.
  • For linear operators with large null spaces, the observed failure of K=1 suggests a hybrid: use the cheap one-step control for observed directions and a separate null-space-filling step, such as a short higher-K re-plan or a hard data-consistency projection, for the unobserved directions.
  • Because the remaining horizon shrinks as generation proceeds, the effective step size is naturally finer near the end; varying K adaptively over time could yield further quality gains beyond the fixed-K results reported.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes MPC-Flow, a model predictive control framework for conditioning pre-trained flow-based generative models at inference time. Instead of solving the full trajectory optimal control problem (5), the method solves a sequence of shorter-horizon control sub-problems and applies only the first control increment before re-planning. Two regimes are analyzed: receding-horizon control (RHC) over the full remaining horizon, and a greedy Δt-horizon control with a predicted terminal cost. The authors prove that each regime recovers the globally optimal control under idealizing assumptions (Theorem 3.1 and Theorem 3.2), and they introduce a practical single-step variant (RHC K=1) that avoids backpropagation through the generative model, enabling scaling to FLUX.2 (32B) in 4-bit quantization. Experiments cover controlled toy and CT settings, CelebA image restoration (denoising, deblurring, super-resolution, inpainting, nonlinear deblurring), and large-model style transfer/colorization tasks.

Significance. If the formal issues are corrected, MPC-Flow is a useful unifying framework for training-free guidance of flow models. The paper is commendable for its breadth: it includes a self-contained existence proof, toy validation of the RHC optimality claim, a wide set of inverse-problem benchmarks, and a genuinely impressive scalability demonstration on a 32B-parameter model. The authors also disclose the failure mode of the K=1 variant on linear inverse problems with null space, which is a strength relative to many papers in this area. The main weaknesses are that the key theorem for the Δt-horizon variant contains a scaling error, and the practical no-backpropagation variants are not covered by the formal guarantees as advertised.

major comments (3)
  1. [Section 3.2, Theorem 3.2, Eq. (8), Eq. (16), Appendix B.2.2] The statement of Theorem 3.2 is incorrect as written. The Δt-horizon objective in Eq. (8) is min_u [∫_t^{t+Δt} ||u||² dτ + λ Φ_MPC(x(t+Δt), t+Δt)]. The value function V(t,x) defined in Section 3.1 already contains the terminal penalty λ Φ(x(1)). Setting Φ_MPC = V in Eq. (8) gives a terminal term λ V(t+Δt, x(t+Δt)), whereas the Bellman recursion used in Eq. (16) is V(t,x) = min_u {∫_t^{t+Δt} ||u||² dτ + V(t+Δt, x(t+Δt))}. The two problems coincide only when λ = 1. For the λ values used in the experiments (e.g., λ ≈ 10^4 in Appendix D), the Δt-horizon control is not the optimal control for problem (5). The theorem should state Φ_MPC = V/λ, or the λ in Eq. (8) should be absorbed into the value function definition. This scaling error propagates into Eq. (9) and Algorithm 2, where Φ_MPC is approximated by Φ(x+(1−t)vθ(x,t)) without the λ factor.
  2. [Section 3.1, Section 3.2, Eq. (9), Appendix E.1, Table 2] The formal guarantees do not cover the scalable variants that are central to the paper's practical claims. Theorem 3.1 requires solving each sub-problem (7) exactly over [t,1]; Theorem 3.2 requires the true value function as the terminal cost. The regimes that avoid backpropagation through the model — RHC K=1 and MPC-Δt with Eq. (9) — both replace these exact objects by a single-Euler-step, uncontrolled approximation of the value function. For a quadratic terminal loss, the K=1 gradient (Eq. (18)) has no component in the null space of A, and Table 2 shows the method collapses on super-resolution and random inpainting (PSNR 18.02 and 18.31 vs. PnP-Flow's 31.49 and 33.54), a failure the authors acknowledge in Appendix E.1. The abstract and introduction advertise 'theoretical guarantees' connecting MPC-Flow to the underlying optimal control objective, but those guarantees do not apply to th
  3. [Proposition B.1 / Appendix B.1] The existence result is stated for problem (10), whose terminal cost is Φ(x(1)) with no λ weight, whereas the main problem (5) has terminal cost λΦ(x(1)). This is a minor mismatch — the proof still works with λΦ — but it should be stated consistently. Also, the final inequality in the proof writes ∫ ||u|| dt rather than ∫ ||u||² dt, which appears to be a typo.
minor comments (4)
  1. [Section 3.1, Algorithm 3] The single-step RHC variant (K=1) is described in Section 3.1 but the corresponding Algorithm 3 is relegated to the appendix without a pointer in the main text. Add a reference for readability.
  2. [Table 2] The 'Time/img' column is measured only for the denoising task, but this is not stated in the table caption. Since runtimes are a key claimed advantage, clarify the measurement protocol.
  3. [Figure 3] The observation that MPC-RHC with finite K outperforms the 'global control' solution is explained by local minima in the global nonconvex optimization. This is fine, but the wording invites confusion; it should be made explicit that this is not a violation of Theorem 3.1 but an artifact of the optimizer.
  4. [Section 2.2, Eq. (6)] The index in the discrete dynamics is written 'k = 1, ..., N−1' after defining t_k = k/N, but the state update uses x_{t_k} with k starting at 0. Adjust the index range for consistency.

Circularity Check

0 steps flagged

No significant circularity: Bellman-principle theorems are self-contained, and the fast value-function approximation is explicitly heuristic.

full rationale

The derivation chain is not circular. Theorem 3.1 is a direct application of Bellman's principle: the proof in Appendix B.2.1 splits the global cost and observes that the receding-horizon sub-problem (7) is exactly the future-cost subproblem, so the claimed optimality is a consequence of dynamic programming, not an assumption of the result. Theorem 3.2 likewise states the standard dynamic-programming identity for the Δt-horizon problem when the terminal cost is the cost-to-go V. One caveat is a correctness issue, not a circularity: eq. (8) writes the terminal term as λΦ_MPC while the Bellman recursion (16) uses V, which already contains λΦ(x(1)); as written the two problems are not identical unless Φ_MPC = V/λ or the λ is absorbed. This makes the proof of Theorem 3.2 invalid as stated, but it does not constitute fitting a prediction to its own input. The single-Euler-step value approximation (eq. 9) is explicitly labeled a 'general-purpose heuristic', and Appendix E.1 candidly documents its failure on null-space-dominated linear inverse problems; this honesty is a limitation rather than hidden reuse of the target result. Empirical comparisons use external baselines (FlowGrad, OC-Flow, PnP-Flow, D-Flow, Flow-Priors, OT-ODE, FlowChef) with hyperparameters reported in Appendix E, so no central claim is justified solely by a self-citation. The self-citations present (Barbano et al. 2025; Denker et al. 2024) are contextual and not load-bearing. The Section G variational-regularisation reading is a mathematical reformulation of (5), not an assumption of the conclusion. Overall, the central optimality and approximation claims are self-contained; any minor self-citations do not support the derivation.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

No new physical entities or forces are postulated. The control u is an auxiliary optimization variable, not an invented entity. The main unproven ingredients are standard optimal-control assumptions plus the single-Euler-step value approximation, which is a heuristic rather than a derived result.

free parameters (5)
  • λ (data-consistency weight) = Varies by task: e.g., 15 (MPC-Δt denoising), 0.1 (RHC K=1 denoising), 0.063 (RHC K=1 deblurring)
    Controls trade-off between measurement fidelity and prior/control regularization; tuned per task on a validation subset (Table E1).
  • ρ (path regularization for FLUX.2 style transfer) = 4.0 to 1024.0, with 8.0 used for the shown Figure 5 examples
    Reparameterized control regularization strength; tuned to balance style similarity and content preservation (Appendix F).
  • Learning rate for inner control optimization = 0.1 (MPC-Δt), 0.05 or 0.1 (RHC K=3), 0.5 or 0.2 (FLUX.2)
    Affects convergence of the inner subproblem solve; chosen per task.
  • Nctrl (number of control optimization steps) = 20 for most CelebA experiments, 20 for FLUX.2
    Number of inner gradient steps used to solve the MPC subproblem; a practical design choice.
  • K (RHC discretisation steps) = 1 or 3 in main experiments; up to 10 in toy example
    Number of steps in the receding-horizon subproblem; higher K improves fidelity but increases cost, lower K enables the backprop-free fast variant.
axioms (5)
  • domain assumption The learned flow field v_θ is continuous, uniformly Lipschitz in x, and has at most linear growth (Appendix B.1).
    Needed for existence of the optimal control (Proposition B.1). In practice, neural network vector fields may not satisfy these globally.
  • domain assumption The terminal cost Φ is lower semicontinuous (and differentiable where gradients are used).
    Required for existence of an optimal control and for gradient-based inner solves.
  • standard math Bellman's principle of optimality applies to the deterministic optimal control problem (5).
    Used in the proofs of Theorems 3.1 and 3.2.
  • ad hoc to paper The value function V(t,x) can be approximated by a single Euler step: V(t,x) ≈ Φ(x + (1-t) v_θ(x,t)) (eq. 9).
    This heuristic is central to MPC-Δt and the K=1 variant; the paper notes it is exact only for affine linear probability paths (Tweedie). Its failure for curved flows is a risk.
  • domain assumption The pretrained flow model provides a good prior for the data distribution and is well-matched to the inverse problem.
    Standard assumption for training-free generative priors; if the prior is poor, guidance cannot recover good reconstructions.

pith-pipeline@v1.3.0-alltime-deepseek · 3837 in / 3784 out tokens · 157613 ms · 2026-08-03T06:08:06.307795+00:00 · methodology

0 comments
read the original abstract

Flow-based generative models provide strong unconditional priors for inverse problems, but guiding their dynamics for conditional generation remains challenging. Recent work casts training-free conditional generation in flow models as an optimal control problem; however, solving the resulting trajectory optimisation is computationally and memory intensive, requiring differentiation through the flow dynamics or adjoint solves. We propose MPC-Flow, a model predictive control framework that formulates inverse problem solving with flow-based generative models as a sequence of control sub-problems, enabling practical optimal control-based guidance at inference time. We provide theoretical analysis linking MPC-Flow to the underlying optimal control objective and show how different algorithmic choices yield a spectrum of guidance algorithms, including regimes that avoid backpropagation through the generative model trajectory. We evaluate MPC-Flow on benchmark image restoration tasks, spanning linear and non-linear settings such as in-painting, deblurring, and super-resolution, and demonstrate strong performance and scalability to massive state-of-the-art architectures via training-free guidance of FLUX.2 (32B) in a quantised setting on consumer hardware.

Figures

Figures reproduced from arXiv: 2601.23231 by Alexander Denker, Andrew J Reader, George Webber, Riccardo Barbano.

Figure 1
Figure 1. Figure 1: MPC-Flow strategy for guiding flow-based generative models toward a target objective. Starting from the current state, MPC-Flow plans a sequence of velocity adjustments that steer the flow toward the objective while keeping the intervention small (A). Only the initial part of the plan is applied (B), after which we re-plan from the new state and repeat the process (C). and OC-Flow, employ an explicit Euler… view at source ↗
Figure 2
Figure 2. Figure 2: RHC optimises over the full horizon [t, 1], discretised into K steps; ∆t-Horizon uses a 1-step lookahead horizon [t, t + ∆t]. For MPC–RHC, the original terminal loss Φ is used, whereas MPC–∆t employs the projected loss ΦMPC. problem up to the terminal time t = 1, the intermediate loss function is simply chosen to be ΦMPC(x, 1) := Φ(x). To analyse the theoretical properties of this scheme, we make use of th… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of MPC-RHC with varying K compared to the global optimal control solution. All approaches use the same initial value x0 and λ = 2500. degradation operators, on CelebA (Yang et al., 2015). Then, we show that we can scale MPC-RHC with K = 1 to the recently released FLUX.2 (Labs, 2025) model and consider style transfer and image colouration. 4.1. Computed Tomography We train a flow-based model on t… view at source ↗
Figure 4
Figure 4. Figure 4: Qualitative results on the CelebA dataset for the image super-resolution task with noise level σ = 0.05 and ×2 upscaling. Method Denoising Deblurring Super-res. Rand. inpaint. Box inpaint. Nonl. Debl. Time/img σ = 0.2 σ = 0.05, σb = 1.0 σ = 0.05, ×2 σ = 0.01, 70% σ = 0.05, 40 × 40 σ = 0.05 [s] PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM Degraded 20.00 0.348 27.67 0.740 10.17 -0.182 11.82 -0… view at source ↗
Figure 5
Figure 5. Figure 5: Example images generated by FLUX.2 with training-free style transfer guidance by FlowChef (middle row) and MPC (bottom row). The top row shows the image generated without conditioning as well as the prompt used and the reference style image. Original image Degraded (luminance image) PSNR 20.22 dB FLUX.2 conditional PSNR 27.11 dB MPC-Flow PSNR 22.20 dB FlowChef [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Example re-colouration images generated with FLUX.2. Prompt used: ”colorize this luminance image”. vary the balance between terminal loss and control regu￾larisation λ; for FlowChef, we vary the learning rate on the terminal loss (see Appendix F). As the conditioning strength increases, FlowChef’s outputs move closer to the style image, but do so by substantially deviating from the original generation traj… view at source ↗
Figure 7
Figure 7. Figure 7: Column 1: (Top) An image generated by Flux.2. (Bottom) A reference style image. Remaining columns: Images generated using training-free style transfer with Flux.2 with FlowChef (top) and our method MPC-RHC (K=1) (bottom). From left to right, the methods use increased conditioning on the style reward; the path regularisation introduced by our method enables greater adherence to the original generation path … view at source ↗
Figure 8
Figure 8. Figure 8: Style-content trade-off for MPC-Flow and FlowChef. Metrics recorded over 45 images (5 prompts × 9 style images). 5. Conclusion & Further Work We introduce MPC-Flow, a model predictive control frame￾work for conditional generation with flow-based generative models. By decomposing the global trajectory optimisation into a sequence of control subproblems, MPC enables prac￾tical and memory-efficient guidance a… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Stability Benchmark of Generative Regularizers for Inverse Problems

    eess.IV 2026-05 unverdicted novelty 5.0

    Numerical benchmarks indicate generative regularizers deliver strong reconstructions in some imaging inverse problem settings but can be unstable or problematic under imperfect conditions compared to variational methods.

Reference graph

Works this paper leans on

58 extracted references · 27 linked inside Pith · cited by 1 Pith paper

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

  2. [2]

    Albergo, M. S. and Vanden-Eijnden, E. Building normalizing flows with stochastic interpolants. arXiv preprint arXiv:2209.15571, 2022

    Pith/arXiv arXiv 2022

  3. [3]

    H., Arridge, S., Maass, P., Jin, B., and Ye, J

    Barbano, R., Denker, A., Chung, H., Roh, T. H., Arridge, S., Maass, P., Jin, B., and Ye, J. C. Steerable conditional diffusion for out-of-distribution adaptation in medical image reconstruction. IEEE Transactions on Medical Imaging, 2025

    2025

  4. [4]

    Conditional image generation with score-based diffusion models

    Batzolis, G., Stanczuk, J., Sch \"o nlieb, C.-B., and Etmann, C. Conditional image generation with score-based diffusion models. arXiv preprint arXiv:2111.13606, 2021

    Pith/arXiv arXiv 2021

  5. [5]

    D-flow: Differentiating through flows for controlled generation

    Ben-Hamu, H., Puny, O., Gat, I., Karrer, B., Singer, U., and Lipman, Y. D-flow: Differentiating through flows for controlled generation. arXiv preprint arXiv:2402.14017, 2024

    Pith/arXiv arXiv 2024

  6. [6]

    An optimal control perspective on diffusion-based generative modeling

    Berner, J., Richter, L., and Ullrich, K. An optimal control perspective on diffusion-based generative modeling. arXiv preprint arXiv:2211.01364, 2022

    Pith/arXiv arXiv 2022

  7. [7]

    T., Rubanova, Y., Bettencourt, J., and Duvenaud, D

    Chen, R. T., Rubanova, Y., Bettencourt, J., and Duvenaud, D. K. Neural ordinary differential equations. Advances in neural information processing systems, 31, 2018

    2018

  8. [8]

    Training deep nets with sublinear memory cost

    Chen, T., Xu, B., Zhang, C., and Guestrin, C. Training deep nets with sublinear memory cost. arXiv preprint arXiv:1604.06174, 2016

    Pith/arXiv arXiv 2016

  9. [9]

    T., Klasky, M

    Chung, H., Kim, J., Mccann, M. T., Klasky, M. L., and Ye, J. C. Diffusion posterior sampling for general noisy inverse problems. arXiv preprint arXiv:2209.14687, 2022

    Pith/arXiv arXiv 2022

  10. [10]

    J., and Lio, P

    Denker, A., Vargas, F., Padhy, S., Didi, K., Mathis, S., Barbano, R., Dutordoir, V., Mathieu, E., Komorowska, U. J., and Lio, P. Deft: Efficient fine-tuning of diffusion models by learning the generalised h -transform. Advances in Neural Information Processing Systems, 37: 0 19636--19682, 2024

    2024

  11. [11]

    and Nichol, A

    Dhariwal, P. and Nichol, A. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34: 0 8780--8794, 2021

    2021

  12. [12]

    Domingo-Enrich, C., Drozdzal, M., Karrer, B., and Chen, R. T. Adjoint matching: Fine-tuning flow and diffusion generative models with memoryless stochastic optimal control. arXiv preprint arXiv:2409.08861, 2024

    Pith/arXiv arXiv 2024

  13. [13]

    J., et al

    Doucet, A., De Freitas, N., Gordon, N. J., et al. Sequential Monte Carlo methods in practice, volume 1. Springer, 2001

    2001

  14. [14]

    W., Hanke, M., and Neubauer, A

    Engl, H. W., Hanke, M., and Neubauer, A. Regularization of Inverse Problems. Mathematics and Its Applications. Springer, Dordrecht, 1996

    1996

  15. [15]

    E., and Schindler, K

    Erbach, J., Narnhofer, D., Dombos, A., Schiele, B., Lenssen, J. E., and Schindler, K. Solving inverse problems with flair. arXiv preprint arXiv:2506.02680, 2025

    arXiv 2025

  16. [16]

    Scaling rectified flow transformers for high-resolution image synthesis

    Esser, P., Kulal, S., Blattmann, A., Entezari, R., M \"u ller, J., Saini, H., Levi, Y., Lorenz, D., Sauer, A., Boesel, F., et al. Scaling rectified flow transformers for high-resolution image synthesis. In Forty-first international conference on machine learning, 2024

    2024

  17. [17]

    Fleming, W. H. and Rishel, R. W. Deterministic and stochastic optimal control, volume 1. Springer Science & Business Media, 2012

    2012

  18. [18]

    E., Prett, D

    Garcia, C. E., Prett, D. M., and Morari, M. Model predictive control: Theory and practice—a survey. Automatica, 25 0 (3): 0 335--348, 1989

    1989

  19. [19]

    Reinforcement learning with deep energy-based policies

    Haarnoja, T., Tang, H., Abbeel, P., and Levine, S. Reinforcement learning with deep energy-based policies. In International conference on machine learning, pp.\ 1352--1361. PMLR, 2017

    2017

  20. [20]

    Denoising diffusion probabilistic models

    Ho, J., Jain, A., and Abbeel, P. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33: 0 6840--6851, 2020

    2020

  21. [21]

    J., Shen, Y., Wallis, P., Allen-Zhu, Z., Li, Y., Wang, S., Wang, L., Chen, W., et al

    Hu, E. J., Shen, Y., Wallis, P., Allen-Zhu, Z., Li, Y., Wang, S., Wang, L., Chen, W., et al. Lora: Low-rank adaptation of large language models. ICLR, 1 0 (2): 0 3, 2022

    2022

  22. [22]

    S., and Ye, J

    Kim, J., Kim, B. S., and Ye, J. C. Flowdps: Flow-driven posterior sampling for inverse problems. arXiv preprint arXiv:2503.08136, 2025

    Pith/arXiv arXiv 2025

  23. [23]

    Kingma, D. P. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014

    Pith/arXiv arXiv 2014

  24. [24]

    Labs, B. F. FLUX.2: Frontier Visual Intelligence . https://bfl.ai/blog/flux-2, 2025

    2025

  25. [25]

    Derivative-free guidance in continuous and discrete diffusion models with soft value-based decoding

    Li, X., Zhao, Y., Wang, C., Scalia, G., Eraslan, G., Nair, S., Biancalani, T., Ji, S., Regev, A., Levine, S., et al. Derivative-free guidance in continuous and discrete diffusion models with soft value-based decoding. arXiv preprint arXiv:2408.08252, 2024

    Pith/arXiv arXiv 2024

  26. [26]

    Calculus of variations and optimal control theory: a concise introduction

    Liberzon, D. Calculus of variations and optimal control theory: a concise introduction. Princeton university press, 2011

    2011

  27. [27]

    T., Ben-Hamu, H., Nickel, M., and Le, M

    Lipman, Y., Chen, R. T., Ben-Hamu, H., Nickel, M., and Le, M. Flow matching for generative modeling. arXiv preprint arXiv:2210.02747, 2022

    Pith/arXiv arXiv 2022

  28. [28]

    T., Lopez-Paz, D., Ben-Hamu, H., and Gat, I

    Lipman, Y., Havasi, M., Holderrieth, P., Shaul, N., Le, M., Karrer, B., Chen, R. T., Lopez-Paz, D., Ben-Hamu, H., and Gat, I. Flow matching guide and code. arXiv preprint arXiv:2412.06264, 2024

    Pith/arXiv arXiv 2024

  29. [29]

    Rectified flow: A marginal preserving approach to optimal transport

    Liu, Q. Rectified flow: A marginal preserving approach to optimal transport. arXiv preprint arXiv:2209.14577, 2022

    Pith/arXiv arXiv 2022

  30. [30]

    Flow straight and fast: Learning to generate and transfer data with rectified flow

    Liu, X., Gong, C., and Liu, Q. Flow straight and fast: Learning to generate and transfer data with rectified flow. arXiv preprint arXiv:2209.03003, 2022

    Pith/arXiv arXiv 2022

  31. [31]

    Flowgrad: Controlling the output of generative odes with gradients

    Liu, X., Wu, L., Zhang, S., Gong, C., Ping, W., and Liu, Q. Flowgrad: Controlling the output of generative odes with gradients. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.\ 24335--24344, 2023

    2023

  32. [32]

    Z., Domingo-Enrich, C., Liu, W., and Zhang, D

    Liu, Z., Xiao, T. Z., Domingo-Enrich, C., Liu, W., and Zhang, D. Value gradient guidance for flow matching alignment. arXiv preprint arXiv:2512.05116, 2025

    arXiv 2025

  33. [33]

    A variational perspective on solving inverse problems with diffusion models

    Mardani, M., Song, J., Kautz, J., and Vahdat, A. A variational perspective on solving inverse problems with diffusion models. arXiv preprint arXiv:2305.04391, 2023

    Pith/arXiv arXiv 2023

  34. [34]

    Pnp-flow: Plug-and-play image restoration with flow matching

    Martin, S., Gagneux, A., Hagemann, P., and Steidl, G. Pnp-flow: Plug-and-play image restoration with flow matching. arXiv preprint arXiv:2410.02423, 2024

    Pith/arXiv arXiv 2024

  35. [35]

    M., Draxler, F., Karaletsos, T., and Mandt, S

    Pandey, K., Sofian, F. M., Draxler, F., Karaletsos, T., and Mandt, S. Variational control for guidance in diffusion models. arXiv preprint arXiv:2502.03686, 2025

    Pith/arXiv arXiv 2025

  36. [36]

    N., and Yang, Y

    Patel, M., Wen, S., Metaxas, D. N., and Yang, Y. Steering rectified flow models in the vector field for controlled image generation. arXiv preprint arXiv:2412.00100, 2024

    Pith/arXiv arXiv 2024

  37. [37]

    J., Chen, R

    Pokle, A., Muckley, M. J., Chen, R. T., and Karrer, B. Training-free linear image inverses via flows. arXiv preprint arXiv:2310.04432, 2023

    Pith/arXiv arXiv 2023

  38. [38]

    Qin, S. J. and Badgwell, T. A. A survey of industrial model predictive control technology. Control engineering practice, 11 0 (7): 0 733--764, 2003

    2003

  39. [39]

    W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., et al

    Radford, A., Kim, J. W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., et al. Learning transferable visual models from natural language supervision. In International conference on machine learning, pp.\ 8748--8763. PmLR, 2021

    2021

  40. [40]

    B., Mayne, D

    Rawlings, J. B., Mayne, D. Q., Diehl, M., et al. Model predictive control: theory, computation, and design, volume 2. Nob Hill Publishing Madison, WI, 2020

    2020

  41. [41]

    Rb-modulation: Training-free personalization of diffusion models using stochastic optimal control

    Rout, L., Chen, Y., Ruiz, N., Kumar, A., Caramanis, C., Shakkottai, S., and Chu, W.-S. Rb-modulation: Training-free personalization of diffusion models using stochastic optimal control. arXiv preprint arXiv:2405.17401, 2024

    Pith/arXiv arXiv 2024

  42. [42]

    M., Zhang, Z., Hu, X., Qu, Q., and Shen, L

    Song, B., Kwon, S. M., Zhang, Z., Hu, X., Qu, Q., and Shen, L. Solving inverse problems with latent diffusion models via hard data consistency. arXiv preprint arXiv:2307.08123, 2023

    Pith/arXiv arXiv 2023

  43. [43]

    P., Kumar, A., Ermon, S., and Poole, B

    Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., and Poole, B. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456, 2020

    Pith/arXiv arXiv 2011

  44. [44]

    Stuart, A. M. Inverse problems: a bayesian perspective. Acta numerica, 19: 0 451--559, 2010

    2010

  45. [45]

    Improving and generalizing flow-based generative models with minibatch optimal transport

    Tong, A., Fatras, K., Malkin, N., Huguet, G., Zhang, Y., Rector-Brooks, J., Wolf, G., and Bengio, Y. Improving and generalizing flow-based generative models with minibatch optimal transport. arXiv preprint arXiv:2302.00482, 2023

    Pith/arXiv arXiv 2023

  46. [46]

    T., Phung, Q., and Hoai, M

    Tran, P., Tran, A. T., Phung, Q., and Hoai, M. Explore image deblurring via encoded blur kernel space. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp.\ 11956--11965, 2021

    2021

  47. [47]

    Inference-time alignment in diffusion models with reward-guided generation: Tutorial and review

    Uehara, M., Zhao, Y., Wang, C., Li, X., Regev, A., Levine, S., and Biancalani, T. Inference-time alignment in diffusion models with reward-guided generation: Tutorial and review. arXiv preprint arXiv:2501.09685, 2025

    Pith/arXiv arXiv 2025

  48. [48]

    Denoising diffusion samplers

    Vargas, F., Grathwohl, W., and Doucet, A. Denoising diffusion samplers. arXiv preprint arXiv:2302.13834, 2023

    Pith/arXiv arXiv 2023

  49. [49]

    Diffusers: State-of-the-art diffusion models

    von Platen, P., Patil, S., Lozhkov, A., Cuenca, P., Lambert, N., Rasul, K., Davaadorj, M., Nair, D., Paul, S., Berman, W., Xu, Y., Liu, S., and Wolf, T. Diffusers: State-of-the-art diffusion models. https://github.com/huggingface/diffusers, 2022

    2022

  50. [50]

    Dmplug: A plug-in method for solving inverse problems with diffusion models

    Wang, H., Zhang, X., Li, T., Wan, Y., Chen, T., and Sun, J. Dmplug: A plug-in method for solving inverse problems with diffusion models. Advances in Neural Information Processing Systems, 37: 0 117881--117916, 2024 a

    2024

  51. [51]

    Training free guided flow matching with optimal control

    Wang, L., Cheng, C., Liao, Y., Qu, Y., and Liu, G. Training free guided flow matching with optimal control. arXiv preprint arXiv:2410.18070, 2024 b

    Pith/arXiv arXiv 2024

  52. [52]

    Wang, Y., Lu, J., Jaitly, N., Susskind, J., and Bautista, M. A. Simplefold: Folding proteins is simpler than you think. arXiv preprint arXiv:2509.18480, 2025

    arXiv 2025

  53. [53]

    Medmnist classification decathlon: A lightweight automl benchmark for medical image analysis

    Yang, J., Shi, R., and Ni, B. Medmnist classification decathlon: A lightweight automl benchmark for medical image analysis. In IEEE 18th International Symposium on Biomedical Imaging (ISBI), pp.\ 191--195, 2021

    2021

  54. [54]

    Medmnist v2-a large-scale lightweight benchmark for 2d and 3d biomedical image classification

    Yang, J., Shi, R., Wei, D., Liu, Z., Zhao, L., Ke, B., Pfister, H., and Ni, B. Medmnist v2-a large-scale lightweight benchmark for 2d and 3d biomedical image classification. Scientific Data, 10 0 (1): 0 41, 2023

    2023

  55. [55]

    From facial parts responses to face detection: A deep learning approach

    Yang, S., Luo, P., Loy, C.-C., and Tang, X. From facial parts responses to face detection: A deep learning approach. In Proceedings of the IEEE international conference on computer vision, pp.\ 3676--3684, 2015

    2015

  56. [56]

    Adding conditional control to text-to-image diffusion models

    Zhang, L., Rao, A., and Agrawala, M. Adding conditional control to text-to-image diffusion models. In Proceedings of the IEEE/CVF international conference on computer vision, pp.\ 3836--3847, 2023

    2023

  57. [57]

    and Chen, Y

    Zhang, Q. and Chen, Y. Path integral sampler: a stochastic control approach for sampling. arXiv preprint arXiv:2111.15141, 2021

    Pith/arXiv arXiv 2021

  58. [58]

    N., and Leong, O

    Zhang, Y., Yu, P., Zhu, Y., Chang, Y., Gao, F., Wu, Y. N., and Leong, O. Flow priors for linear inverse problems via iterative corrupted trajectory matching. Advances in Neural Information Processing Systems, 37: 0 57389--57417, 2024

    2024