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Flow matching for stochastic linear control systems

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper establishes that a feedback law given by the conditional expectation of the control inputs of stochastic bridges of a linear system steers any initial distribution to any target distribution through that system.

desk verdict The stochastic-case construction is sound and useful, but the paper overclaims the deterministic case: conditional-expectation feedback cannot split atoms, so 'any pair of distributions' is false for ε=0. read the letter →

arxiv 2412.00617 v2 pith:J5V6OSTE submitted 2024-11-30 math.OC

Yuhang Mei , Mohammad Al-Jarrah , Amirhossein Taghvaei , Yongxin Chen This is my paper · ORCID
classification math.OC MSC 93E2093B0560J60
keywords flowmatchingstochasticlinearcontrolsystemsprobabilitydensitysteeringbridgefeedbackcontrollabilityGramianGaussianmixturemean-field
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that flow matching still works when the flow is constrained to a linear control system: instead of freely choosing a velocity field, one can only push the state through the given control channel. Its central claim is that a feedback law equal to the conditional expectation of the bridge control inputs steers the state distribution from any initial law to any target law, for both deterministic and stochastic linear systems. This matters for applications where agents or particles are manipulated only through control actions, such as robot swarms and stochastic thermodynamics, because it provides a constructive controller without solving an optimal control problem. For Gaussian and Gaussian-mixture targets the paper derives an explicit formula for the feedback law, and otherwise it proposes a regression-based approximation.

What carries the argument

The central object is the stochastic bridge $X^z_t$ of the linear system, whose mean is $R_t x + S_t y$ and whose covariance is $\epsilon^2\Sigma_t$, together with its input $u^z_t = B^\top e^{(1-t)A^\top}\Phi_{1-t}^{-1}(y - e^{(1-t)A}X^z_t)$. The mechanism that carries the argument is the identity $\bar{k}(t,\xi) = \mathbb{E}[u^z_t \mid X^z_t = \xi]$: averaging the bridge controls over all endpoints that pass through $\xi$ produces a feedback law whose drift matches the bridge's drift, so the closed-loop marginals coincide with the bridge marginals at every time. The controllability Gramian $\Phi_t$, through the inverse $\Phi_{1-t}^{-1}$, is what makes the bridge reach the prescribed endpoint and is therefore the reason the controllability assumption is needed.

What would settle it

Fix a controllable linear system, take $P_0$ and $P_1$ to be two-point distributions, compute the exact conditional expectation (10) by replacing the bridge law with a fine histogram, and simulate the closed loop; the theorem predicts the time-marginals coincide with the bridge marginals at every time, so any systematic mismatch is a direct refutation of the claimed law-matching property.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 7. For the stochastic linear system $dX_t = AX_t\,dt + B(u_t\,dt + \epsilon\,dW_t)$, if one builds, for each paired endpoint $z=(x,y)$ drawn from any coupling of $P_0$ and $P_1$, the stochastic bridge $X^z_t$ with bridge control $u^z_t$, then the closed-loop system driven by $\bar{k}(t,\xi) = \mathbb{E}[u^z_t \mid X^z_t = \xi]$ has the same time-marginal law as $X^z_t$. In particular, $X_1$ follows $P_1$. The proof compares the evolution of expectations under the closed loop and under the bridge process, and the only identity needed is the tower property of conditional expectation. The same law-matching statement holds in the deterministic limit, where the bridge is the minimum-norm interpolating trajectory of the linear system.

Load-bearing premise

The system must be controllable, and the noise must enter through exactly the same input channels as the control; otherwise the bridge formula the whole construction is built on is unavailable.

Editorial extensions

If this is right

  • Any distribution pair can be connected through the constrained linear dynamics, not only through an unconstrained velocity field, and the terminal law is matched exactly in the ideal setting where the conditional expectation is known.
  • For Gaussian initial and Gaussian or mixture-of-Gaussians target laws, the feedback law is given explicitly by formula (14), so no iterative optimization is needed in that case.
  • The stochastic and deterministic problems are solved by exactly the same feedback law, and the stochastic bridge tends to the deterministic interpolant as the noise intensity goes to zero.
  • When the exact conditional expectation is unavailable, the proposed least-squares regression (15) approximates it, and the numerical experiments report small normalized MMD and $W_2$ distances in two-, four-, and eight-dimensional examples.
  • The argument extends to control-affine systems whenever a sampler for the corresponding stochastic bridge exists.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: because the coupling $\Pi$ between the endpoint distributions is arbitrary, one can choose it to shape trajectories or reduce control effort without changing the theorem, for example by taking an optimal-transport coupling.
  • Editorial inference: the proof's mechanism is essentially the tower property of conditional expectation, so in principle the same feedback construction would apply to any bridge process one can sample from; the paper's explicit linear-system formulas are what make the construction directly usable in that setting.
  • Editorial inference: a quantitative extension would be a finite-sample bound relating the regression error in (15) to the error in the terminal distribution, a step the paper does not attempt.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper extends flow matching to linear control systems of the form dX_t = AX_t dt + B(u_t dt + ε dW_t). The authors construct stochastic bridges between pairs (x,y) using the controllability Gramian, derive the explicit bridge control (7) and marginal (8), and then propose the feedback law (10), defined as the conditional expectation of the bridge control given the bridge state. Theorem 7 claims that this feedback law steers any initial distribution P0 to any target distribution P1, and that the controlled process has the same law as the bridge process. Analytical formulas are given for Gaussian and mixture-of-Gaussian targets, a numerical regression procedure is proposed, and experiments are shown for several 2D, 4D, and 8D systems.

Significance. The stochastic construction is conceptually attractive: it provides an explicit, parameter-free feedback law in the stochastic setting, avoiding the solution of an optimal control problem, and it extends flow matching to systems with constrained control channels. The tower-property argument in Theorem 7 is elegant and, under suitable well-posedness assumptions, gives a clean law-matching proof. The paper also ships reproducible code and compares the generated samples with the bridge samples using MMD and W2 distances. However, the claimed validity for arbitrary distributions in the deterministic case is false as stated, and the mixture-of-Gaussians formula in Corollary 8 contains a concrete error. These issues need correction before the paper can be accepted.

major comments (3)
  1. [Section 3, Theorem 7 and Abstract] The claim that Problem 3 is solved for 'any pair of distributions' in the deterministic case is false. For the scalar system dX_t/dt = u_t (A=0, B=1, which is controllable), take P0 = δ_0 and P1 = (δ_{-1}+δ_1)/2. The bridge formulas (7)-(8) give X^z_t = ty and u^z_t = y for z=(0,y). The control law (10) at t=0 is kbar(0,0)=E[y|X^z_0=0]=(−1+1)/2=0, so the deterministic closed-loop ODE has the unique trajectory X_t≡0 and cannot realize the two-point target P1. Thus the statement in Section 3 that the deterministic construction is 'identical' to the stochastic one is not correct as written, and the theorem/abstract need a restriction (e.g., ε>0, or non-atomic P0 with additional regularity for ε=0).
  2. [Corollary 8, Eq. (14)] The formula for the mixture weights w'_l is missing the component-dependent normalizing constants. Writing C_l = R_t Q0 R_t^T + S_t Q_l S_t^T + ε^2 Σ_t, the correct posterior weight is proportional to w_l |C_l|^{-1/2} exp(−(1/2)(ξ−R_t m0−S_t m_l)^T C_l^{-1}(ξ−R_t m0−S_t m_l)), with the determinant factor depending on l. As printed, the displayed formula is only correct when all Q_l are identical; otherwise the weights are incorrectly normalized and the conditional expectation (14) is wrong. This should be corrected or the determinant factors should be incorporated into the definition of w'_l.
  3. [Theorem 7 proof] The proof shows that if a solution X_t of (6) with the feedback law (10) exists, then its law satisfies the same Fokker-Planck equation as the bridge process X^z_t. It does not establish existence or uniqueness of such a solution for arbitrary P0,P1. The drift kbar(t,ξ) can be undefined or irregular, as the deterministic atomic example in the first comment shows, and no well-posedness assumptions are stated. The theorem should be restated with explicit conditions under which the SDE (6) with drift kbar has a unique solution, or it should be formulated as a formal result pending those conditions.
minor comments (6)
  1. [Section 3, first paragraph] There is a duplicated article in 'through a a deterministic or stochastic linear system'; it should be 'through a deterministic or stochastic linear system'.
  2. [Equation (12)] The notation 'E(y|X^z_t=ξ]' has a mismatched parenthesis; it should be 'E[y|X^z_t=ξ]'.
  3. [Theorem 7 proof] There are small typos: 'due the the equality' should be 'due to the equality', and 'It ˆo rule' contains a formatting artifact (the hat is misplaced).
  4. [Corollary 8 proof] 'spacial case' should be 'special case'.
  5. [Figure 1 caption] The caption lists three systems (a)-(c) but does not explain what the colors or curves represent, nor how the ε values are distinguished; please expand the caption.
  6. [Section 4.1] The text says 'The left panel shows...' and 'The second panel compares...', but the figure has three panels per row and the second and third rows are described only by row; the panel references should be made explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the control law is defined as a conditional expectation of constructed bridge controls, and the law-matching theorem is a direct generator verification rather than a fitted input.

full rationale

The paper's central derivation is self-contained and non-circular. The feedback law (10), kbar(t,xi)=E[u^z_t|X^z_t=xi], is a construction from the stochastic bridge (7)-(8), and Theorem 7 proves law matching by comparing the generators of X_t and X^z_t, using the tower property and the definition of kbar. No fitted parameter is renamed as a prediction: the Gaussian and mixture formulas in Corollary 8 are derived from standard conditional-Gaussian identities, and the general formula does not depend on tuning to the target distribution. The cited prior work, including Chen and Georgiou (2015), is not load-bearing because Proposition 4 supplies an in-paper proof of the bridge and control law. The numerical experiments fit a neural network to the conditional expectation (15) and then compare simulated trajectories to the training samples; this is an internal consistency check of the approximation and is accompanied by a direct comparison to the exact target density, so it does not reduce the claimed result to its own input. The paper's stated limitations, notably Remark 6 restricting the noise to enter through the control channel B and the acknowledged reliance on regression without distributional guarantees, concern scope and approximation error rather than circularity. The deterministic-case objection raised by the skeptic is a correctness/well-posedness concern about atomic initial distributions at epsilon=0, not a circularity of the derivation. Overall, the derivation chain is independent of its conclusions and merits a circularity score of 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The theoretical result uses no fitted constants; the only fitted quantities are neural-network weights in the numerical approximation. The main domain assumptions are controllability and matched noise/control channels, both stated explicitly in the paper. No new physical entities, forces, or conserved quantities are introduced.

free parameters (1)
  • Neural network weights = learned, not reported in the paper
    The numerical approximation in Section 4 solves the regression problem (15) over a parameterized network class; the resulting control approximates k-bar. These weights are an algorithmic fitting artifact, not a parameter of the theoretical control law.
assumptions (4)
  • domain assumption (A,B) is controllable.
    Assumption 1; ensures the controllability Gramian Phi_t is invertible for t > 0 and that the explicit bridge formulas (3)-(4) and (7)-(8) exist.
  • domain assumption Noise enters only through the same input matrix B as the control.
    Remark 6 restricts the exposition to stochastic systems where Brownian noise is multiplied by B; the time-reversal bridge derivation and the control law rely on this matching structure.
  • standard math Standard SDE regularity for the Ito formula and conditional expectation.
    The proof of Theorem 7 uses the Ito rule and the tower property; it assumes sufficient regularity of k-bar and the test function class.
  • standard math Gaussian sum filter weighting for mixture conditionals.
    Corollary 8 extends the L=1 Gaussian conditioning to mixtures L>1 by likelihood-weighted Gaussian components; the derivation is omitted and the paper cites Gaussian sum filters.

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Cite this review

Pith. "Pith review of Flow matching for stochastic linear control systems." pith.science (2026). https://pith.science/paper/J5V6OSTE

@misc{pith2026241200617,
  author       = {Pith},
  title        = {Pith review of: Flow matching for stochastic linear control systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J5V6OSTE}},
  note         = {Machine review of arXiv:2412.00617}
}
read the original abstract

This paper addresses the problem of steering an initial probability distribution to a target probability distribution through a deterministic or stochastic linear control system. Our proposed approach is inspired by the flow matching methodology, with the difference that we can only affect the flow through the given control channels. The motivation comes from applications such as robotic swarms and stochastic thermodynamics, where agents or particles can only be manipulated through control actions. The feedback control law that achieves the task is characterized as the conditional expectation of the control inputs for the stochastic bridges that respect the given control system dynamics. Explicit forms are derived for special cases, and a numerical procedure is presented to approximate the control law, illustrated with examples.

Figures

Figures reproduced from arXiv: 2412.00617 by the authors.

Figure 1
Figure 1. Interpolations for three linear control systems with modeling parameters specified in (9). concluding the formula (8). In order to obtain the formula for the control input, consider the uncontrolled process Xet satis￾fying dXet = −AXet dt + ϵB dWft , Xe0 = y. The probability distribution of Xet is Gaussian N (mt , Qt) with mean mt = e −Aty and covariance matrix Qt = ϵ 2 Z t 0 e −sABB⊤e −sA⊤ = ϵ 2 e −tAΦte −tA⊤ . The… view at source ↗
Figure 2
Figure 2. Numerical results for three 2-dimensional linear control systems, with model parame￾ters (9)-(a)-(b)-(c). First row: numerical result for (9)-(a). The left panel shows the tra￾jectories {Xi t} N′ i=1 generated from the prediction stage of the algorithm. The second panel compares the exact target density with a kernel density approximation of the generated samples {Xi 1 } N′ i=1. The right panel shows the MMD distanc… view at source ↗
Figure 3
Figure 3. Numerical results for four dimensional (first row) and eight dimensional (second row) mass-springs systems. The left panel shows the approximated and exact densities, at t = 1, projected onto the last two components. The panel at the center shows the MMD distance, and the right panel shows the W2 distance between the generated samples and training samples as a function of time t. 4.2. Numerical results for higher di… view at source ↗

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

40 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    Building normalizing flows with stochastic interpolants

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

    arXiv 2022

  2. [2]

    Nonlinear bayesian estimation using gaussian sum approximations

    Daniel Alspach and Harold Sorenson. Nonlinear bayesian estimation using gaussian sum approximations. IEEE transactions on automatic control, 17 0 (4): 0 439--448, 1972

    work page 1972

  3. [3]

    Reverse-time diffusion equation models

    Brian DO Anderson. Reverse-time diffusion equation models. Stochastic Processes and their Applications, 12 0 (3): 0 313--326, 1982

    1982

  4. [4]

    Lecture notes on control system theory and design

    Tamer Basar, Sean Meyn, and William R Perkins. Lecture notes on control system theory and design. arXiv preprint arXiv:2007.01367, 2020

    arXiv 2007

  5. [5]

    Mean field games and mean field type control theory, volume 101

    Alain Bensoussan, Jens Frehse, Phillip Yam, et al. Mean field games and mean field type control theory, volume 101. Springer, 2013

    2013

  6. [6]

    Notes on the control of the liouville equation

    Roger Brockett. Notes on the control of the liouville equation. In Control of partial differential equations, pages 101--129. Springer, 2012

    work page 2012

  7. [7]

    Optimal control of the liouville equation

    Roger W Brockett. Optimal control of the liouville equation. AMS IP Studies in Advanced Mathematics, 39: 0 23, 2007

    work page 2007

  8. [8]

    Probabilistic theory of mean field games with applications I-II

    Ren \'e Carmona, Fran c ois Delarue, et al. Probabilistic theory of mean field games with applications I-II. Springer, 2018

    work page 2018

Show all 40 references
  1. [9]

    Mean field deep reinforcement learning for fair and efficient UAV control

    Dezhi Chen, Qi Qi, Zirui Zhuang, Jingyu Wang, Jianxin Liao, and Zhu Han. Mean field deep reinforcement learning for fair and efficient UAV control. IEEE Internet of Things Journal, 8 0 (2): 0 813--828, 2020

    2020

  2. [10]

    Density control of interacting agent systems

    Yongxin Chen. Density control of interacting agent systems. IEEE Transactions on Automatic Control, 69 0 (1): 0 246--260, 2023

    2023

  3. [11]

    Stochastic bridges of linear systems

    Yongxin Chen and Tryphon Georgiou. Stochastic bridges of linear systems. IEEE Transactions on Automatic Control, 61 0 (2): 0 526--531, 2015

    2015

  4. [12]

    Optimal steering of a linear stochastic system to a final probability distribution, part ii

    Yongxin Chen, Tryphon T Georgiou, and Michele Pavon. Optimal steering of a linear stochastic system to a final probability distribution, part ii. IEEE Transactions on Automatic Control, 61 0 (5): 0 1170--1180, 2015

    2015

  5. [13]

    Optimal transport over a linear dynamical system

    Yongxin Chen, Tryphon T Georgiou, and Michele Pavon. Optimal transport over a linear dynamical system. IEEE Transactions on Automatic Control, 62 0 (5): 0 2137--2152, 2016 a

    2016

  6. [14]

    On the relation between optimal transport and schr \"o dinger bridges: A stochastic control viewpoint

    Yongxin Chen, Tryphon T Georgiou, and Michele Pavon. On the relation between optimal transport and schr \"o dinger bridges: A stochastic control viewpoint. Journal of Optimization Theory and Applications, 169: 0 671--691, 2016 b

    2016

  7. [15]

    Steering the distribution of agents in mean-field games system

    Yongxin Chen, Tryphon T Georgiou, and Michele Pavon. Steering the distribution of agents in mean-field games system. Journal of Optimization Theory and Applications, 179 0 (1): 0 332--357, 2018

    2018

  8. [16]

    Stochastic control and nonequilibrium thermodynamics: Fundamental limits

    Yongxin Chen, Tryphon T Georgiou, and Allen Tannenbaum. Stochastic control and nonequilibrium thermodynamics: Fundamental limits. IEEE transactions on automatic control, 65 0 (7): 0 2979--2991, 2019

    2019

  9. [17]

    Optimal transport for applications in control and estimation

    Yongxin Chen, Johan Karlsson, and Axel Ringh. Optimal transport for applications in control and estimation. IEEE Control Systems Magazine, 41 0 (4): 0 28--33, 2021. doi:10.1109/MCS.2021.3076390

    2021

  10. [18]

    A stochastic control approach to reciprocal diffusion processes

    Paolo Dai Pra. A stochastic control approach to reciprocal diffusion processes. Applied mathematics and Optimization, 23 0 (1): 0 313--329, 1991

    1991

  11. [19]

    Mean-field models in swarm robotics: A survey

    Karthik Elamvazhuthi and Spring Berman. Mean-field models in swarm robotics: A survey. Bioinspiration & Biomimetics, 15 0 (1): 0 015001, 2019

    2019

  12. [20]

    Mean-field optimal control

    Massimo Fornasier and Francesco Solombrino. Mean-field optimal control. ESAIM: Control, Optimisation and Calculus of Variations, 20 0 (4): 0 1123--1152, 2014

    2014

  13. [21]

    Maximal power output of a stochastic thermodynamic engine

    Rui Fu, Amirhossein Taghvaei, Yongxin Chen, and Tryphon T Georgiou. Maximal power output of a stochastic thermodynamic engine. Automatica, 123: 0 109366, 2021

    2021

  14. [22]

    Vincent Poor

    Hao Gao, Wonjun Lee, Yuhan Kang, Wuchen Li, Zhu Han, Stanley Osher, and H. Vincent Poor. Energy-efficient velocity control for massive numbers of uavs: A mean field game approach. IEEE Transactions on Vehicular Technology, 71 0 (6): 0 6266--6278, 2022. doi:10.1109/TVT.2022.3158896

    2022

  15. [23]

    Control and estimation of ensembles via structured optimal transport

    Isabel Haasler, Johan Karlsson, and Axel Ringh. Control and estimation of ensembles via structured optimal transport. IEEE Control Systems Magazine, 41 0 (4): 0 50--69, 2021

    2021

  16. [24]

    Time reversal of diffusions

    Ulrich G Haussmann and Etienne Pardoux. Time reversal of diffusions. The Annals of Probability, pages 1188--1205, 1986

    1986

  17. [25]

    Large population stochastic dynamic games: closed-loop mckean-vlasov systems and the nash certainty equivalence principle

    Minyi Huang, Roland P Malham \'e , and Peter E Caines. Large population stochastic dynamic games: closed-loop mckean-vlasov systems and the nash certainty equivalence principle. Communications in Information & Systems, 6 0 (3): 0 221 -- 252, 2006

    2006

  18. [26]

    Mean field games

    Jean-Michel Lasry and Pierre-Louis Lions. Mean field games. Japanese journal of mathematics, 2 0 (1): 0 229--260, 2007

    2007

  19. [27]

    Flow matching for generative modeling

    Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling. arXiv preprint arXiv:2210.02747, 2022

    2022 arXiv

  20. [28]

    Generalized schr " odinger bridge matching

    Guan-Horng Liu, Yaron Lipman, Maximilian Nickel, Brian Karrer, Evangelos A Theodorou, and Ricky TQ Chen. Generalized schr " odinger bridge matching. arXiv preprint arXiv:2310.02233, 2023

    2023 arXiv

  21. [29]

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

    Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. arXiv preprint arXiv:2209.03003, 2022

    2022 arXiv

  22. [30]

    A mean field game approach to swarming robots control

    Zhiyu Liu, Bo Wu, and Hai Lin. A mean field game approach to swarming robots control. In 2018 Annual American Control Conference (ACC), pages 4293--4298. IEEE, 2018

    2018

  23. [31]

    Time-reversal solution of bsdes in stochastic optimal control: a linear quadratic study

    Yuhang Mei and Amirhossein Taghvaei. Time-reversal solution of bsdes in stochastic optimal control: a linear quadratic study. arXiv preprint arXiv:2410.04615, 2024

    2024 arXiv

  24. [32]

    Stochastic Thermodynamic Treatment of Thermal Anisotropy

    Olga Movilla Miangolarra. Stochastic Thermodynamic Treatment of Thermal Anisotropy. University of California, Irvine, 2023

    2023

  25. [33]

    Stochastic control and nonequilibrium thermodynamical systems

    Michele Pavon. Stochastic control and nonequilibrium thermodynamical systems. Applied Mathematics and Optimization, 19: 0 187--202, 1989

    1989

  26. [34]

    On free energy, stochastic control, and schr \"o dinger processes

    Michele Pavon and Anton Wakolbinger. On free energy, stochastic control, and schr \"o dinger processes. In Modeling, Estimation and Control of Systems with Uncertainty: Proceedings of a Conference held in Sopron, Hungary, September 1990, pages 334--348. Springer, 1991

    1990

  27. [35]

    Stochastic Thermodynamics: An Introduction

    Luca Peliti and Simone Pigolotti. Stochastic Thermodynamics: An Introduction. Princeton University Press, 2021

    2021

  28. [36]

    Collective stochastic discrete choice problems: A min-lqg dynamic game formulation

    Rabih Salhab, Roland P Malham \'e , and Jerome Le Ny. Collective stochastic discrete choice problems: A min-lqg dynamic game formulation. IEEE Transactions on Automatic Control, 65 0 (8): 0 3302--3316, 2019

    2019

  29. [37]

    Stochastic thermodynamics, fluctuation theorems and molecular machines

    Udo Seifert. Stochastic thermodynamics, fluctuation theorems and molecular machines. Reports on progress in physics, 75 0 (12): 0 126001, 2012

    2012

  30. [38]

    Stochastic energetics, volume 799

    Ken Sekimoto. Stochastic energetics, volume 799. Springer, 2010

    2010

  31. [39]

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

    Alexander Tong, Kilian Fatras, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector-Brooks, Guy Wolf, and Yoshua Bengio. Improving and generalizing flow-based generative models with minibatch optimal transport. arXiv preprint arXiv:2302.00482, 2023

    2023 arXiv

  32. [40]

    Optimal transport on networks

    Haomin Zhou. Optimal transport on networks. IEEE Control Systems Magazine, 41 0 (4): 0 70--81, 2021

    2021

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