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Open problems in Flow matching

Scalable and stable training for flow matching1 paper raises it · 1 addresses it
Large-scale flow models struggle with training stability and integration error due to stochastic source-target pairing.
- Scaling Online Optimal Transport Minibatches for Flow Matchingraises · addresses
Efficient few-step generation without fixed discretization1 paper raises it · 1 addresses it
Few-step generation often requires heuristic timestep discretization, limiting flexibility and performance.
- Few-Step Generation via Data-Space Iterationraises · addresses
Data-free and simulation-free distillation for diffusion/flow models1 paper raises it · 1 addresses it
Distillation often relies on curated datasets or costly teacher rollouts, complicating training and scaling.
- Consistent Distribution Matching for Data-Free Diffusion Distillationraises · addresses
Dynamics-aligned latent representations for PDEs1 paper raises it · 1 addresses it
Reconstruction-based learning for PDEs often fails to capture the evolution of physical fields accurately.
- PDE-JEPA: Predictive Representation Learning of Latent Dynamics Modeling for Parametric PDEsraises · addresses
Detecting memorization in diffusion models1 paper raises it · 1 addresses it
Memorization in diffusion models can lead to failure in proper distributional learning.
- Detection of Memorization in Diffusion Models Through Thermodynamic Singularityraises · addresses

Named by an AI from these papers' abstracts.

Open problems

Recently funded

Topics in this field

flow matching · flow · generative modeling292 papers29 this week
physics informed neural · neural operators · informed neural networks291 papers25 this week
diffusion priors · posterior sampling · diffusion posterior73 papers14 this week

Topics in this field

Active contributors · 90 days

Anima AnandkumarNeural Operator Splitting for Fluid Dynamics in Complex Geometry5
Stefan T. RadevConditional Mean Flows: Few-Step Densities for Simulation-Based Inference5
Taiji SuzukiFinite-Sample Approximation of Hessian-Guided Perturbed Wasserstein Gradient Flows5
Jing XuSame Target, Different Origins: Aligning Latent Rollout for Consistent PDE Emulation4
Jun ZhangEqual Path Cost, Unequal Output Effects: Understanding Perturbation Propagation in Diffusion Models4
Alexander KorotinDiscrete Action Matching: Learning Stochastic Dynamics from Samples via State Graphs4

Active contributors

Activity · 12 weeks

Activity

Recent papers

All papers in Research

arXiv cs.LGOct 2
Hyperparameter selection for equation learning with biologically-informed neural networks
William Lavery, Jodie A. Cochrane, John T. Nardini, Sara Hamis
Biologically-informed neural networks (BINNs) have emerged as a flexible subclass of physics-informed neural networks (PINNs) for learning terms in partial differential equations from data. BINNs are particularly suited for biological systems, where the governing equations are highly nonlinear and only partially known a priori, and where data observations are often sparse, noisy, and incomplete. However, applying BINNs effectively in practice depends critically on hyperparameter selection, which remains a central challenge in equation-learning frameworks. Hyperparameters are often chosen heuristically and only cursorily documented, which limits the reproducibility of results and the transferability of methods. We present a diagnostic workflow for hyperparameter selection that can be used when the ground-truth equations are not known. The workflow is guided by three main questions: (1) Are the benefits of greater network capacity worth the cost? (2) Do more training epochs keep reducing the validation loss? (3) Do the learned terms stop changing as network capacity and training increase? We apply our workflow to synthetic systems of varying complexity with known ground truth,…
PDF arXiv
arXiv cs.LGOct 2
Tangent Schrödinger Bridge Matching: Learning Stochastic Transport with Mechanistic Sensitivities
Jowaria Khan, Elizabeth Bondi-Kelly
Predicting how stochastic systems respond to changes in viscosity, reaction rates, or external forces requires costly simulations, motivating reusable learned models. Yet matching observed outcome distributions does not ensure accurate intervention responses. We introduce Tangent Schrödinger Bridge Matching (Tangent-SBM), which learns stochastic transports from endpoint observations and mechanistic sensitivities. It propagates parameter derivatives alongside trajectories and supervises them against supplied targets. For average-response targets, single-rollout squared error also penalizes response variability; our objective uses two independent rollouts to match the mean without this additional penalty. We establish conditions under which sensitivity accuracy bounds finite-change prediction error and decision regret. Across Gaussian, stochastic double-well, PDEBench reaction--diffusion, and stochastic Navier--Stokes systems, Tangent-SBM improves sensitivity and finite-change prediction over matched conditional-bridge baselines while maintaining comparable endpoint and distributional accuracy. Controls examine target correctness, response objectives, and simulator-budget…
PDF arXiv
arXiv eess.IVOct 2
One Photon, Many Worlds: Posteriors and Predictions with Single-Photon Cameras
Haejoon Lee, Mohit Gupta, Vijayakumar Bhagavatula, Aswin C. Sankaranarayanan
Single-photon avalanche diode (SPAD) cameras operate fundamentally differently from conventional cameras due to their photon-counting nature. Each frame produces a binary image: pixels report zero if no photons arrived during exposure, and one if one or more photons arrived. Reconstructing a scene or inferring its properties from a single binary frame is difficult because many different images could produce the same measurement; thus, the inverse problem is fundamentally one-to-many. As we gather more binary measurements, the inherent uncertainty associated with the inverse problem and any associated inference diminishes. With sufficient photon counts, photon noise becomes negligible relative to the signal mean, enabling near-deterministic scene recovery and inference. This work characterizes the transition from stochastic to near-deterministic scene understanding as photon budget increases, analyzing how the stochasticity in photon arrival affects downstream inference tasks. Technically, we develop a conditional generative framework based on a Hypergeometric frame-thinning process for accumulated binary SPAD measurements. Generative models capture the one-to-many nature of…
PDF arXiv
arXiv stat.MLOct 2
Generalization Bounds for Flow-matching Generative Models for Intrinsically Low-dimensional Data
Saptarshi Chakraborty, Quentin Berthet, Peter L. Bartlett
Despite the remarkable empirical success of flow-matching models, their statistical generalization guarantees remain underdeveloped. Existing analyses often impose restrictive assumptions on the estimated velocity field and yield convergence rates that fail to reflect the intrinsic low-dimensional structure common in real data, such as natural images and molecular geometries. In this work, we study the statistical generalization of flow-matching models for learning an unknown distribution $P_{\mathrm{data}}$ from finitely many samples. We derive finite-sample error bounds on the learned generative distribution, measured in the Wasserstein-$p$ distance, for all $p\geq 1$. Specifically, given $n$ i.i.d. samples from $P_{\mathrm{data}}$, we show that, for every $d>d_p^\ast(P_{\mathrm{data}})$ and appropriately chosen network architectures and hyperparameters, the learned distribution $\widehat{P}^{\mathrm{FM}}$ satisfies $ \mathbb{W}_p(\widehat{P}^{\mathrm{FM}},P_{\mathrm{data}}) \lesssim n^{-1/d}+n^{-1/(2p)}\bigl(\log(1/ξ)\bigr)^{1/(2p)}$ with probability at least $1-ξ$, where $d_p^\ast(P_{\mathrm{data}})$ denotes the Wasserstein-$p$ dimension of the target measure. Our results…
PDF arXiv
arXiv math.OCOct 2
A hybrid CNN-adjoint optimization framework for reconstruction of viscoelastic tissue properties in magnetic resonance elastography
Anwesa Dey, Johann Rudi, Elena Cherkaev
Magnetic resonance elastography (MRE) is a noninvasive imaging modality for quantifying the viscoelastic properties of soft tissues from shear wave propagation. Recovering the complex-valued shear modulus from measured displacement fields leads to a severely ill-posed inverse problem, particularly in the presence of noise and limited boundary excitations. We investigate adjoint-based optimization, convolutional neural network (CNN) reconstruction, and a hybrid framework combining both approaches. The forward model is based on a scalar form of the modified stationary Stokes system with a complex shear modulus. We establish well-posedness of the forward problem, existence of minimizers, and first-order optimality conditions for the adjoint-based formulation, and implement a nonlinear conjugate-gradient method with Armijo line search. While PDE-constrained optimization can accurately refine coefficient reconstructions, its performance depends strongly on initialization. We therefore construct a two-dimensional CNN that maps complex-valued displacement measurements to spatially varying complex shear modulus fields and provides rapid, informative initial reconstructions. The proposed…
PDF arXiv
arXiv math.NAOct 1
Learning Closure of Dynamical Systems with Kernel Ridge Regression
Evan Habbershaw, John Harlim, Senwei Liang
We develop a closure modeling framework for identifying missing components of dynamical systems using Kernel Ridge Regression (KRR). The framework addresses two classes of closure problems: difference-equation closures arising in ODE and PDE settings, and algebraic closures arising from moment closure in kinetic equations. For the first class, we derive an error bound in an ODE setting that quantifies contributions from time integration, approximation of unresolved scales, and interpolation required to couple unresolved-scale effects to the resolved solver. Numerical experiments on the Lorenz-63 system and the Kuramoto-Sivashinsky equation demonstrate accurate long-horizon predictions and substantial improvements over an LSTM-based closure model. For the second class, we consider moment closure for a one-dimensional kinetic equation by modeling discrepancies between kinetic and macroscopic fluxes as a function of the resolved macroscopic variables. We compare global KRR models based on PCA coordinates with spatially local models. While the global model performs well for unimodal initial conditions, its accuracy deteriorates for bimodal initial conditions. Spatially local models…
PDF arXiv

Papers

Topics in this field

flow matching · flow · generative modeling292 papers29 this week
physics informed neural · neural operators · informed neural networks291 papers25 this week
diffusion priors · posterior sampling · diffusion posterior73 papers14 this week

Active contributors · 90 days

Anima AnandkumarNeural Operator Splitting for Fluid Dynamics in Complex Geometry5
Stefan T. RadevConditional Mean Flows: Few-Step Densities for Simulation-Based Inference5
Taiji SuzukiFinite-Sample Approximation of Hessian-Guided Perturbed Wasserstein Gradient Flows5
Jing XuSame Target, Different Origins: Aligning Latent Rollout for Consistent PDE Emulation4
Jun ZhangEqual Path Cost, Unequal Output Effects: Understanding Perturbation Propagation in Diffusion Models4
Alexander KorotinDiscrete Action Matching: Learning Stochastic Dynamics from Samples via State Graphs4

Recent papers

All papers in Research

arXiv cs.LGOct 2
Hyperparameter selection for equation learning with biologically-informed neural networks
William Lavery, Jodie A. Cochrane, John T. Nardini, Sara Hamis
Biologically-informed neural networks (BINNs) have emerged as a flexible subclass of physics-informed neural networks (PINNs) for learning terms in partial differential equations from data. BINNs are particularly suited for biological systems, where the governing equations are highly nonlinear and only partially known a priori, and where data observations are often sparse, noisy, and incomplete. However, applying BINNs effectively in practice depends critically on hyperparameter selection, which remains a central challenge in equation-learning frameworks. Hyperparameters are often chosen heuristically and only cursorily documented, which limits the reproducibility of results and the transferability of methods. We present a diagnostic workflow for hyperparameter selection that can be used when the ground-truth equations are not known. The workflow is guided by three main questions: (1) Are the benefits of greater network capacity worth the cost? (2) Do more training epochs keep reducing the validation loss? (3) Do the learned terms stop changing as network capacity and training increase? We apply our workflow to synthetic systems of varying complexity with known ground truth,…
PDF arXiv
arXiv cs.LGOct 2
Tangent Schrödinger Bridge Matching: Learning Stochastic Transport with Mechanistic Sensitivities
Jowaria Khan, Elizabeth Bondi-Kelly
Predicting how stochastic systems respond to changes in viscosity, reaction rates, or external forces requires costly simulations, motivating reusable learned models. Yet matching observed outcome distributions does not ensure accurate intervention responses. We introduce Tangent Schrödinger Bridge Matching (Tangent-SBM), which learns stochastic transports from endpoint observations and mechanistic sensitivities. It propagates parameter derivatives alongside trajectories and supervises them against supplied targets. For average-response targets, single-rollout squared error also penalizes response variability; our objective uses two independent rollouts to match the mean without this additional penalty. We establish conditions under which sensitivity accuracy bounds finite-change prediction error and decision regret. Across Gaussian, stochastic double-well, PDEBench reaction--diffusion, and stochastic Navier--Stokes systems, Tangent-SBM improves sensitivity and finite-change prediction over matched conditional-bridge baselines while maintaining comparable endpoint and distributional accuracy. Controls examine target correctness, response objectives, and simulator-budget…
PDF arXiv
arXiv eess.IVOct 2
One Photon, Many Worlds: Posteriors and Predictions with Single-Photon Cameras
Haejoon Lee, Mohit Gupta, Vijayakumar Bhagavatula, Aswin C. Sankaranarayanan
Single-photon avalanche diode (SPAD) cameras operate fundamentally differently from conventional cameras due to their photon-counting nature. Each frame produces a binary image: pixels report zero if no photons arrived during exposure, and one if one or more photons arrived. Reconstructing a scene or inferring its properties from a single binary frame is difficult because many different images could produce the same measurement; thus, the inverse problem is fundamentally one-to-many. As we gather more binary measurements, the inherent uncertainty associated with the inverse problem and any associated inference diminishes. With sufficient photon counts, photon noise becomes negligible relative to the signal mean, enabling near-deterministic scene recovery and inference. This work characterizes the transition from stochastic to near-deterministic scene understanding as photon budget increases, analyzing how the stochasticity in photon arrival affects downstream inference tasks. Technically, we develop a conditional generative framework based on a Hypergeometric frame-thinning process for accumulated binary SPAD measurements. Generative models capture the one-to-many nature of…
PDF arXiv
arXiv stat.MLOct 2
Generalization Bounds for Flow-matching Generative Models for Intrinsically Low-dimensional Data
Saptarshi Chakraborty, Quentin Berthet, Peter L. Bartlett
Despite the remarkable empirical success of flow-matching models, their statistical generalization guarantees remain underdeveloped. Existing analyses often impose restrictive assumptions on the estimated velocity field and yield convergence rates that fail to reflect the intrinsic low-dimensional structure common in real data, such as natural images and molecular geometries. In this work, we study the statistical generalization of flow-matching models for learning an unknown distribution $P_{\mathrm{data}}$ from finitely many samples. We derive finite-sample error bounds on the learned generative distribution, measured in the Wasserstein-$p$ distance, for all $p\geq 1$. Specifically, given $n$ i.i.d. samples from $P_{\mathrm{data}}$, we show that, for every $d>d_p^\ast(P_{\mathrm{data}})$ and appropriately chosen network architectures and hyperparameters, the learned distribution $\widehat{P}^{\mathrm{FM}}$ satisfies $ \mathbb{W}_p(\widehat{P}^{\mathrm{FM}},P_{\mathrm{data}}) \lesssim n^{-1/d}+n^{-1/(2p)}\bigl(\log(1/ξ)\bigr)^{1/(2p)}$ with probability at least $1-ξ$, where $d_p^\ast(P_{\mathrm{data}})$ denotes the Wasserstein-$p$ dimension of the target measure. Our results…
PDF arXiv
arXiv math.OCOct 2
A hybrid CNN-adjoint optimization framework for reconstruction of viscoelastic tissue properties in magnetic resonance elastography
Anwesa Dey, Johann Rudi, Elena Cherkaev
Magnetic resonance elastography (MRE) is a noninvasive imaging modality for quantifying the viscoelastic properties of soft tissues from shear wave propagation. Recovering the complex-valued shear modulus from measured displacement fields leads to a severely ill-posed inverse problem, particularly in the presence of noise and limited boundary excitations. We investigate adjoint-based optimization, convolutional neural network (CNN) reconstruction, and a hybrid framework combining both approaches. The forward model is based on a scalar form of the modified stationary Stokes system with a complex shear modulus. We establish well-posedness of the forward problem, existence of minimizers, and first-order optimality conditions for the adjoint-based formulation, and implement a nonlinear conjugate-gradient method with Armijo line search. While PDE-constrained optimization can accurately refine coefficient reconstructions, its performance depends strongly on initialization. We therefore construct a two-dimensional CNN that maps complex-valued displacement measurements to spatially varying complex shear modulus fields and provides rapid, informative initial reconstructions. The proposed…
PDF arXiv
arXiv math.NAOct 1
Learning Closure of Dynamical Systems with Kernel Ridge Regression
Evan Habbershaw, John Harlim, Senwei Liang
We develop a closure modeling framework for identifying missing components of dynamical systems using Kernel Ridge Regression (KRR). The framework addresses two classes of closure problems: difference-equation closures arising in ODE and PDE settings, and algebraic closures arising from moment closure in kinetic equations. For the first class, we derive an error bound in an ODE setting that quantifies contributions from time integration, approximation of unresolved scales, and interpolation required to couple unresolved-scale effects to the resolved solver. Numerical experiments on the Lorenz-63 system and the Kuramoto-Sivashinsky equation demonstrate accurate long-horizon predictions and substantial improvements over an LSTM-based closure model. For the second class, we consider moment closure for a one-dimensional kinetic equation by modeling discrepancies between kinetic and macroscopic fluxes as a function of the resolved macroscopic variables. We compare global KRR models based on PCA coordinates with spatially local models. While the global model performs well for unimodal initial conditions, its accuracy deteriorates for bimodal initial conditions. Spatially local models…
PDF arXiv

Activity · 12 weeks

Open problems in Flow matching

Scalable and stable training for flow matching1 paper raises it · 1 addresses it
Large-scale flow models struggle with training stability and integration error due to stochastic source-target pairing.
- Scaling Online Optimal Transport Minibatches for Flow Matchingraises · addresses
Efficient few-step generation without fixed discretization1 paper raises it · 1 addresses it
Few-step generation often requires heuristic timestep discretization, limiting flexibility and performance.
- Few-Step Generation via Data-Space Iterationraises · addresses
Data-free and simulation-free distillation for diffusion/flow models1 paper raises it · 1 addresses it
Distillation often relies on curated datasets or costly teacher rollouts, complicating training and scaling.
- Consistent Distribution Matching for Data-Free Diffusion Distillationraises · addresses
Dynamics-aligned latent representations for PDEs1 paper raises it · 1 addresses it
Reconstruction-based learning for PDEs often fails to capture the evolution of physical fields accurately.
- PDE-JEPA: Predictive Representation Learning of Latent Dynamics Modeling for Parametric PDEsraises · addresses
Detecting memorization in diffusion models1 paper raises it · 1 addresses it
Memorization in diffusion models can lead to failure in proper distributional learning.
- Detection of Memorization in Diffusion Models Through Thermodynamic Singularityraises · addresses

Named by an AI from these papers' abstracts.

Fields · AI · updated 36m ago

Flow matching

Found by the topic model: flow matching, physics informed neural, diffusion priors.

68papers this week−56.1% vs last week

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