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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

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Topics in this field

flow matching · flow · generative modeling292 papers40 this week
physics informed neural · neural operators · informed neural networks290 papers31 this week
diffusion priors · posterior sampling · diffusion posterior71 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

Coming with more sources: repos

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Recent papers

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arXiv cs.LGOct 15
The importance of tensor handling and implementation of physics-informed DeepONets
Nahum Banks, Gustavo A. M. de Almeida, Sergio Maldonado
Physics-informed neural networks (PINNs) reduce or eliminate the need for large labelled datasets by enforcing governing equations during training. Physics-informed Deep Operator Networks (PI-DeepONets) extend this approach to learning solution operators, enabling rapid prediction across families of problems. Despite these advantages, we show that their application to higher-dimensional problems remains limited by the computational cost of training. To address this issue, we propose a modified PI-DeepONet architecture that reduces this cost by removing redundancy in how the Branch network processes input functions across collocation points. The proposed formulation is mathematically equivalent to the original architecture but reduces input tensor sizes and the number of Branch network evaluations. We analyse the computational complexity of both architectures and derive theoretical scaling laws for training time with respect to the number of collocation points and input functions. The proposed architecture is evaluated on 1D, 2D, and 3D linear advection problems. In all cases, it achieves comparable predictive accuracy to the original PI-DeepONet while consistently reducing…
PDF arXiv
arXiv cs.AIOct 15
Learnable Splines for Unstructured Neural Operators
Vignesh Gopakumar, Zhuoyuan Wang, Ander Gray, Valentin Duruisseaux, Josh Williams +10
Neural operators approximate solution operators of partial differential equations at a fraction of the cost of numerical solvers. Efficient grid-based architectures such as the Fourier neural operator (FNO) assume uniform rectangular grids, whereas engineering simulations use large, irregular, and locally refined meshes. We introduce L-SplineFNO, a memory-efficient neural operator that couples an FNO backbone with a sparse, learnable tensor-product B-spline transfer between unstructured meshes and a latent Cartesian grid. A local normalised scatter avoids the geometry-dependent solve of analytic spline transformation while preserving compact support, partition of unity, and spline smoothness by construction. The neural operator, conditioned on global design and operating parameters, learns the mapping of geometry to the field solution. Across six three-dimensional engineering benchmarks, L-SplineFNO achieves strong predictive accuracy while requiring substantially less memory and compute than baseline operators, scaling to 26.6 million nodes in training and 46 million in inference on a single A100 GPU. It also supports zero-shot super-resolution and efficient closed-form spatial…
PDF arXiv
arXiv cs.LGOct 15
Learn Only Where You Can't Compute: Structured Corrections to Gaussian Process Posteriors
Tobias Weber, Carla Sagebiel, Philipp Hennig
Physics-informed Gaussian processes (GPs) provide a principled framework for scientific inference by expressing uncertainty over empirical measurements, boundary conditions, and governing equations. However, they struggle to scale to larger datasets and nonlinear information. Conversely, diffusion models excel at learning complex distributions over functions from massive datasets, but struggle in scientific contexts where training data are scarce and physical or observational information must be incorporated. We propose a structured posterior-correction framework that combines a tractable Gaussian reference with data-driven corrections via generative learning. After a mean correction and covariance calibration, a diffusion model learns a targeted correction in a low-dimensional subspace selected to capture discrepancies from the reference, while the complement retains its Gaussian law. Across three scientific inference tasks, these corrections recover hidden modes, improve sample structure, and reduce prediction error relative to the Gaussian reference. Comparisons with full-field generative models reveal task-dependent benefits and tradeoffs between predictive accuracy and…
PDF arXiv
arXiv stat.MLOct 15
SelectPFN: Amortized Bayesian Variable Selection for Sparse Nonlinear Regression with Prior-Data Fitted Networks
Bin Luo
Tabular foundation models based on prior-data fitted networks (PFNs) predict on a new data set in a single forward pass, but they cannot report which inputs their predictions depend on. We propose SelectPFN, a PFN that performs Bayesian variable selection and prediction jointly for sparse nonlinear regression. In one forward pass it returns posterior inclusion probabilities for every variable, and its predictions are computed from sampled subsets only, so excluded variables provably cannot affect them; we show this is exactly what the Bayesian model prescribes. The network is pretrained once on synthetic tasks from a meta-prior built around a group spike-and-slab sparse-input neural network. Column-permutation equivariance combined with random interleaving of original and knockoff columns yields knockoff statistics with exact finite-sample false discovery rate control, even when the meta-prior is wrong. We develop theory for amortized selection: a Bayes-regret bound for the amortized median probability model, a characterization showing that the information needed to detect a variable grows like $(2/λ)\log(1/ω)$ in the prior weight $ω$ of structures that can use it, where $λ\le1$…
PDF arXiv
arXiv stat.MLOct 15
Robust Covariance Estimation for Entry-wise Contaminated and Partially Observed Data via Expectation Maximization Algorithm
Matys Savidan, Mohammed Nabil El Korso
Robust covariance estimation under entry-wise contamination and missing data is challenging in various signal processing applications. Under the Student-$t$ model, observation-level reweighting induces a uniform attenuation across samples, which can lead to cumulative information loss even when most coordinates remain informative. The alternative Student-$t^*$ model enables coordinate-wise robustness, but its E-step is not available in closed form. We extend inference for the Student-$t^*$ model to incomplete observations through a stochastic approximation EM (SAEM) algorithm operating on a reduced latent space: stochastic updates of the coordinate-wise scales are combined with Rao-Blackwellized Gaussian conditional moments, which avoids explicit sampling of latent Gaussian variables while preserving a likelihood-based framework. Experiments on synthetic and real sensor data show a lower Fisher-Rao error than Gaussian, Student-$t$, imputation-based and detect-then-estimate estimators at moderate to high outlier rates under diffuse contamination.
PDF arXiv
arXiv cs.LGOct 15
Classical methods for modern applications: synthetic tabular data generation using lightweight discrete response continuous latent variable model
Bartosz Barzdajn, Mateusz Krzysztoń
In this paper, we present a lightweight algorithm for modelling tabular data containing both continuous and discrete variables. The method uses empirical marginal distributions and a selected quantile function to transform the data into a continuous latent space, where a distribution of choice, e.g. a multivariate Gaussian, can be efficiently estimated and applied. While this approach is closely related to classical well-established machine learning methods, we present a novel adaptation that fits well within the contemporary computational ecosystem. Unlike in the case of deep architectures, it results in a simple implementation with few dependencies, robust training/estimation, as well as efficient sampling. The transformation preserves complex marginal distributions and correlations, providing high-fidelity models, especially for data with unimodal distributions. One of the key characteristics of the proposed approach is its continuous representation of discrete variables. While highly desirable, this introduces a significant challenge, since a single observation corresponds to a region rather than a point in the latent space. We show that this ambiguity naturally leads to a…
PDF arXiv

Papers

Topics in this field

flow matching · flow · generative modeling292 papers40 this week
physics informed neural · neural operators · informed neural networks290 papers31 this week
diffusion priors · posterior sampling · diffusion posterior71 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

Activity · 12 weeks

Coming with more sources: repos

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.

Recent papers

All in Research →

arXiv cs.LGOct 15
The importance of tensor handling and implementation of physics-informed DeepONets
Nahum Banks, Gustavo A. M. de Almeida, Sergio Maldonado
Physics-informed neural networks (PINNs) reduce or eliminate the need for large labelled datasets by enforcing governing equations during training. Physics-informed Deep Operator Networks (PI-DeepONets) extend this approach to learning solution operators, enabling rapid prediction across families of problems. Despite these advantages, we show that their application to higher-dimensional problems remains limited by the computational cost of training. To address this issue, we propose a modified PI-DeepONet architecture that reduces this cost by removing redundancy in how the Branch network processes input functions across collocation points. The proposed formulation is mathematically equivalent to the original architecture but reduces input tensor sizes and the number of Branch network evaluations. We analyse the computational complexity of both architectures and derive theoretical scaling laws for training time with respect to the number of collocation points and input functions. The proposed architecture is evaluated on 1D, 2D, and 3D linear advection problems. In all cases, it achieves comparable predictive accuracy to the original PI-DeepONet while consistently reducing…
PDF arXiv
arXiv cs.AIOct 15
Learnable Splines for Unstructured Neural Operators
Vignesh Gopakumar, Zhuoyuan Wang, Ander Gray, Valentin Duruisseaux, Josh Williams +10
Neural operators approximate solution operators of partial differential equations at a fraction of the cost of numerical solvers. Efficient grid-based architectures such as the Fourier neural operator (FNO) assume uniform rectangular grids, whereas engineering simulations use large, irregular, and locally refined meshes. We introduce L-SplineFNO, a memory-efficient neural operator that couples an FNO backbone with a sparse, learnable tensor-product B-spline transfer between unstructured meshes and a latent Cartesian grid. A local normalised scatter avoids the geometry-dependent solve of analytic spline transformation while preserving compact support, partition of unity, and spline smoothness by construction. The neural operator, conditioned on global design and operating parameters, learns the mapping of geometry to the field solution. Across six three-dimensional engineering benchmarks, L-SplineFNO achieves strong predictive accuracy while requiring substantially less memory and compute than baseline operators, scaling to 26.6 million nodes in training and 46 million in inference on a single A100 GPU. It also supports zero-shot super-resolution and efficient closed-form spatial…
PDF arXiv
arXiv cs.LGOct 15
Learn Only Where You Can't Compute: Structured Corrections to Gaussian Process Posteriors
Tobias Weber, Carla Sagebiel, Philipp Hennig
Physics-informed Gaussian processes (GPs) provide a principled framework for scientific inference by expressing uncertainty over empirical measurements, boundary conditions, and governing equations. However, they struggle to scale to larger datasets and nonlinear information. Conversely, diffusion models excel at learning complex distributions over functions from massive datasets, but struggle in scientific contexts where training data are scarce and physical or observational information must be incorporated. We propose a structured posterior-correction framework that combines a tractable Gaussian reference with data-driven corrections via generative learning. After a mean correction and covariance calibration, a diffusion model learns a targeted correction in a low-dimensional subspace selected to capture discrepancies from the reference, while the complement retains its Gaussian law. Across three scientific inference tasks, these corrections recover hidden modes, improve sample structure, and reduce prediction error relative to the Gaussian reference. Comparisons with full-field generative models reveal task-dependent benefits and tradeoffs between predictive accuracy and…
PDF arXiv
arXiv stat.MLOct 15
SelectPFN: Amortized Bayesian Variable Selection for Sparse Nonlinear Regression with Prior-Data Fitted Networks
Bin Luo
Tabular foundation models based on prior-data fitted networks (PFNs) predict on a new data set in a single forward pass, but they cannot report which inputs their predictions depend on. We propose SelectPFN, a PFN that performs Bayesian variable selection and prediction jointly for sparse nonlinear regression. In one forward pass it returns posterior inclusion probabilities for every variable, and its predictions are computed from sampled subsets only, so excluded variables provably cannot affect them; we show this is exactly what the Bayesian model prescribes. The network is pretrained once on synthetic tasks from a meta-prior built around a group spike-and-slab sparse-input neural network. Column-permutation equivariance combined with random interleaving of original and knockoff columns yields knockoff statistics with exact finite-sample false discovery rate control, even when the meta-prior is wrong. We develop theory for amortized selection: a Bayes-regret bound for the amortized median probability model, a characterization showing that the information needed to detect a variable grows like $(2/λ)\log(1/ω)$ in the prior weight $ω$ of structures that can use it, where $λ\le1$…
PDF arXiv
arXiv stat.MLOct 15
Robust Covariance Estimation for Entry-wise Contaminated and Partially Observed Data via Expectation Maximization Algorithm
Matys Savidan, Mohammed Nabil El Korso
Robust covariance estimation under entry-wise contamination and missing data is challenging in various signal processing applications. Under the Student-$t$ model, observation-level reweighting induces a uniform attenuation across samples, which can lead to cumulative information loss even when most coordinates remain informative. The alternative Student-$t^*$ model enables coordinate-wise robustness, but its E-step is not available in closed form. We extend inference for the Student-$t^*$ model to incomplete observations through a stochastic approximation EM (SAEM) algorithm operating on a reduced latent space: stochastic updates of the coordinate-wise scales are combined with Rao-Blackwellized Gaussian conditional moments, which avoids explicit sampling of latent Gaussian variables while preserving a likelihood-based framework. Experiments on synthetic and real sensor data show a lower Fisher-Rao error than Gaussian, Student-$t$, imputation-based and detect-then-estimate estimators at moderate to high outlier rates under diffuse contamination.
PDF arXiv
arXiv cs.LGOct 15
Classical methods for modern applications: synthetic tabular data generation using lightweight discrete response continuous latent variable model
Bartosz Barzdajn, Mateusz Krzysztoń
In this paper, we present a lightweight algorithm for modelling tabular data containing both continuous and discrete variables. The method uses empirical marginal distributions and a selected quantile function to transform the data into a continuous latent space, where a distribution of choice, e.g. a multivariate Gaussian, can be efficiently estimated and applied. While this approach is closely related to classical well-established machine learning methods, we present a novel adaptation that fits well within the contemporary computational ecosystem. Unlike in the case of deep architectures, it results in a simple implementation with few dependencies, robust training/estimation, as well as efficient sampling. The transformation preserves complex marginal distributions and correlations, providing high-fidelity models, especially for data with unimodal distributions. One of the key characteristics of the proposed approach is its continuous representation of discrete variables. While highly desirable, this introduces a significant challenge, since a single observation corresponds to a region rather than a point in the latent space. We show that this ambiguity naturally leads to a…
PDF arXiv

Fields · AI · updated 48m ago

Flow matching

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

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