Normalizing Flows

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Overview

Normalizing flows learn reversible transformations from a simple distribution, such as a Gaussian, into a complex one, allowing both samples and exact probability densities to be computed. Common approaches include coupling and autoregressive layers, spline transformations, continuous flows, and structured or conditional flows for tasks such as image generation, Bayesian inference, anomaly detection, and control. Current work also combines flows with compact latent representations, flow matching, and physics-informed models, extending their use in scientific simulation and uncertainty estimation. A central tension is balancing expressive, multimodal models against efficient and stable computation, while ensuring that good likelihoods translate into reliable mode coverage and calibrated uncertainty.

Latest papers

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73 papers7 papers in the last four weeks, down 12% on the four weeks before. 0.0% of all new papers.

Jun 29, 2026

Factorizable Normalizing Flows for parameter-dependent density morphing

Normalizing Flows excel at modeling a single fixed density, yet many problems across the sciences, such as high energy physics, instead require modeling how that density deforms as a function of continuous parameters: the strength of a physical effect, a calibration constant, or a source of systematic uncertainty. Learning a separate flow for every parameter configuration quickly becomes intractable, since the number of joint settings grows exponentially with the number of parameters. We introduce Factorizable Normalizing Flows (FNFs), which represent the parameter-dependent density as a fixed, high-fidelity flow for a reference configuration composed with a learnable transformation that is polynomial in the parameters and factorized over them. This structure has a practical consequence: each parameter's effect is learned in isolation, from samples in which that parameter alone is varied. The combined response of many parameters is then recovered by summation at inference, without ever sampling their combinatorially large joint space. On a controlled problem with two interpretable deformations applied jointly to the data, the learned transformation reproduces the true deformations and matches the optimal likelihood, while optional interaction terms capture residual correlations when several parameters vary strongly at once. The resulting model is interpretable, scales linearly with the number of parameters, and keeps the likelihood tractable. This provides a general tool for any inference workflow requiring continuous density morphing, and directly enables the next generation of unbinned likelihood fits in high energy physics.

Davide Valsecchi, Mauro Donegà, Rainer Wallny

Institute for Particle Physics and Astrophysics, D-PHYS, ETH Zurich, Zurich, Switzerland

Normalizing FlowsConditional Distribution Estimation
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Jun 29, 2026

Simplifying Flow Matching Transformations with Low-Rank Mixture Models

Normalizing flows are powerful generative models that learn an invertible mapping between complex data distributions and simple latent distributions, typically a standard normal density. However, this choice of latent density can impose unnecessary complexity on the learned flow transformation due to the topological mismatch between the latent and data densities, leading to slower training and suboptimal performance. In this work, we propose using mixtures of probabilistic principal component analyzers (MPPCA) as the latent density for normalizing flows. We simplify the learned flow transformation by learning a latent distribution that more closely aligns with the data distribution in terms of KL divergence, thus enabling faster convergence and improved generative performance. Critically, MPPCA models can be fit quickly and cheaply using the expectation-maximization algorithm, making them a practical choice for initializing latent distributions even in high-dimensional generative tasks. We validate our method on both tabular and image datasets, demonstrating consistent gains in training efficiency and generation quality compared to baselines.

Liam A. Kruse, Houjun Liu, Alexandros E. Tzikas +2

Stanford Intelligent Systems Laboratory in the Department of Aeronautics and Astronautics at Stanford University, Stanford, CA 94305, USA · Industrial and Systems Engineering Department at King Fahd University of Petroleum and Minerals (KFUPM), Dhahran, 31261, KSA

Normalizing FlowsFlow-Based Generative Modeling
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Jun 25, 2026

DeCoFlow: Structural Decomposition of Normalizing Flows for Continual Anomaly Detection

In industrial environments, new product categories arrive sequentially, requiring continual anomaly detection without access to past data. Normalizing Flows (NFs) provide exact density estimation but suffer from catastrophic forgetting as parameter updates across tasks distort the density manifold. While parameter isolation can prevent interference, it must preserve the strict invertibility and Jacobian validity of NFs. To satisfy these requirements, we exploit the inherent property that affine coupling layers maintain transformation validity regardless of subnet parameterization. Based on this, we propose DeCoFlow, which decomposes subnets into a frozen universal base and task-specific low-rank adapters to isolate updates. We further introduce Task-Specific Alignment, Auxiliary Coupling Layers, and Tail-Aware Loss to compensate for frozen-base rigidity. DeCoFlow achieves state-of-the-art image-level AUROCs of 98.40% on MVTec-AD and 93.00% on VisA, while maintaining parameter-level zero forgetting (0.00% FM under correct routing) with only 2.27M parameters per task.

Hun Im, Jungi Lee, Subeen Cha +1

Seoul National University, Seoul, Korea

Normalizing FlowsContinual Anomaly DetectionIndustrial Anomaly DetectionAnomaly Detection
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Jun 24, 2026

MIMFlow: Integrating Masked Image Modeling with Normalizing Flows for End-to-End Image Generation

Normalizing Flows (NFs) are powerful generative models capable of exact density estimation and sampling. However, their strict invertibility often forces the model to exhaust its capacity on low-level pixel details, hindering the capture of high-level semantic structures. While Masked Image Modeling (MIM) has excelled in representation learning, its integration into generative pipelines has remained largely modular and disjointed. In this paper, we propose MIMFlow, a unified end-to-end framework that jointly optimizes latent semantics, pixel reconstruction, and generative flow. By employing a VAE encoder to infer semantic latent from masked images, MIMFlow achieves a principled decoupling of the generative task: the Normalizing Flow focuses on modeling a simplified, low-frequency semantic manifold, while a specialized decoder handles high-frequency synthesis. This design effectively resolves the inherent capacity bottleneck of NFs, allowing the model to prioritize global structural coherence over redundant noise. Empirical results on ImageNet 256×\times256 show that MIMFlow-L reaches 71.3% linear probing accuracy and an FID of 2.50. Despite using only 128 tokens (50% fewer than standard models), it yields a 32.8% performance gain over similar-scale NF baselines. Our code is available at https://github.com/MCG-NJU/MIMFlow.

Yang Chen, Xiaowei Xu, Shuai Wang +4

State Key Laboratory for Novel Software Technology, Nanjing University · Alibaba Group · Shanghai AI Lab

Normalizing FlowsImage GenerationFlow-Based Generative ModelingMasked Image Modeling
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Jun 13, 2026

The Reverse Telescoping Coordinate System for Positive Definite Matrices: Geometry, Computation, and Generative Modeling

We design a new unconstrained coordinate system where a p×pp\times p symmetric positive definite (SPD) matrix ΘΘ is represented by a reverse telescoping map Θ(x)=RT(x)Θ(x)=\rm{RT}(x), with x=(v,d,r)∈R×R(p−1)×Rp(p−1)/2x=(v,d,r)\in\mathbb{R}\times\mathbb{R}^{(p-1)}\times\mathbb{R}^{p(p-1)/2}, representing respectively the log volume or log determinant; and the shape, as encoded by log relative diagonal scales and partial covariances among the nodes. This construction results in important properties not available in other charts, e.g., matrix logarithm, such as Jacobian depending on only the log-determinant. A useful feature of our construction is xx contains a lossless symbolic representation of both the matrix and its inverse. Many important computations involving a matrix and its inverse can be performed in O(p2)O(p^2) in the transformed domain, while it is the rendering of results in matrix forms (on demand) that must incur an O(p3)O(p^3) cost. Moreover, two unit-determinant matrices in the transformed domain can be joined by a straight line with pathwise unit determinant. For generative modeling, this allows designing a split volume-shape flow model trained by conditional flow matching for transporting the shape over the unit-determinant path, with a separate one-dimensional flow for transporting the volume or the determinant. The forbidding SPD constraint, tamed thus into a powerful guiding force, leads to the surprising insight that it is in some sense easier to design a volume-normalized shape flow for SPD compared to the unconstrained Rp×p\mathbb{R}^{p\times p}, with no intrinsic notion of volume to aid normalization, unlike the determinant of SPD matrices. We apply our construction for up to p=200p=200 in generative modeling of SPD matrices on a difficult synthetic bimodal target, and in generating brain connectivity networks by models trained on fMRI data; as well as in intrinsic diffusion on the SPD manifold.

Anindya Bhadra

Department of Statistics, Purdue University

Normalizing FlowsGenerative Modeling
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Jun 8, 2026

Learning the Universe: Posterior Reliability of Neural Generative Models in High-Dimensional Field-Level Inference of Cosmic Initial Conditions

Accurate posterior estimation is central to scientific inference, as uncertainties determine what can be reliably learned from observational data. While Markov chain Monte Carlo methods provide asymptotic convergence guarantees, they are computationally demanding in high-dimensional settings. Neural network-based generative models for entire discretized 3D fields enable fast amortized inference but often lack convergence guarantees and principled accuracy assessment. Using Hamiltonian Monte Carlo to obtain reference posterior samples, we conduct a controlled field-level evaluation of an implicit generative model (Stochastic Interpolants) and an explicit likelihood-based model (GLOW normalizing flows). This comparison, unavailable in typical applications, enables the detection of posterior geometry failures that standard metrics cannot capture. As a case study, we consider the cosmological inverse problem of inferring cosmic initial conditions from present-day large-scale structure. To match the precision of modern cosmological data, this problem increasingly relies on complex, non-linear, and non-differentiable simulators, which are incompatible with gradient-based inference frameworks. Generative models offer a route to address these challenges, provided their inferred posteriors are reliable. In this work, we show that matching posterior means, marginal distributions, or achieving high cross-correlation does not imply correct uncertainty structure, as revealed by posterior variance fields and sample-based evaluations. Through this work, we aim to raise awareness of the challenges of uncertainty estimation in high-dimensional field-level settings, highlighting the importance of careful design and validation of neural generative approaches for scientific applications.

Ludvig Doeser, Jens Jasche

The Oskar Klein Centre, Department of Physics, Stockholm University, AlbaNova University Centre, SE 106 91 Stockholm, Sweden · Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA

Simulation-Based InferenceNormalizing FlowsUncertainty QuantificationBayesian Inverse ProblemsConditional Generative Modeling
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Jun 7, 2026

Active Flow Expansion for Out-of-Distribution Discovery: from Theory to Molecules

Standard flow and diffusion pre-training matches the distribution of available data (e.g., molecules), which often covers only a small fraction of the valid design space. In generative discovery, however, one aims to sample valid new-to-nature designs, assigned negligible probability under, and thus inaccessible to, standard models fitted to the observed data. To overcome this limitation, we depart from data distribution matching and view a generative model through its generable set: the region it covers with non-negligible probability. This allows to introduce a new learning principle for out-of-distribution flow modeling: enlarging a model's generable set to increase coverage of the valid design space. We propose Active Flow Expansion (ActFlow), a continued pre-training method that employs verifier feedback to expand a pre-trained model over new valid regions by iteratively adapting to synthetic data generated through active exploration in the learned flow representation. Theoretically, we establish to our knowledge first-of-their-kind statistical learning guarantees for out-of-distribution flow modeling, analyzing generable set expansion as a local-to-global reachability process over a learned representation. Empirically, we assess ActFlow with suitable out-of-distribution generative modeling metrics across small organic molecules, mid-sized drug-like molecules, therapeutic peptides, and protein sequence design tasks. Results show that ActFlow expands valid coverage far beyond the region modeled by the initial pre-trained model, significantly outperforming widely adopted synthetic flow pre-training methods.

Riccardo De Santi, Bruce Lee, Cristian Perez Jensen +6

ETH Zurich ETH AI Center · University of Pennsylvania · Caltech FutureHouse

Synthetic Data PretrainingNormalizing FlowsGenerative ModelingMolecular GenerationFlow-Based Generative Modeling
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Jun 4, 2026

Latent Reasoning with Normalizing Flows

Large language models often improve reasoning by generating explicit chain-of-thought (CoT), demonstrating the importance of intermediate computation. However, textual CoT forces this computation through a discrete, serial, and communication-oriented token stream: each reasoning step must be verbalized before the model can proceed, even when the underlying update is semantic, uncertain, or only partially formed. Latent reasoning offers a higher-bandwidth alternative by performing intermediate computation in compact continuous states before committing to text. Yet existing latent-reasoning methods often sacrifice key advantages that make CoT effective in autoregressive language models, including native left-to-right generation, probabilistic sampling, compatibility with KV-cache decoding, and tractable likelihood estimation. We propose NF-CoT, a latent reasoning framework that preserves these advantages by modeling continuous thoughts with normalizing flows. NF-CoT instantiates a TARFlow-style normalizing flow inside the LLM backbone, defining a tractable probability model over compact continuous thoughts distilled from explicit CoT. Continuous-thought positions are generated by an NF head, while text positions are generated by the standard LM head within the same causal stream. This design provides exact likelihoods for latent thoughts, enables probabilistic left-to-right decoding with the original KV cache, and supports direct policy-gradient optimization in the latent reasoning space. On code-generation benchmarks, NF-CoT improves pass rates over explicit-CoT and prior latent-reasoning baselines while substantially reducing intermediate-reasoning cost.

Guancheng Tu, Xiangjun Fu, Suhao Yu +5

University of Pennsylvania · UC San Diego · Meta

Normalizing FlowsContinuous Latent ReasoningEfficient Language Model InferenceLatent Reasoning
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Jun 3, 2026

Neural Galerkin Normalizing Flows for Bayesian Inference of Diffusions with Inaccessible Boundaries

One of the primary challenges in Bayesian inference on the parameters of a diffusion model from discrete observations is the unavailability of an analytical expression for the transition density function between consecutive observation times, which is needed to derive the likelihood function. Extending previous studies that solve Fokker-Planck (FP) type partial differential equations with Normalizing Flows, we propose a new Normalizing Flow architecture to learn the transition density function of the diffusion process between two observation times. We do so by solving in a Neural Galerkin framework the associated FP equation with a Dirac mass as initial condition, over a specified training distribution of the initial datum and the coefficients of the diffusion. We specifically focus on processes whose diffusion matrix vanishes in certain inaccessible boundary regions, such as Stochastic Volatility models that satisfy a Feller condition. The product of the obtained transition densities evaluated along the observed trajectory approximates the likelihood function, thereby enabling cheap posterior sampling via Markov chain Monte Carlo (MCMC). After the offline training phase, inference becomes significantly more efficient, as it avoids the need to solve the FP equation in real time for each parameter proposed by the MCMC sampler or to rely on other likelihood-free methods for Bayesian inference that involve repeated simulation of diffusion bridges.

Riccardo Saporiti, Fabio Nobile

CSQI École Polytechnique Fédérale de Lausanne Lausanne, Switzerland 1015

Normalizing FlowsNeural PDE SolversBayesian Inference
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May 31, 2026

Non-Vacuous Certification of Transport MCMC via Oscillation-Controlled Normalizing Flows

Transport MCMC trains a normalizing flow to precondition Metropolis--Hastings proposals, achieving high empirical efficiency on challenging posteriors; yet no prior work produces a numerically non-vacuous, rigorous spectral-gap bound for such samplers. We establish the first such bounds. For independence MH on the banana family we certify (γ^\ast = 0.828) at (D = 2) (covering in the original space) and (γ^\ast \ge 7.6\times 10^{-4}) at (D = 5) (covering in an analytically unwarped Gaussian space with a grid-certified gradient bound under the stated numerical Lipschitz certification), both rigorous at 95% confidence. The framework rests on three pillars: (i) spectral normalization with reduced scale clips constrains the flow Lipschitz constant from (10^{47}) to (10^4); (ii) a coverage-based empirical oscillation bound replaces the vacuous analytical bound with a data-dependent certificate; and (iii) oscillation-regularised training cuts the empirical oscillation by 60--90% at no cost to density fit, extending practical certificates through (D = 20) ((γ^\ast \ge 1.7\times 10^{-4})). Tests on four further targets (Gaussian mixture, shear-building, Neal's funnel, Bayesian logistic regression) identify three precise barriers: boundary curvature, target stiffness, and tail-coverage mismatch. An affine-vs-spline comparison shows that simpler architectures yield tighter certificates at identical NLL, inverting the usual expressiveness hierarchy.

Jun Hu

China Sanya Science and Education Innovation Park, Wuhan University of Technology, Sanya 572025, China · School of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, China

Markov Chain Monte CarloNormalizing FlowsBayesian InferenceConvergence Analysis
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May 18, 2026

SRC-Flow: Compact Semantic Representations Enable Normalizing Flows for Image Generation

Normalizing flows (NFs) provide exact likelihoods and deterministic invertible sampling, but have historically lagged behind diffusion models for large-scale image generation. We identify a key obstacle: NFs are required to learn a single invertible transport over the full ambient space, making them highly sensitive to high-dimensional representations. This leads to a semantic-capacity mismatch in modern visual representation spaces, where semantic information is compact but encoded in overcomplete features. We propose SRC-Flow, which introduces a Semantic Representation Compressor (SRC) to compact high-dimensional RAE features into a low-dimensional semantic space before flow modeling and preserve reconstruction through the frozen RAE decoder. This compact space reduces the modeling burden of NFs and enables effective likelihood-based generation in semantic representation space. We further adopt constant noise regularization tailored to the fixed unconditional bijection learned by flows. On ImageNet 256×256256 \times 256 and 512×512512 \times 512, SRC-Flow achieves state-of-the-art generation quality among normalizing flow methods, with gFID scores of 1.65 and 2.07 under classifier-free guidance, while retaining exact likelihood computation in the compact semantic representation space and deterministic invertible sampling at the flow level. Codes and models will be available at https://github.com/longtaojiang/SRC-Flow.

Longtao Jiang, Jianmin Bao, Zhendong Wang +4

University of Science and Technology of China · Kling Team, Kuaishou Technology

Normalizing FlowsEmbedding CompressionFlow-Based Generative Modeling
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May 11, 2026

Operator-Guided Model Reduction for Generative Sampling in Lattice Field Theory

Neural generative samplers for lattice field theory can be costly to train and evaluate. When they miss modes or assign them incorrect relative weights, biased observables do not reveal which collective variables are responsible. We project a trained flow-matching velocity onto vector fields built from lattice operators and Fourier modes. In two-dimensional lattice φ4φ^4 theory, the projection separates changes in the overall magnetization from the lowest nonzero-momentum fluctuations and guides an explicit invertible proposal that treats them separately. Allowing the amplitude of the lowest nonzero-momentum fluctuations to depend on the magnetization improves the overlap between the proposal and target distributions, while the same two-parameter modification at higher momenta gives smaller improvements. The Metropolis--Hastings correction defines a Markov chain with the target Boltzmann distribution as its stationary law, and the normalized proposal density yields finite-volume partition-function estimates consistent with an independent HMC calculation. At the larger tested volume, the overlap between the proposal and target distributions deteriorates substantially, limiting the range over which the same parameterization remains effective.

Moxian Qian

Johannes Gutenberg University Mainz, 55128 Mainz, Germany

Flow MatchingMarkov Chain Monte CarloNormalizing FlowsHigh-Energy PhysicsLattice Field Theory
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May 11, 2026

Language Modeling with Hyperspherical Flows

Discrete Diffusion Language Models progressed rapidly as an alternative to autoregressive (AR) models, motivated by their parallel generation abilities. However, for tractability, discrete diffusion models sample from a factorized distribution, which is less expressive than AR. Recent Flow Language Models (FLMs) apply continuous flows to language, transporting noise to data with a deterministic ODE that avoids factorized sampling. FLMs operate on one-hot vectors whose dimension scales with the vocabulary size, making FLMs costly to train. Moreover, since all distinct one-hot embeddings are equidistant in ℓ2\ell_2, adding Gaussian noise does not have a clear semantic interpretation (unlike images, where Gaussian noise progressively degrades structure). We introduce S\mathbb{S}-FLM, a latent FLM in the hypersphere. S\mathbb{S}-FLM generates sequences by rotating vectors in Sd−1\mathbb{S}^{d-1} along a velocity field learned with cross-entropy, avoiding the overhead of materializing one-hot vectors. Previous FLMs match AR in Generative Perplexity (Gen.\ PPL), but samples with high likelihood are not necessarily correct in verifiable domains such as math and code. S\mathbb{S}-FLM substantially improves continuous flow language models on large-vocabulary reasoning and closes the gap to masked diffusion under standard-temperature sampling (T=1T=1), while a gap remains under optimized low-temperature (T=0.1T=0.1) decoding.

Justin Deschenaux, Caglar Gulcehre

EPFL, Lausanne, Switzerland

Normalizing FlowsContinuous Normalizing FlowsLanguage ModelingFlow-Based Generative Modeling
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May 8, 2026

Normalizing Trajectory Models

Diffusion-based models decompose sampling into many small Gaussian denoising steps -- an assumption that breaks down when generation is compressed to a few coarse transitions. Existing few-step methods address this through distillation, consistency training, or adversarial objectives, but sacrifice the likelihood framework in the process. We introduce Normalizing Trajectory Models (NTM), which models each reverse step as an expressive conditional normalizing flow with exact likelihood training. Architecturally, NTM combines shallow invertible blocks within each step with a deep parallel predictor across the trajectory, forming an end-to-end network trainable from scratch or initializable from pretrained flow-matching models. Its exact trajectory likelihood further enables self-distillation: a lightweight denoiser trained on the model's own score produces high-quality samples in four steps. On text-to-image benchmarks, NTM matches or outperforms strong image generation baselines in just four sampling steps while uniquely retaining exact likelihood over the generative trajectory.

Jiatao Gu, Tianrong Chen, Ying Shen +3

Apple · UIUC

Normalizing FlowsDiffusion Model DistillationFew-Step Diffusion SamplingDiffusion-Based Image GenerationFlow-Based Generative ModelingDiffusion Models
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May 7, 2026

Expressivity of Bi-Lipschitz Normalizing Flows: A Score-Based Diffusion Perspective

Many normalizing flow architectures impose regularity constraints, yet their distributional approximation properties are not fully characterized. We study the expressivity of bi-Lipschitz normalizing flows through the lens of score-based diffusion models. For the probability flow ODE of a variance-preserving diffusion, Lipschitz regularity of the score induces a flow of bi-Lipschitz diffeomorphic transport maps. This ODE bridge allows us to analyze the distributional approximation power of bi-Lipschitz normalizing flows and, conversely, derive deterministic convergence guarantees for diffusion-based transport. Our key idea is to use the probability flow ODE to link regularity of the score to regularity of the induced transport maps. We verify score regularity for broad target densities, including compactly supported densities, Gaussian convolutions of compactly supported measures and finite Gaussian mixtures. We obtain a universal distributional approximation result: Gaussian pullbacks induced by bi-Lipschitz variance-preserving transport maps are L1L^1-dense among all probability densities. For Gaussian convolution targets, we further obtain convergence in Kullback-Leibler divergence without early stopping.

Meira Iske, Carola-Bibiane Schönlieb

Center for Industrial Mathematics, University of Bremen, Bremen, Germany · Department of Theoretical Physics and Applied Mathematics, University of Cambridge, Cambridge, U.K.

Normalizing FlowsDiffusion Models
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May 6, 2026

Joint Treatment Effect Estimation from Incomplete Healthcare Data: Temporal Causal Normalizing Flows with LLM-driven Evolutionary MNAR Imputation

Target trial emulation (TTE) enables causal questions to be studied with observational data when randomized controlled trials (RCTs) are infeasible. Yet treatment-effect methods often address causal estimation, missingness, and temporal structure separately, limiting their robustness in electronic health records (EHRs), where time-varying confounding and missing-not-at-random (MNAR) biomarkers can reach 50%--80%. We propose a two-stage pipeline for treatment effect estimation from incomplete longitudinal EHRs. First, CausalFlow-T, a directed acyclic graph (DAG)-constrained normalizing flow with long short-term memory (LSTM)-encoded patient history, performs exact invertible counterfactual inference, avoiding approximation errors from variational inference and separating confounding through explicit causal structure. Ablations on four synthetic and one semi-synthetic benchmark with known counterfactuals show that DAG constraints and exact inference address distinct failure modes: neither compensates for the other. Second, because CausalFlow-T requires completed inputs, we introduce an LLM-driven evolutionary imputer that proposes executable imputation operators rather than individual entries, and evaluate it with three large language model (LLM) backends, including two open-source models. Across 30%--80% MNAR missingness, this imputer achieves the best pooled rank over biomarker and causal metrics, leading in point-wise accuracy and temporal extrapolation while preserving average treatment effect (ATE) recovery as statistical baselines degrade. On Swiss primary-care EHRs from adults with type 2 diabetes initiating a GLP-1 receptor agonist or SGLT-2 inhibitor, the pipeline estimates a per-protocol weight-loss difference of -0.98 kg [95% CI -1.01, -0.96] favoring GLP-1 receptor agonists, consistent with randomized evidence and obtained from realistically incomplete real-world EHRs.

Olivia Jullian Parra, Sara Zoccheddu, David Catalan Cerezo +7

Department of Mathematical Modeling and Machine Learning University of Zürich Winterthurerstrasse 190, Zürich 8057 · Department of Mathematics ETH Zürich Rämistrasse 101, Zürich 8092 · Epidemiology, Biostatistics & Prevention Institute (EBPI), University of Zürich Hirschengraben 84, Zürich 8001 +2

Causal Effect EstimationNormalizing FlowsIncomplete Data ImputationHealthcareLongitudinal EHR Modeling
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May 3, 2026

Robust Conditional Conformal Prediction via Branched Normalizing Flow

Conformal prediction (CP) constructs prediction sets with marginal coverage guarantees under the assumption that the calibration and test distributions are identical. However, under distribution shift, existing approaches primarily align marginal conformal score distributions, which is sufficient to preserve marginal coverage but does not control the conditional coverage error at individual test inputs. As a consequence, CP can remain unreliable in regions where the conditional score distributions are mismatched. In this work, we bound the conditional invalidity of CP under distribution shift in terms of the Wasserstein distance between the calibration and test distributions. This result highlights the role of invertible transport in mitigating conditional coverage degradation. Motivated by this insight, we introduce Branched Normalizing Flow (BNF), a two-branch architecture that normalizes a test input to the calibration distribution and transforms the prediction set of the normalized input back to the test distribution while preserving conditional guarantees. Empirically, BNF consistently improves conditional coverage robustness on nine datasets across a wide range of confidence levels.

Rui Xu, Xingyuan Chen, Wenxing Huang +4

Information Hub, The Hong Kong University of Science and Technology (Guangzhou) · The Second Affiliated Hospital of Guangzhou Medical University

Distribution Shift RobustnessNormalizing FlowsGroup-Conditional Conformal PredictionConformal Prediction
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Apr 21, 2026

Normalizing Flows with Iterative Denoising

Normalizing Flows (NFs) are a classical family of likelihood-based methods that have received revived attention. Recent efforts such as TARFlow have shown that NFs are capable of achieving promising performance on image modeling tasks, making them viable alternatives to other methods such as diffusion models. In this work, we further advance the state of Normalizing Flow generative models by introducing iterative TARFlow (iTARFlow). Unlike diffusion models, iTARFlow maintains a fully end-to-end, likelihood-based objective during training. During sampling, it performs autoregressive generation followed by an iterative denoising procedure inspired by diffusion-style methods. Through extensive experiments, we show that iTARFlow achieves competitive performance across ImageNet resolutions of 64, 128, and 256 pixels, demonstrating its potential as a strong generative model and advancing the frontier of Normalizing Flows. In addition, we analyze the characteristic artifacts produced by iTARFlow, offering insights that may shed light on future improvements. Code is available at https://github.com/apple/ml-itarflow.

Tianrong Chen, Jiatao Gu, David Berthelot +2

Apple Machine Learning Research.

Normalizing FlowsAutoregressive Image GenerationFlow-Based Generative Modeling
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Apr 21, 2026

Neural posterior estimation of the neutrino direction in IceCube using transformer-encoded normalizing flows on the sphere

IceCube is a cubic-kilometer-scale neutrino detector located at the geographic South Pole. A precise directional reconstruction of IceCube neutrinos is vital for associations with astronomical objects. In this context, we discuss neural posterior estimation of the neutrino direction via a transformer encoder that maps to a normalizing flow on the 2-sphere. It achieves a new state-of-the-art angular resolution for the two main event morphologies in IceCube - tracks and showers - while being significantly faster than traditional B-spline-based likelihood reconstructions. All-sky scans can be performed within seconds rather than hours, and take constant computation time, regardless of whether the posterior extent is arc-minutes or spans the whole sky. We utilize a combination of C2C^2-smooth rational-quadratic splines, scale transformations and rotations to define a novel spherical normalizing-flow distribution whose parameters are predicted as a whole as the output of the transformer encoder. We test several structural choices diverting from the vanilla transformer architecture. In particular, we find dual residual streams, nonlinear QKV projection and a separate class token with its own cross-attention processing to boost test-time performance. The angular resolution for both showers and tracks improves substantially over the whole trained energy range from 100 GeV to 100 PeV. At 100 TeV deposited energy, for example, the median angular resolution improves by a factor of 1.31.3 for throughgoing tracks, by a factor of 1.71.7 for showers and by a factor of 2.52.5 for starting tracks compared to state-of-the art likelihood reconstructions based on B-splines. While previous machine-learning (ML) efforts have managed to obtain competitive shower resolutions, this is the first time an ML-based method outperforms likelihood-based muon reconstructions above 100 GeV.

R. Abbasi, M. Ackermann, J. Adams +411

Normalizing FlowsTransformerDirection-of-Arrival EstimationHigh-Energy Physics
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Apr 20, 2026

Complex normalizing flows can almost be information Kähler-Ricci flows

We develop interconnections between the complex normalizing flow for data drawn from Borel probability measures on the twofold realification of the complex manifold and a nonlinear flow nearly Kähler-Ricci. The complex normalizing flow relates the initial and target realified densities under the complex change of variables, necessitating the log determinant of the ensemble of Wirtinger Jacobians. The Ricci curvature of a Kähler manifold is the second order mixed Wirtinger partial derivative of the log of the local density of the volume form. Therefore, we reconcile these two facts by drawing forth the connection that the log determinant used in the complex normalizing flow matches a Ricci curvature term under differentiation and conditions. The log density under the normalizing flow is kindred to a spatial information metric under an augmented Jacobian and a Bayesian perspective to the parameter, thus under the continuum limit the log likelihood matches a Fisher metric, or more closely a Kähler cross-entropy Hessian. This recovers a Kähler-Ricci flow variation up to a time derivative and expectation, or an average-valued Kähler-Einstein flow. Using this framework, we establish other relevant results, attempting to bridge the statistical and ordinary behaviors of the complex normalizing flow to the geometric features of our derived Kähler flow.

Andrew Gracyk

Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States

Information GeometryNormalizing Flows
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Mar 31, 2026

Lévy-Flow Models: Heavy-Tail-Aware Normalizing Flows for Financial Risk Management

We introduce Lévy-Flows, a class of normalizing flow models that replace the standard Gaussian base distribution with Lévy process-based distributions, specifically Variance Gamma (VG) and Normal-Inverse Gaussian (NIG). These distributions naturally capture heavy-tailed behavior while preserving exact likelihood evaluation and efficient reparameterized sampling. We establish theoretical guarantees on tail behavior, showing that for regularly varying bases the tail index is preserved under asymptotically linear flow transformations, and that identity-tail Neural Spline Flow architectures preserve the base distribution's tail shape exactly outside the transformation region. Empirically, we evaluate on S&P 500 daily returns and additional assets, demonstrating substantial improvements in density estimation and risk calibration. VG-based flows reduce test negative log-likelihood by 69% relative to Gaussian flows and achieve exact 95% VaR calibration, while NIG-based flows provide the most accurate Expected Shortfall estimates. These results show that incorporating Lévy process structure into normalizing flows yields significant gains in modeling heavy-tailed data, with applications to financial risk management.

Rachid Drissi

Normalizing FlowsHeavy-Tailed DistributionsQuantitative Finance
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Mar 26, 2026

Calorimeter Shower Superresolution with Conditional Normalizing Flows: Implementation and Statistical Evaluation

In High Energy Physics, detailed calorimeter simulations and reconstructions are essential for accurate energy measurements and particle identification, but their high granularity makes them computationally expensive. Developing data-driven techniques capable of recovering fine-grained information from coarser readouts, a task known as calorimeter superresolution, offers a promising way to reduce both computational and hardware costs while preserving detector performance. This thesis investigates whether a generative model originally designed for fast simulation can be effectively applied to calorimeter superresolution. Specifically, the model proposed in arXiv:2308.11700 is re-implemented independently and trained on the CaloChallenge 2022 dataset based on the Geant4 Par04 calorimeter geometry. Finally, the model's performance is assessed through a rigorous statistical evaluation framework, following the methodology introduced in arXiv:2409.16336, to quantitatively test its ability to reproduce the reference distributions.

Andrea Cosso

Università degli Studi di Genova Scuola di Scienze Matematiche, Fisiche e Naturali Laurea Magistrale in Fisica

High-Energy PhysicsImage Super-ResolutionConditional Generative ModelingNormalizing Flows
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Mar 23, 2026

Motion Semantics Guided Normalizing Flow for Privacy-Preserving Video Anomaly Detection

As embodied perception systems increasingly bridge digital and physical realms in interactive multimedia applications, the need for privacy-preserving approaches to understand human activities in physical environments has become paramount. Video anomaly detection is a critical task in such embodied multimedia systems for intelligent surveillance and forensic analysis. Skeleton-based approaches have emerged as a privacy-preserving alternative that processes physical world information through abstract human pose representations while discarding sensitive visual attributes such as identity and facial features. However, existing skeleton-based methods predominantly model continuous motion trajectories in a monolithic manner, failing to capture the hierarchical nature of human activities composed of discrete semantic primitives and fine-grained kinematic details, which leads to reduced discriminability when anomalies manifest at different abstraction levels. In this regard, we propose Motion Semantics Guided Normalizing Flow (MSG-Flow) that decomposes skeleton-based VAD into hierarchical motion semantics modeling. It employs vector quantized variational auto-encoder to discretize continuous motion into interpretable primitives, an autoregressive Transformer to model semantic-level temporal dependencies, and a conditional normalizing flow to capture detail-level pose variations. Extensive experiments on benchmarks (HR-ShanghaiTech & HR-UBnormal) demonstrate that MSG-Flow achieves state-of-the-art performance with 88.1% and 75.8% AUC respectively.

Yang Liu, Boan Chen, Yuanyuan Meng +5

Tongji Univ. · SJTU · Shanghai Creative Studies Institute +4

Video Anomaly DetectionInterpretable Anomaly DetectionNormalizing Flows
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Mar 19, 2026

Neural Galerkin Normalizing Flow for Transition Probability Density Functions of Diffusion Models

We propose a new Neural Galerkin Normalizing Flow framework to approximate the transition probability density function of a diffusion process by solving the corresponding Fokker-Planck equation with an atomic initial distribution, parametrically with respect to the location of the initial mass. By using Normalizing Flows, we look for the solution as a transformation of the transition probability density function of a reference stochastic process, ensuring that our approximation is structure-preserving and automatically satisfies positivity and mass conservation constraints. By extending Neural Galerkin schemes to the context of Normalizing Flows, we derive a system of ODEs for the time evolution of the Normalizing Flow's parameters. Adaptive sampling routines are used to evaluate the Fokker-Planck residual in meaningful locations, which is of vital importance to address high-dimensional PDEs. Numerical results show that this strategy captures key features of the true solution and enforces the causal relationship between the initial datum and the density function at subsequent times. After completing an offline training phase, online evaluation becomes significantly more cost-effective than solving the PDE from scratch. The proposed method serves as a promising surrogate model, which could be deployed in many-query problems associated with stochastic differential equations, like Bayesian inference, simulation, and diffusion bridge generation.

Riccardo Saporiti, Fabio Nobile

CSQI, École Polytechnique Féderale de Lausanne, Switzerland

Normalizing FlowsNeural PDE SolversPDE SolvingPDE Surrogate ModelingFokker–Planck Equation
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Mar 18, 2026

Tackling the Sign Problem in the Doped Hubbard Model with Normalizing Flows

The Hubbard model at finite chemical potential is a cornerstone for understanding doped correlated systems, but simulations are severely limited by the sign problem. In the auxiliary-field formulation, the spin basis mitigates the sign problem, yet severe ergodicity issues have limited its use. We extend recent advances with normalizing flows at half-filling to finite chemical potential by introducing an annealing scheme enabling ergodic sampling. Compared to state-of-the-art hybrid Monte Carlo in the charge basis, our approach accurately reproduces exact diagonalization results while reducing statistical uncertainties by an order of magnitude, opening a new path for simulations of doped correlated systems.

Dominic Schuh, Lena Funcke, Janik Kreit +2

Transdisciplinary Research Area (TRA) Matter, University of Bonn, Bonn, Germany · Helmholtz Institute for Radiation and Nuclear Physics (HISKP), University of Bonn, Bonn, Germany · Institute for Advanced Simulation 4 (IAS-4), Forschungszentrum Jülich, Jülich, Germany

Quantum Many-Body PhysicsMarkov Chain Monte CarloNormalizing Flows
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Mar 15, 2026

Sampling Boltzmann distributions via normalizing flow approximation of transport maps

In a celebrated paper \cite{noe2019boltzmann}, Noé, Olsson, Köhler and Wu introduced an efficient method for sampling high-dimensional Boltzmann distributions arising in molecular dynamics via normalizing flow approximation of transport maps. Here, we place this approach on a firm mathematical foundation. We prove the existence of a normalizing flow between the reference measure and the true Boltzmann distribution up to an arbitrarily small error in the Wasserstein distance. This result covers general Boltzmann distributions from molecular dynamics, which have low regularity due to the presence of interatomic Coulomb and Lennard-Jones interactions. The proof is based on a rigorous construction of the Moser transport map for low-regularity endpoint densities and approximation theorems for neural networks in Sobolev spaces. Numerical simulations for a simple model system and for the alanine dipeptide molecule confirm that the true and generated distributions are close in the Wasserstein distance. Moreover we observe that the RealNVP architecture does not just successfully capture the equilibrium Boltzmann distribution but also the metastable dynamics.

Zia Ur Rehman, Gero Friesecke

Department of Mathematics, School of Computation, Information and Technology, Technical University of Munich, Boltzmannstrasse 3, 85748 Garching, Germany

Boltzmann SamplingNormalizing FlowsMolecular Dynamics SimulationOptimal Transport
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Mar 12, 2026

Anomaly detection in time-series via inductive biases in the latent space of conditional normalizing flows

Deep generative models for anomaly detection in multivariate time-series are typically trained by maximizing observed data likelihood. However, likelihood in observation space measures marginal density rather than conformity to structured temporal dynamics, and therefore can assign high probability to anomalous or out-of-distribution samples. We address this structural limitation by relocating the notion of anomaly to a prescribed latent space. We introduce explicit inductive biases in conditional normalizing flows, modeling time-series observations within a discrete-time state-space framework that constrains latent representations to evolve according to prescribed temporal dynamics. Under this formulation, expected behavior corresponds to compliance with a specified distribution over latent trajectories, while anomalies are defined as violations of these dynamics. Anomaly detection is consequently reformulated as a statistically grounded compliance test, such that observations are mapped to latent space and evaluated via goodness-of-fit tests against the prescribed latent evolution. This yields a principled decision rule that remains effective even in regions of high observation likelihood. Experiments on synthetic and real-world time-series demonstrate reliable detection of anomalies in frequency, amplitude, and observation noise, while providing interpretable diagnostics of model compliance.

David Baumgartner, Eliezer de Souza da Silva, Iñigo Urteaga

Department of Computer Science, Norwegian University of Science and Technology, Norway · BCAM — Basque Center for Applied Mathematics, Spain · Department of Informatics Engineering, University of Coimbra, Portugal +2

Normalizing FlowsLatent Dynamics ModelingTime-Series Anomaly Detection
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Mar 12, 2026

NFPO: Stabilized Policy Optimization of Normalizing Flow for Robotic Policy Learning

Deep Reinforcement Learning (DRL) has experienced significant advancements in recent years and has been widely used in many fields. In DRL-based robotic policy learning, however, current de facto policy parameterization is still multivariate Gaussian (with diagonal covariance matrix), which lacks the ability to model multi-modal distribution. In this work, we explore the adoption of a modern network architecture, i.e. Normalizing Flow (NF) as the policy parameterization for its ability of multi-modal modeling, closed form of log probability and low computation and memory overhead. However, naively training NF in online Reinforcement Learning (RL) usually leads to training instability. We provide a detailed analysis for this phenomenon and successfully address it via simple but effective technique. With extensive experiments in multiple simulation environments, we show our method, NFPO could obtain robust and strong performance in widely used robotic learning tasks and successfully transfer into real-world robots.

Diyuan Shi, Yiqi Tang, Zifeng Zhuang +1

Zhejiang University · Westlake University

RL for RoboticsRobotic RLNormalizing Flows
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Mar 11, 2026

RC-NF: Robot-Conditioned Normalizing Flow for Real-Time Anomaly Detection in Robotic Manipulation

Recent advances in Vision-Language-Action (VLA) models have enabled robots to execute increasingly complex tasks. However, VLA models trained through imitation learning struggle to operate reliably in dynamic environments and often fail under Out-of-Distribution (OOD) conditions. To address this issue, we propose Robot-Conditioned Normalizing Flow (RC-NF), a real-time monitoring model for robotic anomaly detection and intervention that ensures the robot's state and the object's motion trajectory align with the task. RC-NF decouples the processing of task-aware robot and object states within the normalizing flow. It requires only positive samples for unsupervised training and calculates accurate robotic anomaly scores during inference through the probability density function. We further present LIBERO-Anomaly-10, a benchmark comprising three categories of robotic anomalies for simulation evaluation. RC-NF achieves state-of-the-art performance across all anomaly types compared to previous methods in monitoring robotic tasks. Real-world experiments demonstrate that RC-NF operates as a plug-and-play module for VLA models (e.g., pi0), providing a real-time OOD signal that enables state-level rollback or task-level replanning when necessary, with a response latency under 100 ms. These results demonstrate that RC-NF noticeably enhances the robustness and adaptability of VLA-based robotic systems in dynamic environments.

Shijie Zhou, Bin Zhu, Jiarui Yang +3

Institute of Trustworthy Embodied AI, Fudan University · Shanghai Key Laboratory of Multimodal Embodied AI · Singapore Management University

Robotic Anomaly DetectionAnomaly DetectionUnsupervised Anomaly DetectionRobustness of VLA ModelsRobotic ManipulationNormalizing Flows
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Mar 10, 2026

Temporal-Conditioned Normalizing Flows for Multivariate Time Series Anomaly Detection

This paper introduces temporal-conditioned normalizing flows (tcNF), a novel framework that addresses anomaly detection in time series data with accurate modeling of temporal dependencies and uncertainty. By conditioning normalizing flows on previous observations, tcNF effectively captures complex temporal dynamics and generates accurate probability distributions of expected behavior. This autoregressive approach enables robust anomaly detection by identifying low-probability events within the learned distribution. We evaluate tcNF on diverse datasets, demonstrating good accuracy and robustness compared to existing methods. A comprehensive analysis of strengths and limitations and open-source code is provided to facilitate reproducibility and future research.

David Baumgartner, Helge Langseth, Kenth Engø-Monsen +1

Department of Computer Science, Norwegian University of Science and Technology.

Time-Series Anomaly DetectionNormalizing Flows
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