Topics — The twenty-eight kinds of claim this anthology files against — the primary axis of both the practice registry and the reading list, and the same twenty-eight for each (ADR-026). The filing rule is the point: take a domain topic when the claim is ABOUT the domain; a claim merely discovered in one still takes its kind. The axis organizes the record, it does not bound it: content the list cannot place is evidence the list is short (ADR-059).
Flows and transport — generative models as learned transport between distributions — flow matching, rectified flow, stochastic interpolants, normalizing flows and neural ODEs, and the paths, couplings and time schedules they are trained on; Gaussian diffusion is the special case (THEORY-106). A subtopic, worn beside generative-modeling.
30 of 737 LIT documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| LIT-447 v2 | SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers | Ma et al. (2024), ARXIV-2401.08740. Holds DiT’s architecture, parameter count and GFLOPs fixed and varies four transport choices one at a time: discrete vs continuous time, score vs velocity prediction, the interpolant connecting the distributions, and the sampler. Velocity prediction and a linear interpolant each help substantially; the sampler’s diffusion coefficient turns out to be choosable after training. | Active |
| LIT-449 v2 | Scaling Rectified Flow Transformers for High-Resolution Image Synthesis | Esser et al. (2024), ARXIV-2403.03206. A 61-way comparison of diffusion and rectified-flow formulations settles on a straight-line path with timesteps drawn from a logit-normal, then scales it to 8B for text-to-image. Adds a resolution-dependent shift of the timestep schedule, and MMDiT, which gives each modality its own weights while letting the two token streams attend jointly. | Active |
| LIT-573 | FLUX.1 Kontext: Flow Matching for In-Context Image Generation and Editing in Latent Space | Black Forest Labs (2025), ARXIV-2506.15742. The only paper describing FLUX.1: a 12B rectified-flow transformer in a 16-channel VAE latent, with double-stream then 38 single-stream blocks, fused feed-forward layers and 3D RoPE. Kontext fine-tunes it for editing by appending context-image latents to the sequence, offset by a “virtual time step” in RoPE. The evidence is human preference against other systems and a reconstruction table. It has no ablations. | Active |
| LIT-619 v2 | Wan: Open and Advanced Large-Scale Video Generative Models | Wan Team, Alibaba (2025), ARXIV-2503.20314. The open Wan2.1 video models, 1.3B and 14B, and the full recipe behind them. The recipe covers data curation, a 127M causal 3D VAE compressing 4×8×8, a cross-attention DiT trained by rectified flow on an image-then-video resolution curriculum, the 2D context parallelism it needs, and a mask-conditioned image-to-video variant. The abstract claims video scaling laws in data and model size, but the report plots no scaling curve. | Active |
| LIT-626 v2 | Movie Gen: A Cast of Media Foundation Models | The Movie Gen team, Meta (2024), ARXIV-2410.13720. A 30B video model built from a Llama 3-style transformer, flow matching and an 8×8×8 temporal autoencoder, with the most controlled ablations in the video line. At 5B, flow matching beats diffusion and video captions beat frame captions. The “LLaMa3 scaling laws” claim rests on image-only runs at four sizes. | Active |
| LIT-630 v3 | Flow Matching for Generative Modeling | Lipman et al., Meta AI and Weizmann (2022), ARXIV-2210.02747. A continuous normalizing flow can be trained without simulation by regressing onto a per-sample conditional vector field. Diffusion paths are special cases. The straight “OT” path beats the diffusion path under the same objective, architecture and epochs, and it needs about 60% of the function evaluations. It used uniform timesteps. | Active |
| LIT-636 v4 | Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow | Liu, Gong and Liu, UT Austin (2022), ARXIV-2209.03003. It names rectified flow: regress a velocity onto x1 − x0 along the straight interpolation, which beats VP and sub-VP probability-flow ODEs on CIFAR-10 at the same architecture (FID 2.58 against 3.93). The “fast” in the title needs reflow or distillation. Without reflow or distillation one-step FID is 378. Distillation alone gives 6.18. One reflow gives 12.21 undistilled and 4.85 distilled, and full-simulation quality gets worse. | Active |
| LIT-644 v2 | Building Normalizing Flows with Stochastic Interpolants | Albergo and Vanden-Eijnden (2022), ARXIV-2209.15571. A continuous normalizing flow can be trained without ODE solves by regressing a velocity onto the time derivative of any interpolant between two densities. It is one of three concurrent origins of this simulation-free objective, with Flow Matching and Rectified Flow. Its path is not the straight line: every experiment uses the trigonometric cos/sin interpolant, which SiT later calls GVP. The image results arrived only in v3, and they are quoted against diffusion baselines, not matched against them. | Active |
| LIT-645 | Stochastic Interpolants: A Unifying Framework for Flows and Diffusions | Albergo, Boffi and Vanden-Eijnden (2023), ARXIV-2303.08797. The earlier interpolant objective is extended with a latent Gaussian term. One learned velocity and score then give an ODE and a family of SDEs, all sharing the interpolant’s marginals, and the SDE’s diffusion coefficient can be chosen after training. For SDEs the regression losses bound the KL, and for the ODE they do not. This is the framework SiT instantiates. The evidence that SDE sampling beats ODE sampling is a 2-D checkerboard and a 128-D Gaussian mixture. The paper reports no image metric and no diffusion baseline. | Active |
| LIT-678 v2 | Diffusion Meets Flow Matching: Two Sides of the Same Coin | Gao, Hoogeboom, Heek, De Bortoli, Murphy and Salimans (2024), a blog post at diffusionflow.github.io. With a Gaussian source, flow matching is a diffusion model with the schedule α_t = 1−t, σ_t = t, and the flow-matching Euler sampler is DDIM. What separates the named recipes is three choices: the loss weighting, the network output and the sampling schedule. “Straight paths” holds only for a single-point target, and a variance-preserving schedule can be straighter for wide data. The content is derivation and toy illustration only: no trained models and no benchmarks. | Active |
| LIT-722 | Score-Based Generative Modeling through Stochastic Differential Equations | Song, Sohl-Dickstein, Kingma, Kumar, Ermon and Poole (2020), ARXIV-2011.13456. The framework this record has been standing on without holding. Classifier guidance is Eq. 14 here, six months before ADM: the conditional reverse-time SDE adds ∇ₓ log pₜ(y|x) to the unconditional score, and §5 gives the recipe for the noise-conditioned classifier that supplies it — sample (x(0), y), noise it through the tractable forward SDE, train with a mixture of cross-entropy losses over timesteps. Also the probability flow ODE with exact likelihoods, predictor-corrector sampling, CIFAR-10 IS 9.89 / FID 2.20 / 2.99 bits/dim, and the first 1024×1024 samples from a score model. Its §I.2 imputation approximation is the “replacement” SOTA-397 exists to reject. | Active |
| LIT-771 | Glow: Generative Flow with Invertible 1x1 Convolutions | Kingma and Dhariwal, OpenAI (2018), ARXIV-1807.03039. A flow step is actnorm (a per-channel affine layer initialized from one batch), a learned invertible 1×1 convolution in place of Real NVP’s fixed channel permutation, and an affine coupling layer. The 1×1 convolution is the one controlled result: lower CIFAR-10 NLL than reversal or random permutation with either coupling, three seeds, about 7% more wall-clock (Fig. 3). Against Real NVP’s published numbers it is lower on all six datasets (CIFAR-10 3.35 against 3.49 bits/dim), with much larger models. The 256×256 faces are trained on 5-bit images and sampled at temperature 0.7. | Active |
| LIT-772 | Neural Ordinary Differential Equations | Chen, Rubanova, Bettencourt and Duvenaud, Toronto and Vector (2018), ARXIV-1806.07366. Parameterize a hidden state’s derivative with a network, integrate it with a black-box solver, and backpropagate by solving the adjoint ODE backwards, at memory constant in depth (§2, Alg. 1). For densities, the log-density of a state moving under dz/dt = f obeys d log p/dt = −tr(∂f/∂z) (Theorem 1), so a continuous normalizing flow needs a trace, not a determinant. The CNF evidence is 2-D toys. An MNIST ODE-Net matches a ResNet at 0.42% against 0.41% error. | Active |
| LIT-773 | Improving Video Generation with Human Feedback | Liu, Liu, Liang et al., CUHK, Tsinghua and Kuaishou (2025), ARXIV-2501.13918. Three parts. A 182k-pair human preference set over videos from 12 text-to-video systems, rated separately for visual quality, motion and text alignment. VideoReward, a Qwen2-VL-2B reward model trained with a Bradley–Terry loss that models ties. Flow-DPO, Diffusion-DPO rewritten for rectified flow. Carried over exactly, the loss weights the KL by β(1 − t)², which goes to zero at the noise end; that variant raises the reward’s quality scores and drops its text alignment below the base model’s. A constant β fixes it. On an internal 1B video model with LoRA, constant-β Flow-DPO beats SFT and Flow-RWR, and humans prefer it to the base 44% to 27% with 29% ties. Training labels and automatic win rates come from the same reward model, and VBench barely moves. | Active |
| LIT-775 | Inductive Moment Matching | Zhou, Ermon and Song, Luma AI and Stanford (2025), ARXIV-2503.07565. A one-network, from-scratch few-step model that maps the interpolant’s marginal at t to the one at s < t. It is trained by matching, with MMD over groups of samples, the distributions reached from t and from a nearby r. Consistency models are its one-sample, first-moment case, and that case collapses on ImageNet. ImageNet-256 FID is 8.05 at one step and 1.99 at eight, with guidance at sampling time, so 16 evaluations. Training is 1.2M steps at batch 4,096. CIFAR-10 2-step 1.98. One run per row. | Active |
| LIT-777 | Flow Matching Guide and Code | Lipman et al., FAIR, MIT and Weizmann (2024), ARXIV-2412.06264. A tutorial and PyTorch library, with no new experiments. It states flow matching as “choose a path, regress its generator” across Euclidean, Riemannian, discrete and general Markov-process state spaces. Its claims useful to practice are proved or attributed. The straight conditional path minimizes a bound on kinetic energy and is optimal transport only for a one-point target. Every affine scheduler gives the same exact-ODE endpoint, so a schedule matters only through training and discretization. Sampling t beats weighting by t, citing SD3. It does not cover few-step generation. | Active |
| LIT-779 v2 | Flow-GRPO: Training Flow Matching Models via Online RL | Liu, Liu et al., CUHK, Tsinghua and Kuaishou (2025), ARXIV-2505.05470. GRPO on a flow-matching image model. The deterministic sampler is swapped for an SDE with the same marginals, built from the model’s own velocity, which makes each step a Gaussian policy with a closed-form KL. Rollouts use 10 steps and evaluation 40. SD3.5-M with LoRA: GenEval 0.63 → 0.95, OCR accuracy 0.59 → 0.92, PickScore 21.72 → 23.31. Without the KL penalty, quality metrics fall on two tasks and diversity collapses on the third. GenEval is trained on its own templates and scored by its own detector. Quality is judged only by reward models, and there is one run per setting. | Active |
| LIT-780 | FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models | Grathwohl, Chen, Bettencourt, Sutskever and Duvenaud, Toronto, Vector and OpenAI (2018), ARXIV-1810.01367. Replaces the CNF’s exact Jacobian trace with Hutchinson’s estimator, one noise vector fixed per solve, so the log-density costs O(D) per evaluation and the dynamics network is unrestricted (§3, Eq. 8). Best reversible model on all five tabular sets, behind the autoregressive MAF-DDSF on four. MNIST 0.99 bits/dim against Glow’s 1.05; CIFAR-10 3.40 against Glow’s 3.35, with under 2% of Glow’s parameters. Training took about five days on six GPUs, and NFE grows during training until it can become “prohibitive” (§6). | Active |
| LIT-781 | Density estimation using Real NVP | Dinh, Sohl-Dickstein and Bengio, Google Brain (2016), ARXIV-1605.08803. The affine coupling layer: half the dimensions pass through unchanged and parameterize a scale and shift of the other half, so the Jacobian is triangular, its log-determinant is a sum, and the inverse costs the same as the forward pass whatever networks compute the scale and shift (§3.2–3.3). Stacked with squeeze and factor-out in a multi-scale architecture, it gives exact likelihood, exact latent inference and one-pass sampling. It does not beat PixelRNN’s likelihood on any dataset it shares with it (CIFAR-10 3.49 against 3.00 bits/dim, Table 1), and sample quality is shown, not measured. | Active |
| LIT-784 | Mean Flows for One-step Generative Modeling | Geng, Deng, Bai, Kolter and He, CMU and MIT (2025), ARXIV-2505.13447. Train a network for the average velocity u(z, r, t) over an interval, using the identity u = v − (t − r)·du/dt with du/dt from a Jacobian-vector product and the target under stop-gradient. With r = t it is flow matching. From scratch on ImageNet-256 with DiT-XL/2 and guidance folded into the target, one-step FID is 3.43 at 240 epochs and two-step 2.20 at 1,000. The best one-step baselines it beats, Shortcut 10.60 and IMM 7.77, are copied rows at other budgets. Behind iCT on CIFAR-10. One run per row. | Active |
| LIT-785 | STARFlow: Scaling Latent Normalizing Flows for High-resolution Image Synthesis | Gu et al., Apple (2025), ARXIV-2506.06276. TarFlow moved into a VAE latent with a deep-shallow split: one deep Transformer flow block and five 2-layer ones. It adds a decoder fine-tuned on noisy latents in place of score denoising, and a guidance rule derived from the Gaussian score. ImageNet 256 FID 2.40 at 1.4B against a retrained pixel TarFlow’s 5.56, with SiT’s quoted 2.06 and MAR’s 1.55 still ahead. ImageNet 512 3.00 against EDM2’s 1.25; COCO zero-shot 9.1; GenEval 0.56. The only head-to-head against diffusion is a plot against an in-house 2.1B DiT trained on 200M images (STARFlow’s schedule is 400M) with a stock decoder. Sampling is about 2.2 s per 256×256 image with guidance. | Active |
| LIT-787 | One Step Diffusion via Shortcut Models | Frans, Hafner, Levine and Abbeel, UC Berkeley (2024), ARXIV-2410.12557. A flow-matching network also conditioned on step size d, trained in one run: the flow-matching loss at d = 0, and a target for step 2d built from two of its own d-steps. On DiT-B at matched compute, one model gives 6.9 / 13.8 / 20.5 FID at 128 / 4 / 1 steps on CelebA-HQ-256 and 15.5 / 28.3 / 40.3 on ImageNet-256. Progressive distillation is better at one step (14.8 and 35.6) and cannot sample at many. The re-run consistency baselines are far below their published numbers. DiT-XL reaches 10.6 in one step. One run per cell. | Active |
| LIT-789 | Multisample Flow Matching: Straightening Flows with Minibatch Couplings | Pooladian et al., Meta AI, NYU and Weizmann (2023), ARXIV-2304.14772. Flow matching with noise and data paired within each minibatch by OT (BatchOT) or a cheaper stable matching keeps the marginals exact. As batch size grows the paths straighten toward the OT map. On ImageNet-64 the FID-20 point drops from 29 Euler steps to 12, and adaptive FID from 13.93 to 12.37, at equal likelihood and 4% more time per iteration. Low-NFE numbers stay far from one-step quality: 38.86 FID at 4 Euler steps on ImageNet-32. One run per row. | Active |
| LIT-790 | Simplifying, Stabilizing and Scaling Continuous-Time Consistency Models | Lu and Song, OpenAI (2024), ARXIV-2410.11081. Continuous-time consistency models had been unstable. A trigonometric flow parameterization, identity time input with positional embeddings, normalized tangents, learned loss weights and a Flash Attention JVP make them train to 1.5B parameters. Continuous time beats every discrete step count tried. Two-step FID is 2.06 on CIFAR-10, 1.48 on ImageNet-64 and 1.88 on ImageNet-512, against teachers at 2.01, 1.33 and 1.73. Every model, “consistency training” included, starts from the pretrained diffusion model. | Active |
| LIT-791 | Improving the Training of Rectified Flows | Lee, Lin and Fanti, CMU and Microsoft Research (2024), ARXIV-2405.20320. One reflow round is enough, argued informally, and the 2-rectified flow then trains better with a U-shaped timestep distribution and an LPIPS-Huber loss. Started from EDM, with no separate distillation stage, one-step FID is 3.07 on CIFAR-10 and 4.31 on ImageNet-64. Its own table lists DMD at 2.62 on ImageNet-64. There is no distillation-alone arm from the same teacher, and many-step quality is not compared with the pre-reflow model. | Active |
| LIT-792 | Discrete Flow Matching | Gat et al., FAIR and Weizmann (2024), ARXIV-2407.15595. Flow matching on discrete tokens as a continuous-time Markov chain, with velocities of the same form as the continuous case, any coupling, a family of schedulers, and corrector sampling that can re-mask emitted tokens. At 150M its generative perplexity beats SEDD’s in float32 (GPT-2 29.3 against 40.5 at 1,024 NFE). In double precision every discrete model’s perplexity rises 2–3.5 times and the gap narrows (97.5 against 109.5). At 1.7B on code it reaches HumanEval pass@1 6.7 against a same-data autoregressive model’s 14.3. | Active |
| LIT-793 | Flow map matching with stochastic interpolants: A mathematical framework for consistency models | Boffi, Albergo and Vanden-Eijnden, Courant Institute (2024), ARXIV-2406.07507. Learn the two-time flow map X_{s,t} of a probability-flow ODE. Distillation from a velocity field can use a Lagrangian loss (new) or an Eulerian one, which is the continuous-time limit of consistency distillation; both bound a Wasserstein distance. Consistency training’s update is shown not to descend any loss. On CIFAR-10 the Lagrangian loss distils far better than the Eulerian: FID 7.13 against 48.32 at two steps, teacher 5.53. Direct training on ImageNet-32 gives 16.90 FID at four steps. Small models; no one-step image table. | Active |
| LIT-794 | Normalizing Flows are Capable Generative Models | Zhai et al., Apple (2024), ARXIV-2412.06329. TarFlow stacks causal Transformers over image patches as affine autoregressive flows, reversing the order between blocks, trained by exact maximum likelihood. A likelihood model reaches 2.99 bits/dim on unconditional ImageNet 64×64, below every diffusion and autoregressive number it quotes (Table 2). Sampling needs three additions: Gaussian input noise about 25× the dequantization noise, a Tweedie denoising step using the flow’s own score, and classifier-free guidance on the affine parameters. Best FIDs are 2.66 at ImageNet 64 and 5.03 at 128, against quoted 1.55 (EDM) and 1.94. Sampling is sequential: 32 samples at 64×64 take about two minutes on one A100. | Active |
| LIT-796 | InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation | Liu, Zhang, Ma, Peng and Liu, UT Austin and Helixon (2023), ARXIV-2309.06380. Reflow Stable Diffusion once, then distil the reflowed model to one step with an LPIPS regression. Against distilling SD 1.4 directly, at the same 100K steps and 3.2M generated pairs, reflow first gives one-step FID 31.0 against 40.9 on COCO-2017-5k and 20.0 against 34.6 on COCO-2014-30k. The scaled model, InstaFlow-0.9B, scores 23.4 and 13.10. The teacher is a diffusion model, not a rectified flow, and the distiller is a regression, not distribution matching. One run per row. | Active |
| LIT-798 | Improving and generalizing flow-based generative models with minibatch optimal transport | Tong et al., Mila (2023), ARXIV-2302.00482. Flow matching trains on any coupling of source and target samples. Pairing each minibatch by exact optimal transport (OT-CFM) gives near-OT paths in 2-D (normalized path energy 0.02–0.09 against 0.2–2.7 for the independent coupling, five seeds) at under 1% overhead. On CIFAR-10 its gain is small: adaptive-solver FID 3.58 against 3.66 at 134 against 146 NFE, and the independent coupling is better at 1,000 Euler steps. Its rerun of the path comparison, at a recipe that beats Flow Matching’s own, has the straight path ahead of VP at 100 Euler steps (4.64 against 7.77). | Active |