CurveFlow: Curvature-Guided Flow Matching

• CurveFlow is a text-to-image generation method that replaces straight interpolation with learned non-linear trajectories by explicitly modeling and regularizing curvature.
• It employs time-dependent coefficient functions combined with a flow matching objective and robust curvature regularization to stabilize trajectory dynamics.
• Empirical evaluations on COCO datasets reveal that CurveFlow improves semantic consistency and FID, outperforming traditional rectified flow baselines.

Searching arXiv for the primary "CurveFlow" paper and closely related flow-matching work. CurveFlow is a text-to-image generation method in the flow matching / rectified flow paradigm that replaces the straight interpolation used by standard rectified flow with smooth non-linear trajectories whose curvature is explicitly modeled and regularized. In the formulation introduced in “CurveFlow: Curvature-Guided Flow Matching for Image Generation” (Luo et al., 20 Aug 2025), the central motivation is that rectified flow typically assumes a linear path between a clean data sample $x_0$ and Gaussian noise $\epsilon$, a choice that enforces zero curvature and may force the generation process through low-probability regions of the data manifold. CurveFlow addresses this by learning time-dependent interpolation coefficients and by introducing a robust curvature regularization term intended to stabilize the intrinsic dynamics of the trajectory, with the stated goal of improving semantic alignment between generated images and conditioning captions (Luo et al., 20 Aug 2025).

1. Definition and conceptual role

CurveFlow is defined within the class of flow matching methods that transport samples from a simple source distribution to a target image distribution through a time-dependent velocity field. In the rectified flow background used by the paper, the baseline trajectory is

$z_t = (1 - t) x_0 + t \epsilon, \quad t \in [0,1].$

This path has constant velocity and satisfies

$\frac{d^2 z_t}{dt^2} = 0,$

so its curvature is zero (Luo et al., 20 Aug 2025).

The method’s central modification is to replace the fixed linear coefficients $(1-t)$ and $t$ with learned functions: $z_t = a_\phi(t) x_0 + b_\psi(t) \epsilon,$ subject to the endpoint constraints

$$a_\phi(0) = 1, \quad b_\psi(0) = 0, \quad a_\phi(1) = 0, \quad b_\psi(1) = 1.$$

This preserves the same endpoints as rectified flow while allowing the path to bend in time (Luo et al., 20 Aug 2025).

The paper explicitly frames CurveFlow as a response to the rigidity of zero-curvature transport. A straight line between $x_0$ and $\epsilon$ is computationally convenient, but the paper argues that natural image distributions lie on non-linear manifolds with diverse structures, styles, and object relationships, so linear interpolation may be poorly aligned with the geometry of the data. This suggests that smooth non-linear trajectories may better remain near high-probability regions of the image manifold, especially in text-conditioned generation where caption fidelity is central (Luo et al., 20 Aug 2025).

2. Trajectory parameterization and curvature

For the learned path

$\epsilon$0

the velocity and acceleration are

$\epsilon$1

The paper uses the standard curvature formula for a parametric curve,

$\epsilon$2

and derives

$\epsilon$3

Hence

$\epsilon$4

while the speed satisfies

$\epsilon$5

The resulting trajectory curvature is

$\epsilon$6

This formula shows that curvature depends on both the sample pair $\epsilon$7 and the coefficient dynamics $\epsilon$8 (Luo et al., 20 Aug 2025).

A key observation in the paper is that directly regularizing the full sample-dependent curvature is unstable, because its empirical estimate varies with the random draw of image–noise pairs. CurveFlow therefore isolates the coefficient-space term

$\epsilon$9

which is the determinant-like factor appearing in the numerator of the full curvature expression. In the paper’s interpretation, this quantity measures abrupt turning in the intrinsic path dynamics without introducing dependence on a specific sampled $z_t = (1 - t) x_0 + t \epsilon, \quad t \in [0,1].$0 (Luo et al., 20 Aug 2025).

3. Training objective and regularization

The baseline flow matching loss in the paper is written as

$z_t = (1 - t) x_0 + t \epsilon, \quad t \in [0,1].$1

where the rectified-flow target velocity is constant along the straight path. CurveFlow replaces that target with the derivative of the learned non-linear path (Luo et al., 20 Aug 2025).

The CurveFlow data-fitting term is

$z_t = (1 - t) x_0 + t \epsilon, \quad t \in [0,1].$2

This trains the velocity field to match the time derivative of the curved trajectory rather than the constant velocity of rectified flow (Luo et al., 20 Aug 2025).

The method’s distinctive regularizer is

$z_t = (1 - t) x_0 + t \epsilon, \quad t \in [0,1].$3

where $z_t = (1 - t) x_0 + t \epsilon, \quad t \in [0,1].$4 is the curvature regularization weight. The total objective is given as

$z_t = (1 - t) x_0 + t \epsilon, \quad t \in [0,1].$5

The paper describes this as a robust curvature regularization technique that penalizes abrupt changes in the trajectory’s intrinsic dynamics (Luo et al., 20 Aug 2025).

Implementation of this term uses a Riemann-sum approximation over a fixed time grid, with numerical derivatives computed by central finite differences: $z_t = (1 - t) x_0 + t \epsilon, \quad t \in [0,1].$6 and analogously for $z_t = (1 - t) x_0 + t \epsilon, \quad t \in [0,1].$7. The paper gives $z_t = (1 - t) x_0 + t \epsilon, \quad t \in [0,1].$8 as an example of the time-grid resolution used for the regularizer (Luo et al., 20 Aug 2025).

4. Architecture and optimization setup

CurveFlow is implemented as a fine-tuning method over a rectified-flow-based text-to-image backbone. The reported setup initializes from stabilityai/stable-diffusion-3.5-large, uses LoRA fine-tuning with rank $z_t = (1 - t) x_0 + t \epsilon, \quad t \in [0,1].$9 adapters on the MMDiT transformer, and trains approximately $\frac{d^2 z_t}{dt^2} = 0,$0M parameters for $\frac{d^2 z_t}{dt^2} = 0,$1 epochs with batch size $\frac{d^2 z_t}{dt^2} = 0,$2, AdamW, learning rate $\frac{d^2 z_t}{dt^2} = 0,$3, a polynomial learning-rate scheduler with $\frac{d^2 z_t}{dt^2} = 0,$4 warmup steps, bf16 mixed precision, and gradient checkpointing (Luo et al., 20 Aug 2025).

The learned coefficient functions $\frac{d^2 z_t}{dt^2} = 0,$5 and $\frac{d^2 z_t}{dt^2} = 0,$6 are implemented as 3-layer MLPs with a 64-dimensional intermediate feature space (Luo et al., 20 Aug 2025). The paper also reports an ablation over coefficient-network width $\frac{d^2 z_t}{dt^2} = 0,$7, concluding that 32 has insufficient capacity, 64 gives the best trade-off, and 128 degrades performance due to overfitting or optimization instability (Luo et al., 20 Aug 2025).

A plausible implication is that CurveFlow treats trajectory shaping as a low-dimensional control problem layered on top of a large pretrained generative model: the main image generator remains the velocity field $\frac{d^2 z_t}{dt^2} = 0,$8, while the coefficient networks act as a compact geometric controller over the interpolation path.

5. Empirical performance

The reported experiments are conducted on MS COCO 2014 and MS COCO 2017, with evaluation emphasizing both image quality and semantic alignment. Semantic consistency is measured using BLEU-1/2/3/4, METEOR, ROUGE-1/2/L, and CLAIR, with captions for generated images produced by BLIP v2 and compared to reference captions. Image quality and text–image alignment are also evaluated using FID, IS, and CLIPScore (Luo et al., 20 Aug 2025).

On COCO17, the paper reports the following semantic-consistency scores for CurveFlow:

• BLEU-1: $\frac{d^2 z_t}{dt^2} = 0,$9
• BLEU-2: $(1-t)$0
• BLEU-3: $(1-t)$1
• BLEU-4: $(1-t)$2
• METEOR: $(1-t)$3
• ROUGE-1: $(1-t)$4
• ROUGE-2: $(1-t)$5
• ROUGE-L: $(1-t)$6
• CLAIR: $(1-t)$7

These values exceed the reported numbers for Rectified Diffusion and for the standard rectified-flow variants RF w/o Reweighting, RF w/ LogNorm, RF w/ ModeSample, and RF w/ CosMap (Luo et al., 20 Aug 2025).

On COCO14, the paper reports for CurveFlow:

• BLEU-1: $(1-t)$8
• BLEU-2: $(1-t)$9
• BLEU-3: $t$0
• BLEU-4: $t$1
• METEOR: $t$2
• ROUGE-1: $t$3
• ROUGE-2: $t$4
• ROUGE-L: $t$5
• CLAIR: $t$6

Again these are the best reported values among the compared methods in the semantic-consistency tables (Luo et al., 20 Aug 2025).

For image quality, CurveFlow attains the best reported FID on both datasets:

• COCO17: FID $t$7
• COCO14: FID $t$8

Its IS and CLIPScore remain competitive, although the paper’s tables show that some baseline methods obtain slightly higher IS or CLIPScore in certain settings (Luo et al., 20 Aug 2025). This suggests that the primary empirical advantage of CurveFlow lies in semantic consistency and FID rather than in uniformly dominating every metric.

6. Ablation findings

The paper studies the effect of curvature regularization strength $t$9 over the set

$z_t = a_\phi(t) x_0 + b_\psi(t) \epsilon,$0

It states that $z_t = a_\phi(t) x_0 + b_\psi(t) \epsilon,$1 yields worse FID and lower semantic scores, indicating that the regularizer is materially useful. One reported comparison gives METEOR $z_t = a_\phi(t) x_0 + b_\psi(t) \epsilon,$2 and ROUGE-1 $z_t = a_\phi(t) x_0 + b_\psi(t) \epsilon,$3 at $z_t = a_\phi(t) x_0 + b_\psi(t) \epsilon,$4, versus METEOR $z_t = a_\phi(t) x_0 + b_\psi(t) \epsilon,$5 and ROUGE-1 $z_t = a_\phi(t) x_0 + b_\psi(t) \epsilon,$6 at $z_t = a_\phi(t) x_0 + b_\psi(t) \epsilon,$7 (Luo et al., 20 Aug 2025).

The paper contains two different summaries of the best-performing $z_t = a_\phi(t) x_0 + b_\psi(t) \epsilon,$8: one passage emphasizes $z_t = a_\phi(t) x_0 + b_\psi(t) \epsilon,$9 as strongest for METEOR and ROUGE-1, while another says $$a_\phi(0) = 1, \quad b_\psi(0) = 0, \quad a_\phi(1) = 0, \quad b_\psi(1) = 1.$$0 gives the best trade-off between FID, METEOR, and stability (Luo et al., 20 Aug 2025). This suggests that the preferred regularization level depends on whether semantic metrics alone or a broader quality–stability trade-off is prioritized.

A plausible implication is that curvature control acts as a bias–variance trade-off on the path family: too little regularization leaves the coefficient dynamics underconstrained, while too much regularization over-smooths the path and reduces expressive power.

7. Qualitative behavior and interpretation

The paper includes qualitative comparisons in which CurveFlow is described as better preserving caption-critical details. Examples discussed include a woman walking down a street holding an orange umbrella, where baseline variants may omit the umbrella, and a Boston Terrier sitting on a couch with a shoe, where baselines may misidentify posture or object multiplicity (Luo et al., 20 Aug 2025).

These examples are used to support the paper’s larger claim that trajectory geometry affects instructional compliance. The argument is not that CurveFlow adds an explicit semantic loss; rather, it suggests that a better-shaped transport path makes it easier for the model to preserve fine-grained semantic structure during the noise-to-image transformation (Luo et al., 20 Aug 2025).

This suggests a broader interpretation of CurveFlow within generative modeling: path design itself is treated as a semantic variable. In standard rectified flow, semantics are delegated almost entirely to the learned velocity field over a fixed straight path. In CurveFlow, part of the burden is shifted into the geometry of the trajectory.

8. Position within the broader curve-flow literature

Despite its name, CurveFlow is not a geometric PDE for embedded planar curves of the kind studied in classical curve shortening, nonlocal area-preserving flows, centro-affine flows, or higher-order polyharmonic flows. Those topics are represented by works on warped-product mean-curvature-type flows (Cant, 2016), nonlocal planar flows with capacity terms (Caffarelli et al., 2017), non-local area-preserving curvature flows (Sun et al., 23 Feb 2025), centro-equiaffine invariant fourth-order flows (Jiang et al., 16 Apr 2026), sixth-order ideal curve flows (Andrews et al., 2018), and anisotropic constrained flows (Sun, 2023). In that literature, “curve flow” refers to geometric evolution equations for immersed or embedded curves under curvature-driven normal velocities.

CurveFlow (Luo et al., 20 Aug 2025) instead uses “flow” in the flow-matching sense of generative modeling. Its trajectories are curves in the ambient sample space between $$a_\phi(0) = 1, \quad b_\psi(0) = 0, \quad a_\phi(1) = 0, \quad b_\psi(1) = 1.$$1 and $$a_\phi(0) = 1, \quad b_\psi(0) = 0, \quad a_\phi(1) = 0, \quad b_\psi(1) = 1.$$2, and curvature refers to the geometry of those trajectories rather than to Euclidean curvature of an image contour. The connection to the classical literature is therefore analogical rather than direct: both settings investigate how curvature shapes evolution, but one acts on geometric curves in physical space, while the other acts on interpolation paths in a high-dimensional latent or image space.

This distinction is important because the paper’s central object is not a moving embedded curve but a path $$a_\phi(0) = 1, \quad b_\psi(0) = 0, \quad a_\phi(1) = 0, \quad b_\psi(1) = 1.$$3 in distribution transport. A plausible implication is that the naming “CurveFlow” is best understood as a curvature-aware flow-matching framework rather than as an extension of geometric curve-shortening theory.

9. Scope and limitations

The experiments reported in the paper are restricted to MS COCO 2014 and 2017, and the method is evaluated in a LoRA fine-tuning regime built on a rectified-flow-based Stable Diffusion 3.5 large backbone (Luo et al., 20 Aug 2025). The paper does not establish whether the same gains persist under training from scratch, on larger and more heterogeneous text-to-image corpora, or across substantially different flow-matching architectures.

The curvature regularizer introduces additional training overhead because it requires separate evaluation of coefficient functions on a dense time grid and numerical differentiation (Luo et al., 20 Aug 2025). The paper states that inference time remains comparable to rectified flow because CurveFlow uses the same ODE-based sampling framework and the same number of sampling steps (Luo et al., 20 Aug 2025).

The paper also does not present a formal theorem connecting trajectory curvature to semantic fidelity. Its claim is empirical: smoother non-linear trajectories together with curvature regularization correlate with stronger instructional compliance in the reported experiments (Luo et al., 20 Aug 2025). This suggests, but does not prove, that trajectory geometry is a useful control variable for caption faithfulness.

10. Significance

CurveFlow introduces a curvature-aware generalization of rectified flow in which the transport path between data and noise is itself learned and regularized. Its main technical contribution is the combination of a learned trajectory family

$$a_\phi(0) = 1, \quad b_\psi(0) = 0, \quad a_\phi(1) = 0, \quad b_\psi(1) = 1.$$4

with a robust curvature regularizer based on

$$a_\phi(0) = 1, \quad b_\psi(0) = 0, \quad a_\phi(1) = 0, \quad b_\psi(1) = 1.$$5

Its main empirical claim is that this geometry-aware path construction improves semantic consistency in text-to-image generation while maintaining strong image quality, with state-of-the-art results on the reported COCO benchmarks (Luo et al., 20 Aug 2025).

Within the landscape of arXiv work on curve and flow models, CurveFlow is therefore best described as a flow-matching method that imports curvature as an organizing principle into generative transport. It does not study the evolution of physical curves; it studies the evolution of generative trajectories, and argues that their curvature is materially linked to the fidelity with which a model follows complex textual instructions (Luo et al., 20 Aug 2025).