Conditional Guided Flow Matching

• Conditional Guided Flow Matching (CGFM) is a framework unifying conditional flow models with external guidance to transport distributions effectively.
• It integrates auxiliary signals, energy gradients, and classifier-free guidance to refine transport dynamics in applications like time-series forecasting and autonomous driving.
• CGFM’s flexible design supports diverse domains, offering actionable insights on source design, path conditioning, and guided corrections for improved generative performance.

Conditional Guided Flow Matching (CGFM) denotes a class of flow-matching methods in which a continuous-time transport model is both conditional on side information and guided by an additional signal during training, sampling, or both. The term is introduced explicitly for time-series forecasting in "Bridging the Last Mile of Prediction: Enhancing Time Series Forecasting with Conditional Guided Flow Matching" (Xu et al., 9 Jul 2025). Closely related formulations include classifier-free “Guided Flows” for flow matching (Zheng et al., 2023) and the broader FM/CFM/guidance synthesis assembled in "Flow Matching Guide and Code" (Lipman et al., 2024). Taken together, these works suggest that CGFM is best viewed not as a single fixed algorithm, but as a family of conditional flow models augmented by guidance mechanisms such as classifier-free interpolation, energy gradients, operator-derived corrections, internal scoring signals, or differentiable forward-model constraints.

1. Conceptual scope and lineage

The immediate conceptual lineage of CGFM lies in Flow Matching (FM), Conditional Flow Matching (CFM), and classifier-free guidance for continuous normalizing flows. "Guided Flows for Generative Modeling and Decision Making" extends classifier-free guidance from diffusion to flow matching and defines guided sampling through a conditional continuous normalizing flow

$\dot{x}_t = u_t(x_t|y),$

together with the guided vector field

$$\tilde{u}_t(x|y) = (1 - \omega) u_t(x) + \omega u_t(x|y),$$

where $\omega$ is the guidance weight (Zheng et al., 2023). "Flow Matching Guide and Code" presents the same ingredients more systematically: FM as simulation-free training of continuous-time generative flows, endpoint-conditioned CFM, conditional generation with side information $Y$, classifier guidance, and classifier-free guidance (Lipman et al., 2024).

The explicit CGFM formulation in forecasting adds a further step. Rather than transporting only from a fixed noise source, it uses the output of an auxiliary forecaster as the source distribution, conditions both endpoints on the same historical context, and introduces a general affine family of conditional probability paths (Xu et al., 9 Jul 2025). This is the paper in which CGFM is named directly, and it emphasizes a forecasting-specific capability: learning from the errors of an auxiliary model.

Later work broadens the phrase through applications rather than through a single unified theory. In autonomous driving, GuideFlow is described as “closest to a conditional guided flow-matching application,” but also as “not a generic CGFM theory paper” (Liu et al., 24 Nov 2025). In speech synthesis, a conditional flow model is trained to internalize CFG-like behavior without inference-time dual passes (Yu et al., 1 Jul 2026). In robotics, ForesightFlow uses a self-guided conditional flow over augmented action-and-potential states rather than an external critic (Mei et al., 3 Jun 2026). These variants preserve the central CGFM pattern—conditional transport plus guidance—while differing sharply in what counts as the guide.

2. Mathematical structure

At its core, CGFM inherits the FM view that a generative model is a time-dependent vector field whose flow transports a source distribution to a target distribution. In the conditional case, the sampling dynamics take the form

$\dot{x}_t = u_t(x_t|y),$

with $y$ denoting a conditioning variable such as a class label, text prompt, observation history, source image, or scene context (Zheng et al., 2023).

The standard FM/CFM training picture is regression on analytically defined pathwise velocities rather than simulation-based likelihood training. "Flow Matching Guide and Code" writes the basic CFM objective as

$$\mathcal{L}_{CFM}(\theta) = \mathbb{E}_{t,Z,X_t\sim p_{t|Z}} D(u_t(X_t|Z),u_t^\theta(X_t)),$$

and emphasizes the marginalization identity

$u_t(x)=\mathbb{E}[u_t(X_t|Z)\mid X_t=x],$

which makes endpoint-conditioned or latent-conditioned targets sufficient for learning the marginal flow field (Lipman et al., 2024). In the linear conditional OT path, one has

$X_t=tX_1+(1-t)X_0,$

and the target velocity reduces to the familiar straight-line displacement $X_1-X_0$ (Lipman et al., 2024).

The explicit CGFM forecasting formulation generalizes this by building a two-sided conditional probability path

$$\tilde{u}_t(x|y) = (1 - \omega) u_t(x) + \omega u_t(x|y),$$0

with conditional independent coupling

$$\tilde{u}_t(x|y) = (1 - \omega) u_t(x) + \omega u_t(x|y),$$1

and an affine bridge

$$\tilde{u}_t(x|y) = (1 - \omega) u_t(x) + \omega u_t(x|y),$$2

Its marginal conditional velocity can then be written as

$$\tilde{u}_t(x|y) = (1 - \omega) u_t(x) + \omega u_t(x|y),$$3

The same paper derives equivalent $$\tilde{u}_t(x|y) = (1 - \omega) u_t(x) + \omega u_t(x|y),$$4-prediction, $$\tilde{u}_t(x|y) = (1 - \omega) u_t(x) + \omega u_t(x|y),$$5-prediction, and $$\tilde{u}_t(x|y) = (1 - \omega) u_t(x) + \omega u_t(x|y),$$6-prediction parameterizations, and reports that $$\tilde{u}_t(x|y) = (1 - \omega) u_t(x) + \omega u_t(x|y),$$7-prediction works best empirically (Xu et al., 9 Jul 2025).

This mathematical picture already contains the two components that define CGFM in practice. The conditional part enters through $$\tilde{u}_t(x|y) = (1 - \omega) u_t(x) + \omega u_t(x|y),$$8 and through context-dependent endpoint laws or paths. The guided part enters when the field used for sampling is modified beyond the base conditional field, or when the source/path itself is redesigned to encode an informative auxiliary signal.

3. Guidance mechanisms

The most canonical guidance mechanism in FM is classifier-free guidance. For Gaussian paths, "Guided Flows for Generative Modeling and Decision Making" shows that the guided field

$$\tilde{u}_t(x|y) = (1 - \omega) u_t(x) + \omega u_t(x|y),$$9

is the flow-matching analogue of diffusion CFG (Zheng et al., 2023). "Flow Matching Guide and Code" presents the same construction in the broader FM framework and also gives the classifier-guided form

$\omega$0

which is valid for Gaussian paths through the score-to-velocity relation (Lipman et al., 2024).

Subsequent work expands “guidance” well beyond classifier guidance. GuideFlow adds application-specific guidance terms for planning in end-to-end autonomous driving. Its sampling update may include energy guidance

$\omega$1

classifier-free interpolation of conditional and unconditional velocity fields, and two explicit constraint interventions: constraining the velocity field (CVF) and constraining the flow states (CF) (Liu et al., 24 Nov 2025). In that paper, the cleanest standard mechanism is classifier-free conditional guidance, while the most theoretically CGFM-adjacent mechanism is the energy-guided sampler.

In physically constrained time-series generation, HPC-FNO-CFM defines an FNO-derived correction

$\omega$2

where

$\omega$3

Guidance is therefore a learned physics-operator correction, reinforced by hierarchical physical penalties in the total objective (Okita, 9 Oct 2025).

Speech synthesis introduces yet another variant. The unified guidance framework for FM speech generation defines an intrinsic guidance target

$\omega$4

followed by a distillation loss that trains the model to internalize guided behavior without inference-time CFG overhead (Yu et al., 1 Jul 2026). In robotics, ForesightFlow does not use additive gradient guidance at all; instead, it jointly generates an action chunk and a success-potential trajectory, then performs best-of-$\omega$5 inference using the generated potential as an internal ranking signal (Mei et al., 3 Jun 2026).

These examples show that “guided” in CGFM is now used in at least four distinct senses: guidance by conditional/unconditional interpolation, guidance by external gradients, guidance by explicit constraint corrections, and guidance by jointly generated internal scoring variables.

4. Conditioning, path design, and source design

A second defining axis of CGFM research concerns what is being conditioned, what source distribution is used, and what probability path is chosen. The original FM and CFM formulations often use Gaussian sources and endpoint-conditioned linear or Gaussian bridges (Lipman et al., 2024). CGFM methods increasingly depart from that default.

The forecasting CGFM paper is explicit that the source need not be noise. It sets

$\omega$6

with $\omega$7 supplied by an auxiliary forecaster, and interprets the resulting transport as learning from the residual gap between auxiliary prediction and ground truth future (Xu et al., 9 Jul 2025). This is a strong departure from both diffusion-style Gaussian starting points and standard FM formulations that do not exploit prior model outputs.

Coreset-Induced Conditional Velocity Flow Matching pushes source design even further. It replaces the inner isotropic Gaussian source in hierarchical rectified flow with a coreset-induced closed-form surrogate conditional law

$\omega$8

obtained from a Sinkhorn coreset and Gaussian mixture surrogate of the target distribution (Wang et al., 13 May 2026). This is not “guided” in the standard semantic sense, but it is a clear example of a data-informed conditional source distribution shrinking the residual transport problem.

Another generalization replaces endpoint conditioning by latent path conditioning. "Stream-level flow matching with Gaussian processes" defines streams $\omega$9 and trains with the stream-level loss

$Y$0

The paper’s central point is that richer latent trajectory conditioning can reduce the variance of the estimated marginal vector field while preserving the simulation-free character of CFM training (Wei et al., 2024). This suggests a trajectory-level extension of CGFM beyond endpoint or label conditioning.

Condition design also matters. MusFlow aligns image, story, and caption embeddings into CLAP’s audio embedding space before conditioning a latent FM model on the fused representation, using

$Y$1

and a generation loss

$Y$2

The paper’s main lesson is that conditional flow matching depends strongly on the semantic geometry of the conditioning space (Song et al., 18 Apr 2025).

A similar source-versus-condition separation appears in CBCT-to-sCT generation. That work uses a Gaussian source with CBCT concatenated as a static condition,

$Y$3

and reports that the Gaussian-conditioned method generally outperforms both “CBCT w/o Concat” and “CBCT w/ Concat” variants (Peng et al., 6 Mar 2026). A plausible implication is that, in inverse problems, separating the transported state from the conditioning observation can be preferable to transporting the corrupted observation itself.

5. Representative domains

The current literature spans a wide range of domains, but the role played by CGFM varies substantially across them.

Domain	Representative papers	Role of CGFM
Time-series forecasting	(Xu et al., 9 Jul 2025)	Auxiliary-model source, two-sided conditional paths, affine path design
Conditional image, speech, and planning	(Zheng et al., 2023, Yu et al., 1 Jul 2026)	CFG-style guided flows; intrinsic guidance distillation for low-NFE sampling
Autonomous driving	(Liu et al., 24 Nov 2025)	Conditional planner with classifier-free guidance, constraint interventions, and EBM guidance
Physics-informed generation and inverse reconstruction	(Okita, 9 Oct 2025, Li et al., 6 Apr 2026, Ray et al., 30 Oct 2025)	Operator guidance, differentiable forward-model guidance, and measurement-guided conditional transport
Robotics and control	(Mei et al., 3 Jun 2026)	Self-guided action generation via jointly generated success-potential trajectories
Multimodal and structured prediction	(Song et al., 18 Apr 2025, Ye et al., 13 Mar 2026)	Multimodal conditional latent FM; CFM-style regularization for smooth correspondences

In autonomous driving, GuideFlow frames CGFM-like planning as conditional flow generation augmented by classifier-free guidance, energy guidance, and driving-specific constrained generation (Liu et al., 24 Nov 2025). In speech synthesis, the major concern is eliminating CFG overhead while preserving conditional adherence and low-NFE generation (Yu et al., 1 Jul 2026). In physics-informed or inverse settings, guidance often becomes measurement consistency or operator consistency rather than semantic conditioning: HPC-FNO-CFM uses an FNO-derived correction with hierarchical physical constraints (Okita, 9 Oct 2025), granular-flow reconstruction uses a differentiable forward operator and sparsity-aware gradient guidance (Li et al., 6 Apr 2026), and ResMatching conditions a super-resolution flow on the observed low-resolution image to sample an implicit posterior over high-resolution reconstructions (Ray et al., 30 Oct 2025).

Robotics introduces a particularly distinctive form. ForesightFlow jointly models action chunks and success-potential trajectories, then uses the generated potential to rank candidates at inference. Its key objective is decoupled across action and potential coordinates: $Y$4 thereby separating policy improvement from potential calibration (Mei et al., 3 Jun 2026). This is a CGFM realization in which guidance is neither external nor post hoc, but part of the jointly generated state.

6. Distinctions, misconceptions, and open issues

A recurring misconception is that any paper using both “conditional” and “guided” belongs straightforwardly to CGFM. The literature is more heterogeneous. "Fast Flow Matching based Conditional Independence Tests for Causal Discovery" is explicitly described as not a paper about CGFM in the usual generative-modeling sense; its “guided” component refers to CI-query selection in PC skeleton learning, not to guidance in the flow dynamics (Zhao et al., 9 Feb 2026). Likewise, SGMatch uses a CFM-style objective as a regularizer for feature-space correspondence smoothness rather than as an inference-time generative transport model (Ye et al., 13 Mar 2026).

Another misconception is that GuideFlow is a foundational general CGFM reference. The paper itself is explicit that it is “primarily application-specific for autonomous driving planning” and “not a generic CGFM theory paper” (Liu et al., 24 Nov 2025). The same caution applies more broadly: many application papers instantiate conditional and guided flow ideas successfully, but do not attempt a universal formalism.

Theoretical completeness also varies. HPC-FNO-CFM uses generic CFM notation but leaves the precise bridge $Y$5 under-specified, while also acknowledging manual constraint design, higher computational cost, partial observability, and difficulty with non-stationary systems (Okita, 9 Oct 2025). The general FM guide explicitly notes that one assembles CGFM by combining FM, CFM, and guidance ingredients rather than by invoking a single canonical named method (Lipman et al., 2024). This suggests that CGFM currently functions more as a design pattern than as a closed theory.

Taken together, the literature suggests three open directions. First, source and path design are becoming as important as guidance itself, as seen in auxiliary-forecast sources, coreset-induced surrogates, and GP streams (Xu et al., 9 Jul 2025, Wang et al., 13 May 2026, Wei et al., 2024). Second, guidance is broadening from CFG-style interpolation to operator, energy, constraint, and self-scoring mechanisms (Liu et al., 24 Nov 2025, Okita, 9 Oct 2025, Mei et al., 3 Jun 2026). Third, several papers imply that more expressive conditional structures—multimodal embedding alignment, multiple anchors, internal value-like coordinates, and explicit forward models—can improve both sample quality and controllability, but these advances remain fragmented across domains rather than unified under a single standard CGFM framework.