Temporally Expansive Flow Matching

• Temporally expansive flow matching is a generative modeling approach that decouples the ODE-based transformation from a global time axis to support scalable, variable-length sequence generation.
• It introduces innovations like segmentwise velocity networks, discrete frame insertions, and hybrid continuous–discrete flows to enhance efficiency and parallelized synthesis in high-dimensional data.
• Advanced conditioning techniques such as semantic feature alignment and residual feature approximation are employed to improve inference accuracy and reduce computational complexity.

Temporally expansive flow matching refers to a family of generative modeling techniques that relax the strict coupling between continuous ODE-based transformations and the time axis in standard flow matching, enabling improved scalability, variable-length sequence generation, and enhanced temporal context handling. By integrating ideas from continuous flows, stochastic frame or event insertions, and segmentwise velocity parameterization, temporally expansive flow matching supports a variety of non-autoregressive, parallelized, and computationally efficient generative pipelines for high-dimensional timeseries, video, and event-structured data.

1. Mathematical Framework of Temporally Expansive Flow Matching

Temporally expansive flow matching generalizes classical flow matching by decoupling the generative ODE from a strictly global time coordinate, introducing mechanisms such as segmentwise modeling, discrete insertions, and temporally local conditioning. The foundational construction involves learning a velocity field or flow map that transports a simple base distribution toward the data distribution. For a sample $x_t$ along the generative trajectory, the classical continuous-time flow matching objective is:

$$\min_\theta \mathbb{E}_{x_0, x_1, t} \|v(x_t, t) - v_\theta(x_t, t)\|^2,$$

where $v(x_t, t)$ is the target velocity along the continuum $t \in [0, 1]$ interpolating between $x_0$ (noise) and $x_1$ (data) (Park et al., 24 Oct 2025).

Temporally expansive flow matching expands the model's flexibility via several approaches:

• Temporal Segmentation: The interval $[0,1]$ is partitioned into $M$ segments, each assigned a specialist velocity network $v_\theta^{(m)}$ responsible for $[t_{m-1}, t_m)$ (Park et al., 24 Oct 2025). Within each segment, starting and ending points and the velocity target are explicitly defined.
• Discrete Insertions and Variable-Length Flows: In video (Flowception), a global reveal scheduler and stochastic slot-insertion process allow the generative path to interleave continuous denoising with frame insertions. Each frame $X^i$ evolves by its own denoising time $t_i$, and the total sequence length is variable and learned (Ifriqi et al., 12 Dec 2025).
• Segmentwise/ODE–Jump Procedures: Sequences can expand over time, with new elements initialized from noise and then denoised via continuous ODE integration, yielding coarse-to-fine synthesis.

2. Core Components and Algorithms

Key algorithmic innovations underpinning temporally expansive flow matching include blockwise specialization, context-aware feature alignment, and hybrid continuous–discrete dynamical treatment.

Blockwise Flow Matching

In Blockwise Flow Matching (BFM), the domain is split into $M$ temporal blocks. Each block's velocity network $v_\theta^{(m)}(x_t, t, c)$ is trained with a loss:

$$\mathcal{L}_{\text{BFM}}^{(m)}(\theta) = \mathbb{E}_{x_0, x_1, t \in [t_{m-1}, t_m)} \|v_\theta^{(m)}(x_t, t, c) - v_t^{(m)}\|^2,$$

where the target velocity $$v_t^{(m)} = (x_{t_m} - x_{t_{m-1}})/(t_m - t_{m-1})$$ relies on segment endpoints (Park et al., 24 Oct 2025). This modular scheme leads to smaller network footprints, segment-specific inductive bias, and reduced inference complexity.

Frame Insertion and Denoising in Video

Flowception introduces a generative process on videos that alternates between inserting new frames (initialized as $\mathcal{N}(0, I)$) and denoising each active frame via ODE integration. For each frame slot $i$, the probability of insertion per small time step is:

$h \, \rho_\kappa(t_g)\, \lambda_i^\theta(X, t),$

with $\rho_\kappa$ the scheduler-dependent hazard, $\lambda_i^\theta$ the insertion score, and $t_g$ the overall reveal time. Denoising of frames is governed by a velocity head $v_i^\theta$, driving each $X^i$ individually (Ifriqi et al., 12 Dec 2025).

Hybrid Continuous–Discrete Flows in Long Horizon Forecasting

Unified flow matching for event forecasting combines continuous flows for inter-event times and discrete flows for event types. The loss is additive:

$$\mathcal{L}_{\text{total}} = \mathcal{L}_{\text{cont}} + \lambda \mathcal{L}_{\text{disc}},$$

where the discrete component addresses marks via flow on the simplex, enabling joint, non-autoregressive modeling (Shou, 6 Aug 2025).

3. Training Procedures and Conditioning Mechanisms

Efficient and semantically-rich conditioning is central to high-fidelity temporally expansive flow matching.

Feature Alignment and Semantic Feature Guidance

Semantic Feature Guidance modules supply high-level context by aligning the blockwise velocity networks' conditioning features $f_\phi(x_t, c)$ with a frozen pretrained encoder (e.g., DINOv2) via an auxiliary loss:

$$\mathcal{L}_{\text{align}}(\phi, \psi) = \mathbb{E}_{x_0, x_1, t} d(h_\psi(f_\phi(x_t, c)), h^*),$$

where $h_\psi$ is a learnable MLP, $h^*$ is the reference embedding, and $d$ is a similarity metric (Park et al., 24 Oct 2025).

Residual Feature Approximation for Efficient Inference

During inference, computing high-dimensional semantic features at every time step becomes prohibitively expensive. Feature Residual Approximation (FRN) uses small segmentwise residual networks $f_\eta$ to approximate $f_\phi(x_t, c)$, reducing the evaluation cost by orders of magnitude (Park et al., 24 Oct 2025).

Multi-modal and Temporally Aligned Conditioning

In multi-modal flow matching (e.g., JAM-Flow for speech+lip synthesis), temporally scaled rotary positional embeddings (RoPE) synchronize different-length sequences to a common clock. Selective joint attention layers enforce local, diagonal, and temporal masking to couple streams only where necessary, retaining modality-specific inductive biases (Kwon et al., 30 Jun 2025).

4. Computational Complexity and Empirical Results

Temporally expansive flow matching markedly improves the Pareto frontier of FLOPs, real-time throughput, and generative quality across domains.

Method / Model	ODE Steps	GFLOPs	FID (↓) / FVD (↓)	Runtime (s)	Key Dataset
SiT-XL (single net)	246	114.5	2.06 (FID)	44.5	ImageNet 256
BFM-XLₛf (M=6, SemFeat)	246	107.8	1.75 (FID)	40.4	ImageNet 256
BFM-XLₛf-RA (w/ FRN)	246	37.8	2.03 (FID)	19.4	ImageNet 256
Flowception (Ours)	2000	N/A	21.80 (FVD)	—	RealEstate10K

Further, Flowception achieves substantial reductions in training and sampling FLOPs—approximately a factor of $3\times$ over full-sequence flows—while maintaining or improving sample quality (e.g., 19% relative decrease in FVD on Kinetics-600 image-to-video synthesis) (Ifriqi et al., 12 Dec 2025). In long-horizon event forecasting, temporally expansive flow matching provides effective parallel, non-autoregressive generation, reducing sequence-level error by 4–10% versus diffusion baselines, and sample times by factors of 8–12$\times$ (Shou, 6 Aug 2025).

5. Applications and Domain Specific Adaptations

Temporally expansive flow matching has demonstrated high effectiveness across a range of generative modeling and forecasting scenarios:

• Image and Video Generation: Blockwise flow matching and Flowception support efficient high-fidelity image synthesis (ImageNet256 FID 1.75) and variable-length, streaming-capable video with improved FVD and VBench metrics (Park et al., 24 Oct 2025, Ifriqi et al., 12 Dec 2025).
• Multi-modal Synthesis: JAM-Flow synchronizes audio and facial motion in talking head generation by aligning temporal flows across modalities using inpainting-style objectives and joint attention mechanisms (Kwon et al., 30 Jun 2025).
• Temporal Point Process Forecasting: Both continuous event-flow methods (EventFlow, Unified Flow Matching) leverage temporally expansive formulations to sidestep autoregressive error propagation and allow non-autoregressive, parallel sampling of future event trajectories (Kerrigan et al., 9 Oct 2024, Shou, 6 Aug 2025).
• Spatiotemporal PDE Modeling: Operator Flow Matching with Fourier Neural Operators (TempO) attains state-of-the-art long-horizon forecasting on PDE datasets, exploiting the smoothness and efficiency inherent in continuous-time flow matching (Lee et al., 16 Oct 2025).

6. Relation to Consistency Models and Flow Map Matching

Flow map matching (FMM) subsumes traditional consistency models and temporally expansive approaches under a single mathematical umbrella. FMM trains two-time maps $\hat X_{s, t}$ to mimic the flows $X_{s, t}$ of the underlying ODE, either via Lagrangian, Eulerian, or direct interpolant objectives. Key theorems guarantee that sufficiently expressive models minimizing these losses recover the true flow, thus connecting consistency model distillation, few-step sampling, and temporally expansive flows (Boffi et al., 11 Jun 2024).

While temporally expansive flow matching often leverages segmentwise or variable-length structure for scalability, FMM provides the theoretical guarantee and guidance for operator design and error control across all such architectures.

7. Advancements, Limitations, and Future Directions

Temporally expansive flow matching achieves practical computational savings, robustness in low-NFE regimes, and supports tasks (e.g., image-to-video, video interpolation, long-horizon event forecasting) previously inaccessible to strictly global, monolithic flows. Notable advancements include:

• $3\times$–$5\times$ FLOPs reduction in image synthesis at competitive FID (Park et al., 24 Oct 2025).
• Robust variable-length and high-resolution video generation with streaming and local attention compatibility (Ifriqi et al., 12 Dec 2025).
• Fully parallel, non-autoregressive event sequence generation free of cascading errors (Kerrigan et al., 9 Oct 2024, Shou, 6 Aug 2025).
• Theoretical guarantees via FMM and spectral operator control (Boffi et al., 11 Jun 2024, Lee et al., 16 Oct 2025).

Challenges remain in scaling these methods to extremely long sequences, managing the trade-off between block specialization and global coherence, and further integrating hybrid continuous–discrete stochastic processes. Future research directions include adaptive temporal partitioning, joint optimization across blocks or insertion regimes, and broader application to non-Euclidean and irregular temporal data.