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FLUX: Geometry-Aware Longitudinal Flow Matching with Mixture of Experts

FLUX is a geometry-aware longitudinal flow matching framework that utilizes a mixture-of-experts architecture to jointly model continuous transport dynamics and discover latent regime switches from unpaired biological snapshots by learning a data-dependent metric to guide sparse velocity field decomposition.

Original authors: Josue Ortega Caro, Yongxu Zhang, Hannah M Batchelor, Sizhuang He, Jessica Cardin, Shreya Saxena

Published 2026-05-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Josue Ortega Caro, Yongxu Zhang, Hannah M Batchelor, Sizhuang He, Jessica Cardin, Shreya Saxena

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you are trying to understand how a city changes over time. You have a series of photos taken on different days, but there's a catch: you don't know which specific person in Photo A is the same person in Photo B. The people have moved, the lighting has changed, and the camera angles are slightly different. You just have a crowd of people at 9:00 AM, a crowd at 10:00 AM, and a crowd at 11:00 AM, but no names or tracking tags.

This is exactly the problem biologists face when studying things like brain cells or developing embryos. They can take snapshots of thousands of cells, but they can't follow the same cell from one moment to the next because the measurement process often destroys the cell or the cells drift around.

The paper introduces a new tool called FLUX to solve this puzzle. Here is how it works, using simple analogies:

1. The Problem: The "Straight Line" Trap

If you try to guess how the city changed between 9:00 and 10:00 by just drawing a straight line between the two crowds, you might make a mistake.

The Reality: The city is built on a curved hill. To get from point A to point B, you have to walk along the hill.
The Mistake: If you draw a straight line through the air (a "Euclidean" shortcut), you might end up floating over the buildings or sinking into the ground. In biology, this means your model might suggest cells do things that are physically impossible or biologically nonsensical.

FLUX's Solution: It first learns the shape of the "hill" (the manifold) where all the data lives. Instead of flying through the air, FLUX learns to walk along the surface of the data, respecting the natural curves and constraints of the biological system.

2. The Big Twist: The "Traffic Switch"

Biological systems don't just move; they often change how they move.

The Analogy: Imagine driving a car. In the morning, you are driving through a quiet neighborhood (Regime A). Suddenly, you hit rush hour and switch to a highway (Regime B). The rules of the road change completely.
The Challenge: Old models try to find one single set of rules for the whole trip. They get confused when the rules change.
FLUX's Solution: FLUX uses a "Mixture of Experts." Think of this as a smart traffic controller with a team of specialized drivers:
- Driver A knows how to drive in the quiet neighborhood.
- Driver B knows how to drive on the highway.
- The Router: A smart AI that looks at the current situation and says, "Okay, we are in the neighborhood now, let's use Driver A," and then later, "We hit the highway, switch to Driver B!"

FLUX doesn't just guess the path; it figures out when and why the rules of the road changed.

3. What Did They Test It On?

The authors tested FLUX on four different "cities" to see if it worked:

The Bunny (A 3D Shape): They used a digital model of a rabbit. They knew the exact shape of the surface. FLUX successfully figured out how to move points along the rabbit's curved skin without falling off, even when the data was stretched out into a high-dimensional space.
The Lorenz System (A Weather Simulator): This is a famous math model for chaotic weather. It has two distinct "modes" of behavior. FLUX correctly identified exactly when the system switched from one mode to the other, whereas other models just saw a blurry mess.
Mouse Brains (Learning a Task): They watched mice learn a visual task over several days. FLUX figured out that the brain's activity follows one set of rules during the "early learning" phase and a completely different set of rules during the "late learning" phase. It matched this discovery with the actual moment the mice started behaving differently.
Embryo Cells (Growing Up): They watched stem cells turn into specialized cells. FLUX successfully separated the "stem cell" phase from the "specialized cell" phase, identifying the exact transition point where the cells changed their internal programming.

4. The Key Takeaway

The paper claims that geometry matters. You cannot just look at the data points; you have to understand the shape they are sitting on.

If you ignore the shape, you might get the movement right but miss the reason for the movement.
If you ignore the changing rules (regimes), you might get the movement right for one moment but fail to understand the bigger picture.

FLUX combines these two ideas: it learns the shape of the data and it uses a team of specialized "drivers" to figure out when the rules of the game change. This allows scientists to reconstruct the full story of how biological systems evolve, even when they only have unpaired snapshots.

Technical Summary: FLUX – Geometry-Aware Longitudinal Flow Matching with Mixture of Experts

1. Problem Statement

Many biological systems, such as neural circuits and differentiating cell populations, evolve through continuous local dynamics while switching between latent regimes defined by learning, stimulus context, or developmental stages. However, these systems are frequently observed only as unpaired longitudinal snapshots. In scenarios like widefield calcium imaging or single-cell RNA sequencing, the same individual cells or neurons cannot be tracked across timepoints; instead, researchers observe a sequence of marginal distributions ( $\mu_0, \mu_1, \dots, \mu_{T-1}$ ).

This setting presents two coupled challenges:

Geometric Constraints: High-dimensional biological measurements often concentrate near low-dimensional, curved nonlinear manifolds. Standard Euclidean interpolants between unpaired endpoints can pass through low-density regions, violating biophysical constraints and failing to respect the intrinsic geometry of the state space.
Regime Discovery: The underlying transport mechanism (the "flow") often changes over time. For example, neural activity may shift from broad stimulus-reward expectation to specific discrimination, or cells may transition from pluripotent to lineage-committed programs. These are changes in the local dynamics (regimes), not merely spatial clusters. Existing methods often focus on transport accuracy but fail to decompose the velocity field to identify these latent dynamical regimes.

2. Methodology: FLUX

The authors introduce FLUX (Flow matching for Unpaired longitudinal data with miXture-of-experts), a framework that jointly models continuous transport and unsupervised regime discovery. The method operates in three sequential stages:

A. Geometry-Aware Conditional Paths

FLUX extends Metric Flow Matching to longitudinal settings. Instead of using straight Euclidean paths between adjacent marginals, FLUX learns a data-dependent metric from pooled observations (including unlabeled samples).

Metric Learning: A geometry backend (using RBF or RBF-MLP deep-kernel metrics) estimates the distance-to-manifold score, defining a metric $G(x)$ that penalizes paths leaving the data manifold.
Bend Network: A network $B$ parameterizes an approximate geometry-aware path $\gamma_k(\alpha)$ between endpoints $x_k$ and $x_{k+1}$ . This path bends to stay on the manifold, providing training points $z_{k,\alpha}$ and target tangents $\dot{z}_{k,\alpha}$ that respect the local geometry.

B. Regime-Switching Velocity Fields

FLUX decomposes the global velocity field $v_\theta$ into a Mixture of Experts (MoE).

Expert Vector Fields: The model learns $M$ sparse expert vector fields $f_m$ .
Router: A router network $g(t, x)$ maps time and state to expert logits.
Routing Mechanism: During training, a Straight-Through Gumbel-Softmax mechanism selects an active expert for each sample, allowing for differentiable optimization of discrete routing decisions. At inference, the regime is determined by the deterministic hard assignment (argmax).
Objective: The velocity field is trained to match the tangents of the geometry-aware paths. The router is regularized to encourage sparse, temporally coherent assignments and prevent expert collapse.

C. Training Pipeline

Training proceeds in three stages to ensure stability:

Metric Learning: Train the geometry backend on pooled samples.
Bend Network Training: Train the bend network with the frozen metric to generate geometry-aware interpolants.
Velocity Field Training: Freeze the geometry and bend networks; train the velocity experts and router on the generated paths and tangents.

3. Key Contributions

Extension to Longitudinal Marginals: The authors extend geometry-aware flow matching to ordered chains of marginal distributions, emphasizing the role of unsupervised manifold learning when trajectory labels are sparse but unlabeled observations are available.
Regime Discovery via Velocity Decomposition: They introduce a regime-switching velocity field where sparse expert routing discovers latent dynamical regimes directly from the local transport structure, rather than clustering static observations post-hoc.
Geometry-Regime Interplay: The work demonstrates that geometric learning is critical for regime discovery, particularly when regimes are defined by changes in local dynamics rather than static spatial separation.
Biological Validation: The framework is validated on two distinct biological settings:
- Neural Data: Separating early/intermediate training from late training in associative learning, coinciding with behavioral divergence.
- Developmental Biology: Recovering expression-evolution regimes associated with pluripotent versus differentiated cell populations in embryoid bodies.

4. Results

The paper evaluates FLUX on four datasets: Stanford Bunny (controlled manifold), Lorenz attractor (regime-switching dynamics), Widefield Calcium Imaging (neural learning), and Embryoid Body differentiation.

Stanford Bunny: FLUX preserves surface transport in high-dimensional embeddings where Euclidean methods fail to stay on the manifold.
Lorenz Attractor: FLUX achieves perfect regime recovery (ARI=1.0, NMI=1.0) by identifying the switch between chaotic and subcritical regimes. Crucially, FLUX without Manifold Learning (using Euclidean paths) collapses to a single expert, failing to discover the regime despite having the same MoE architecture. This proves that geometry-aware paths expose the curvature and velocity signatures necessary for expert specialization.
Neural Data: FLUX identifies a transition in cortical dynamics that aligns with the emergence of behavioral discrimination (CS+ vs. CS−), outperforming Euclidean routing and switching dynamical baselines.
Embryoid Body: While Euclidean routing performs well here (because stages are spatially separable), FLUX with geometric learning remains competitive, reinforcing that geometry is most critical when regimes are encoded in local dynamics.

Ablation Findings: The paper explicitly states that mixture-of-experts routing alone is insufficient. Without geometric learning, the model can fit local transport but fails to recover regimes when those regimes are encoded in the local dynamics rather than static clusters.

5. Significance and Claims

The authors claim that geometry-aware velocity decomposition provides a general strategy for discovering latent biological state transitions from unpaired longitudinal snapshots.

Scientific Impact: FLUX shifts the focus from purely optimizing transport accuracy (minimizing Wasserstein distance) to recovering interpretable dynamical structures. It suggests that biological systems often reorganize into distinct dynamical regimes during processes like learning and differentiation.
Limitations: The authors acknowledge that FLUX is computationally more expensive than Euclidean flow matching due to training three distinct networks. They also note that the number of experts is fixed in advance, and inferred regimes should be interpreted cautiously when local dynamics are noisy or weakly sampled.
Conclusion: The paper concludes that FLUX offers a framework for interpretable dynamical modeling of unpaired biological snapshots, bridging the gap between geometric manifold learning and regime-switching dynamical systems.