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Trajectory inference via Acceleration Matching

This paper introduces Acceleration Matching (AM), a computationally efficient algorithm for trajectory inference that lifts the interpolation problem to phase space to regress an explicit conditional acceleration field, thereby generating smooth trajectories from unpaired snapshots without requiring expensive preprocessing or simulation-based training.

Original authors: Bartolo Dazzini, Giovanni Conforti, Alain Durmus, Aram-Alexandre Pooladian

Published 2026-08-05

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Bartolo Dazzini, Giovanni Conforti, Alain Durmus, Aram-Alexandre Pooladian

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery, but you only have a few scattered snapshots of the suspects. You have a photo of a person at 9:00 AM, another at 11:00 AM, and a third at 1:00 PM, but you never saw them in between. Your goal is to draw a smooth, continuous line connecting these dots to show exactly how they moved through the city. This is the heart of a problem called trajectory inference, which scientists face in fields ranging from tracking ocean currents to understanding how individual cells in our bodies change over time. The tricky part is that in many of these cases, you can't watch the same person or cell continuously; the data is "unpaired," meaning the snapshot at 9:00 AM might be of a different person than the one at 11:00 AM. To make matters worse, simply drawing a straight line between the photos often looks jerky and unrealistic, like a stop-motion animation that jumps awkwardly. Scientists need a way to fill in the gaps with smooth, natural-looking movement, but the current tools to do this are often slow, expensive, or require messy pre-processing steps to force the movement to look smooth.

Enter a new method called Acceleration Matching (AM), proposed by researchers Bartolo Dazzini, Giovanni Conforti, Alain Durmus, and Aram-Alexandre Pooladian. Think of their approach as a clever trick to stop the "jerky" movement before it even starts. Instead of just guessing where a person is next (position), they decide to guess how fast and in what direction the person is speeding up or slowing down (acceleration). By lifting the problem into a "phase space"—a fancy way of saying they track both position and speed simultaneously—they can use a mathematical rule to ensure the movement is naturally smooth, just like a car that can't instantly teleport or change speed without pressing the gas or brake. The authors found that by training a computer to match these acceleration patterns, they could generate smooth, realistic paths that fit the data perfectly. Their method is faster than previous techniques because it doesn't need to simulate thousands of fake trips to learn the rules, and it doesn't need a messy pre-step to smooth things out. In tests on real-world data, like ocean currents and cell biology, their method suggested it could produce smoother, more accurate trajectories than existing algorithms, offering a fresh, efficient way to solve the mystery of how things move through time.

The Mystery of the Missing Middle

Let's dive deeper into the story. Imagine you are trying to recreate a dance routine, but you only have photos of the dancers at the start, middle, and end of the song. You don't have a video of the dance itself. If you just connect the dots between the photos, the dancers might look like they are teleporting or jerking their limbs in unnatural ways. This is the problem with many current computer methods: they try to learn the dance step-by-step, and without extra help, the result is often a clumsy, bumpy mess.

The researchers behind this paper realized that the secret to smooth movement isn't just knowing where something is, but knowing how it is moving. In physics, we know that objects don't just jump; they have velocity (speed and direction) and acceleration (how that speed changes). If you know the acceleration, you can predict a smooth path. The authors decided to stop trying to guess the position directly and instead try to guess the acceleration.

The "Acceleration Matching" Trick

The core idea of their new algorithm, Acceleration Matching (AM), is to treat the problem like a game of "guess the force." Imagine you are a coach trying to teach a robot how to dance smoothly. Instead of telling the robot, "Move your foot to this spot," you tell it, "Push your foot with this much force right now." If the robot knows how to push and pull correctly, it will naturally glide through the air, creating a smooth curve.

To do this, the authors lifted the problem into a higher dimension. Instead of just looking at a map of where the dancers are (position), they looked at a map of where they are and how fast they are moving (velocity). This is called phase space. In this space, they defined a special rule: the dancers should move according to a specific type of random motion that naturally creates smooth paths, similar to how a car drives on a bumpy road but the passengers feel a smooth ride because of the suspension.

The magic happens when they train a computer network to learn the acceleration field. This is a map that tells the system, "If you are at this spot moving at this speed, you should accelerate in this specific direction." The authors showed that if you learn this acceleration map correctly, the resulting paths will automatically be smooth and will perfectly match the snapshots you started with.

Why This is a Big Deal

Before this paper, scientists had two main ways to solve this puzzle, and both had flaws.

The "Stitching" Method: They would learn how to get from photo A to photo B, then from B to C, and so on, and try to glue the pieces together. The problem? The glue often didn't hold, and the path would look jagged where the pieces met.
The "Simulation" Method: They would run thousands of fake simulations to figure out the best path. This was like trying to learn to ride a bike by crashing into a wall a million times. It was incredibly slow and computationally expensive.

The authors' new method, Acceleration Matching, avoids both of these pitfalls.

No Stitching: Because they are learning the acceleration for the whole journey at once, the path is smooth from start to finish. There are no awkward seams.
No Simulation: This is the biggest win. Their method is "simulation-free." They don't need to run thousands of fake trips to learn the rules. They can learn the acceleration map directly from the data, making it much faster and cheaper to run.

What the Numbers Say

The authors tested their idea on several real-world datasets to see if it actually worked.

Ocean Currents: They used data from the Gulf of Mexico, tracking how water particles moved. Their method produced paths that were smoother and more accurate than other popular methods.
Predator-Prey Models: They tested it on a classic math model of how foxes and rabbits interact. Again, their method generated smoother, more realistic movements.
Single-Cell Biology: This is a high-stakes area where scientists try to understand how cells develop. They tested the method on data from embryoid bodies (clusters of cells) and CITE-seq data (a way to measure cell properties). In these complex, high-dimensional tests, their method was competitive with the best existing tools, often producing lower errors and much more stable results than some other methods that struggled to keep the paths smooth.

The results suggest that Acceleration Matching is a powerful new tool. While it didn't beat every single competitor in every single test (some methods were still slightly better in specific high-dimensional scenarios), it consistently produced smoother trajectories and did so without the heavy computational cost of simulating thousands of fake paths.

The Takeaway

In the end, this paper offers a fresh perspective on an old problem. By shifting the focus from "where" to "how fast and in what direction," the authors found a way to generate smooth, realistic movements from scattered snapshots. It's like realizing that to draw a perfect curve, you don't need to guess every single point; you just need to know how the pen should be pushed. For scientists studying everything from the flow of the ocean to the evolution of life at a cellular level, this new method provides a faster, cleaner, and more efficient way to fill in the missing pieces of the puzzle. The authors suggest that this approach could be a game-changer for making sense of complex, time-based data in the future.

Technical Summary: Trajectory Inference via Acceleration Matching

Problem Statement

Trajectory inference addresses the challenge of reconstructing smooth, continuous trajectories from a collection of unpaired observations (snapshots) collected at discrete time points $0 = t_0 < t_1 < \dots < t_J = 1$ . Given empirical probability distributions $\{\rho_{t_j}\}_{j=0}^J$ , the objective is to find a measure-valued curve $t \mapsto \mu_t$ such that the marginal law of the position at each time point matches the data: $\text{Law}(X_{t_j}) = \rho_{t_j}$ .

This problem arises in domains such as astronomy, economics, and single-cell RNA sequencing, where destructive sampling or other constraints prevent repeated measurements of the same individual over time. Existing approaches often suffer from computational inefficiencies:

Piecewise Construction: Learning independent transport maps between consecutive time steps and "stitching" them often results in trajectories lacking global smoothness due to discontinuities in the velocity field.
Simulation-Heavy Training: Methods that enforce smoothness via stochastic differential equations (SDEs) often require simulating trajectories during the training loop, which is computationally expensive even for low-dimensional data.

Methodology: Acceleration Matching (AM)

The authors propose Acceleration Matching (AM), a novel algorithm that lifts the interpolation problem from position space to phase space (the joint space of positions $x$ and velocities $v$ ). The core idea is to learn an explicit conditional acceleration field that induces smooth, random trajectories satisfying the marginal constraints without requiring trajectory simulation during training.

1. Phase Space Dynamics

The method models trajectories as solutions to a second-order stochastic differential equation (SDE):
$\begin{cases} dX_t = V_t \, dt \\ dV_t = a_t(X_t, V_t) \, dt + \sqrt{\varepsilon} \, dB_t \end{cases}$
where $a_t$ is an acceleration field and $B_t$ is standard Brownian motion. By integrating velocity to obtain position, the resulting trajectories are inherently smooth.

2. Conditional Acceleration Fields and Markovianization

The authors construct a target acceleration field based on kinetic Brownian bridges.

Interpolation Process: They define an interpolation process $P_I$ by sampling position knots $X_{t_j}$ from a coupling of the marginals and sampling the remaining path from a conditional distribution of a reference kinetic Brownian motion.
Bridge Characterization: Using Doob's $h$ -transform, they show that the conditional distribution of the path between two knots $(x_j, v_j)$ and $(x_{j+1}, v_{j+1})$ corresponds to a unique weak solution of an SDE with a specific, explicit acceleration term $a^j_t$ .
Markovianization: Since the interpolation process $P_I$ is non-Markovian (it depends on future knots), the authors define a Markovian acceleration field $a^M_t$ as the conditional expectation of the bridge acceleration given the current state:
$a^M_t(x, v) = \mathbb{E}[a^j_t(X_t, V_t; X_{t_{j+1}}, V_{t_{j+1}}) \mid (X_t, V_t) = (x, v)]$
Theorem 1 establishes that a diffusion process driven by this $a^M_t$ preserves the same marginal laws as the original non-Markovian interpolant.

3. Simulation-Free Training

The learning objective is derived directly from the definition of the Markovian field. The algorithm minimizes the mean squared error between a parametric neural network $a_\theta$ and the explicit target acceleration $a^j_t$ :
$\mathcal{L}_{AM}(a_\theta) = \sum_{j=0}^{J-1} \int_{t_j}^{t_{j+1}} \mathbb{E} \left[ \| a_\theta(t, X_t, V_t) - a^j_t(X_t, V_t; X_{t_{j+1}}, V_{t_{j+1}}) \|^2 \right] dt$
Key Training Features:

No Simulation: The target acceleration $a^j_t$ is computed in closed form using Gaussian conditioning formulas. The training loop does not require rolling out SDEs.
Data Requirements: The loss only requires positional data (the marginals). Velocities $V_t$ are sampled efficiently from a conditional Gaussian distribution given the positions, avoiding the need for ground-truth velocity data.
Initial Velocity Sampling: At inference, since only the initial position distribution is known, a separate conditional sampler $q_\phi(\cdot | x_0)$ is trained to approximate the initial velocity distribution required by the Markovian process.

Key Contributions

Acceleration Matching Algorithm: A new framework for trajectory inference that lifts the problem to phase space to learn second-order dynamics.
Explicit, Simulation-Free Objective: Unlike prior methods (e.g., 3MSBM) that require trajectory simulation during training, AM relies on a regression objective with a closed-form target, significantly reducing computational overhead.
Global Smoothness by Construction: By modeling dynamics via acceleration (second-order SDEs), the method guarantees smooth positional trajectories, avoiding the discontinuities inherent in piecewise flow matching.
Theoretical Guarantees: The paper provides proofs that the learned Markovian process satisfies the prescribed marginal constraints (Theorem 1) and derives the explicit form of the kinetic Brownian bridge acceleration (Lemma 2).

Experimental Results

The authors evaluate AM on several benchmarks, comparing it against Multi-Marginal Flow Matching (MMFM) variants and other state-of-the-art methods like OT-CFM and 3MSBM.

Low-Dimensional Datasets (Gulf of Mexico and Lotka–Volterra):
- AM achieves the lowest training and holdout 2-Wasserstein ( $W_2$ ) errors among the flow-matching variants (MMFM, MMFMconst, MMFMvan).
- While AM performs slightly worse than the reported results of 3MSBM on these specific datasets, it produces visibly smoother trajectories than piecewise flow matching.
- MMFM variants often struggle with stability between marginals or fail to match them exactly (MMFMconst smooths them, MMFMvan is unstable).
High-Dimensional Datasets (Single-Cell Biology):
- Embryoid Bodies (EB5): AM performs comparably to OT-CFM and better than MMFM variants.
- CITE-seq (CITE5 and CITE50): AM outperforms MMFMvan significantly (which is unstable) and is competitive with MMFMconst. However, on the most challenging CITE50 dataset, AM lags behind the reported errors of OT-CFM and OT-MFM.
- Visual Quality: In all cases, AM generates trajectories that are visibly smoother and more consistent with the marginal evolution than those generated by piecewise flow matching.

Significance and Claims

The paper claims that Acceleration Matching offers a principled and scalable alternative for trajectory inference. Its primary significance lies in:

Efficiency: It is entirely simulation-free during training, removing a major computational bottleneck found in Schrödinger bridge and momentum-based methods.
Simplicity: It requires minimal preprocessing (no expensive optimal transport coupling computation between all pairs of marginals) and avoids ad-hoc smoothing constraints.
Smoothness: It naturally enforces global smoothness through the physics of the learned acceleration field, rather than relying on post-processing or piecewise stitching.

The authors acknowledge limitations, noting that while AM is competitive, it does not yet universally outperform all existing methods on high-dimensional single-cell data, and future work could explore improving algorithmic complexity and extending the framework to Riemannian manifolds.