TracingFlow: A Simulation-Free Trajectory Inference Framework Based on Second-Order Dynamics

Inferring continuous system evolution from sparse temporal snapshots is a key challenge in generative modeling and single-cell omics. While Optimal Transport (OT) is popular, existing frameworks are largely restricted to first-order dynamics, assuming memoryless velocity fields. This limits expressiveness, as first-order systems fail to account for regulatory momentum and time-delayed responses inherent in processes like cell differentiation. Here, we introduce TracingFlow, a simulation-free Flow Matching framework generalizing to second-order dynamics. By using neural networks to regress the acceleration field, TracingFlow provides an exact, efficient solution to the Dynamical Optimal Acceleration Transport (DOAT) problem. Unlike first-order methods yielding over-smoothed trajectories, our second-order formulation captures high-curvature transitions and nonlinear evolutions by learning the underlying force fields. Evaluated on complex synthetic and large-scale scRNA-seq datasets, TracingFlow achieves superior accuracy in distributional reconstruction and trajectory faithfulness. Moreover, by integrating lineage tracing priors, it recovers dynamical structures that are both mathematically optimal and biologically plausible.

Trajectory inference via Acceleration Matching

Trajectory inference is a fundamental problem in many scientific domains: given a collection of unpaired snapshots of observations at discrete time points, the goal is to generate smooth trajectories that best resemble and interpolate the data. Existing algorithms exhibit computational challenges: they either rely on preprocessing subroutines to enforce smoothness or on simulation-based training objectives, both of which can be expensive. In order to overcome these limitations, we propose a new algorithm called Acceleration Matching (\texttt{AM}). Our approach consists of lifting the original interpolation problem to phase space and then regressing onto an explicit conditional acceleration field that induces random, smooth trajectories that agree with the prescribed marginals. Importantly, our resulting training algorithm only requires positional data, avoids trajectory simulation during training, and is devoid of expensive preprocessing. We provide ample numerical evidence suggesting that \texttt{AM} is competitive with or superior to existing algorithms on several benchmark problems from the existing literature.

Twisted Schrödinger Bridge Matching

Over the past few years, diffusion-based Schrödinger bridge models have been proposed to approximate optimal transport dynamics between two prescribed boundary distributions, with successful applications to generative modeling. More precisely, these methods aim to estimate a path measure whose initial and terminal marginals match the two boundary distributions, while minimizing the Kullback-Leibler divergence with respect to a reference Markov process. In this work, we consider the generalized Schrödinger bridge problem, in which the reference process is a twisted Brownian motion, that is, a Feynman-Kac transform of a Brownian motion induced by a time-dependent differentiable potential. Building on the Iterative Markovian Fitting (IMF) paradigm, and in particular on its special case Diffusion Schrödinger Bridge Matching (DSBM), which corresponds to the zero potential case, we introduce Twisted Schrödinger Bridge Matching (TSBM), a diffusion-based method designed to handle both continuous- and discrete-time potentials. Unlike previous approaches, TSBM provides a rigorous extension of the IMF scheme to the generalized Schrödinger bridge problem. This derivation leads to a new bridge-matching loss that depends explicitly on the gradient of the potential and recovers the DSBM objective when the potential vanishes, yielding improved performance. We further introduce trajectory-based variance-reduction techniques that substantially stabilize optimization and may be useful beyond the present setting. Finally, we empirically demonstrate the benefits of TSBM for trajectory inference across increasingly high-dimensional settings, including crowd navigation and single-cell data. Code available at https://github.com/maxencenoble/twisted-sb-matching.

SpaCellAgent: A Self-Evolving LLM-Based Multi-Agent Framework for Trajectory Analysis

Spatial and Single-cell transcriptomics are transformative in deciphering cellular dynamics. As the fundamental paradigm for reconstructing cell developmental paths, trajectory inference (TI) is critical. However, existing methods require extensive manual intervention and proficiency in heterogeneous tools, posing a significant barrier to efficient TI analysis. To bridge this gap, we propose SpaCellAgent, an autonomous large language model (LLM) multi-agent framework that automates end-to-end spatiotemporal analysis and narrative generation. SpaCellAgent utilizes a multi-agent architecture for strategic workflow planning, a dynamic tool-orchestration engine for adaptive algorithm selection, and a self-evolution module that iteratively refines performance through feedback. We evaluate SpaCellAgent on six heterogeneous datasets encompassing complex temporal developmental trajectories, diverse sequencing platforms, and spatially-resolved tissue architectures. SpaCellAgent consistently demonstrates over 40\% improvement in analytical efficiency while maintaining expert-aligned performance. By converting natural language specifications into optimized analytical workflows and fully automating the pipeline, SpaCellAgent democratizes advanced spatiotemporal modeling and establishes a scalable, agent-driven paradigm for computational biology. The code and materials are available at https://github.com/LittleXH-shw/SpaCellAgent.

Wasserstein Residuals: Learning Gradient Flows from Population Dynamics

Reconstructing population dynamics is a central problem in the physical and data sciences. Often, the dynamics are modeled as a Wasserstein gradient flow (WGF): a curve of distributions driven by an energy functional. Though there are multiple mathematical characterizations of a WGF, the dominant algorithmic approach relies on the Jordan--Kinderlehrer--Otto (JKO) scheme. JKO-based methods are inflexible to time discretisation and require solving costly optimal transport problems. We take a residual approach, enforcing the continuity equations via a non-negative loss function whose minimum is the WGF. Combined with a data-fitting divergence, this gives a single global objective. This perspective unifies several existing methods and leads to a new particle-based method, stitching, that is simulation-free and robust to large gaps between observations. We demonstrate that the stitching method achieves state-of-the-art performance across trajectory inference benchmarks. For code see github.com/BasisResearch/wasserstein-residuals.

Trajectory Inference of Human Aging from Cross-Sectional DNA Methylation Data

DNA methylation (DNAm) serves as one of the most robust molecular biomarkers of biological aging. While conventional epigenetic clocks accurately predict chronological age from high-dimensional CpG profiles, they treat aging as a static regression task, meaning they can only output a single score rather than simulating how an entire profile continuously changes over time. To reconstruct these continuous dynamics, we frame lifelong human epigenetic aging as a trajectory inference problem across discrete age snapshots derived from widely available cross-sectional data. We introduce a two-stage computational pipeline: first, an age-regularized Variational Autoencoder (VAE) maps high-dimensional CpG profiles onto a chronologically ordered latent manifold while preserving a generative decoder bridge back to the original methylation space. Second, we model the continuous movement across this latent space via Regularized Unbalanced Optimal Transport (RUOT) that unifies deterministic drift, random diffusion, and non-conservative mass changes. By resolving this RUOT formulation using the DeepRUOT framework, our model fluidly accommodates population-level density shifts like survivorship bias and cellular attrition without requiring rigid biological priors. Evaluated on a large-scale, 80-year pan-tissue dataset, our model demonstrates robust distribution interpolation and uncovers a prominent late-life surge in the learned growth field that mathematically captures the variance expansion driven by stochastic epigenetic drift. Finally, by decoding continuous latent paths back to individual CpG sites, we reconstruct and empirically verify distinct biological aging archetypes, offering a rigorous, generative paradigm for simulating human molecular aging.

From Snapshots to Trajectories: Learning Single-Cell Gene Expression Dynamics via Conditional Flow Matching

Single-cell RNA sequencing (scRNA-seq) provides high-dimensional profiles of cellular states, enabling data-driven modeling of cellular dynamics over time. In practice, time-resolved scRNA-seq is collected at only a few discrete time points as unpaired snapshot populations, leaving substantial temporal gaps. This motivates trajectory inference at unmeasured time points. Existing methods mainly follow two directions, optimal-transport (OT) alignment provides distribution-level matching between observed snapshots, while continuous-time generative models support forecasting via learned dynamics. However, two challenges remain: (i) unpaired snapshots render local transitions between adjacent time points ambiguous, leading to unstable supervision; and (ii) long-horizon prediction relies on repeated integration, where small modeling errors compound and cause distribution drift. To address these challenges, we propose single-cell Flow Matching (scFM), a latent generative framework based on coupling-conditioned flow matching. First, we compute entropically regularized OT couplings between adjacent snapshots and use them to construct soft, weighted flow-matching targets for learning time-dependent velocity fields. Second, we learn bidirectional velocity fields and leverage their consistency to refine couplings and improve temporal coherence under sparse supervision. Third, we introduce distribution-level alignment and latent dynamic regularization to anchor long rollouts and mitigate drift. Experiments on real-world time-series scRNA-seq datasets show that scFM consistently improves distributional prediction performance for both temporal interpolation and extrapolation. Moreover, scFM yields more accurate trajectory reconstruction and temporally coherent visualizations where intermediate time points are absent, indicating a more faithful recovery of underlying temporal gene expression dynamics.