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.

Pathology Transport: Optimal-Transport Explanations for Clinical Data, and When Their Heatmaps (Fail to) Localize Disease

Generative models promise a route to explainable clinical AI: rather than probe a classifier, model the distributions of healthy and diseased patients and read explanations off the geometry between them. We build such a system - an optimal-transport rectified flow trained between two clinical distributions - and use it to ask a pointed question the field too rarely tests: do the resulting explanation heatmaps actually localize disease? On tabular tumour biomarkers (Breast Cancer Wisconsin) a single flow yields per-patient counterfactuals, an unsupervised malignancy score (AUROC 0.91; 0.93 +/- 0.01 across five seeds), and a label-free attribution that agrees with a supervised classifier (r ~ 0.5) - a compact, honest interpretability engine, though it never out-predicts logistic regression. Moving to chest X-rays, we show the transport heatmap is a population-level signal, not a localiser; a reconstruction-based, identity-preserving variant does localize synthetic lesions (pointing game 0.52), yet on real RSNA radiologist boxes it collapses to chance while only supervised Grad-CAM stays above it. The central result is a synthetic-to-real gap: label-free heatmaps that look compelling on planted lesions are not evidence of real localisation. We contribute a reusable optimal-transport recipe for generative explanations and a controlled benchmark for stress-testing whether they localize.

A Unified Geometric Framework for Developmental Analysis of Spatial Transcriptomic Data

High-throughput single-cell and spatial transcriptomic technologies provide high-resolution snapshots of heterogeneous cellular states, but their destructive nature prevents repeated measurements of the same cells over time. Consequently, temporal and spatial dynamics must be inferred from independently sampled, unaligned cell populations, making it challenging to reconstruct developmental trajectories. Optimal transport (OT) offers a geometric framework for aligning cell populations and inferring developmental trajectories, but many existing approaches focus on modeling the evolution of distributions of cells in gene expression space rather than the relational structure encoded by gene expression networks. To address this limitation, we introduce a geometric framework for analyzing the spatiotemporal evolution of gene expression networks through embeddings in Gromov--Wasserstein (GW) space. By representing each developmental stage as a graph combining gene expression and spatial proximity, our approach enables comparisons of network structure across time, continuous interpolation between developmental stages via GW geodesics, and quantification of network-level changes using Ollivier-Ricci curvature. We evaluate our framework on a spatiotemporal transcriptomic \textit{Drosophila} dataset and show that GW geodesic interpolations reproduce main trends in curvature dynamics observed in empirical gene expression networks. Agreement with higher-order Co-Optimal Transport (COOT) distances, which jointly represent spatial and temporal information, further validates the framework and suggests that hypernetwork representations successfully record salient biological changes across time. In general, our approach provides a unified geometric approach to study dynamically evolving biological networks.

Conditional Neural Optimal Transport for Predicting Cellular Phenotypes from Molecular Structure

High-content microscopy enables systematic profiling of cellular responses to chemical perturbations, but the scale of the chemical space makes exhaustive phenotypic characterization experimentally infeasible. This motivates computational models that can predict image-derived phenotypes without acquiring the corresponding treated cells. We formulate molecule-induced phenotype prediction as an inductive conditional transport problem in image representation space. Given a negative-control phenotype and the structure of a molecule, we aim to predict the phenotype induced by the corresponding molecule. We first evaluate classical optimal transport baselines and show that static couplings do not yield useful predictions on large-scale phenotypic image datasets. We then introduce a molecule-conditioned Neural Optimal Transport (NOT) model with a Monge-Gap regularization training objective that learns to transport negative-control unperturbed phenotypes toward perturbed phenotypes using molecular structure as conditioning information. NOT recovers molecule-specific phenotypic effects while reducing microscopy-associated technical variation, thereby facilitating comparisons across experimental batches. On unseen active molecules, the model outperforms baseline approaches, demonstrating that chemically conditioned transport can generalize beyond the molecules observed during training. We identified the molecular encoder as the main limitation to this generalization, while transport in a compressed representation space improves performance and scalability. These results establish NOT as a promising framework for predicting cellular phenotypes from molecular structure and negative-control phenotypes, while highlighting the development of more informative molecular representations as a key direction for improving out-of-distribution performance.

Control-Anchored Residual Flow Matching Conditioned on Gene Geometry for Virtual Cell Perturbation Modeling

A central task in virtual cell modeling is predicting single-cell transcriptional responses to unseen genetic perturbations and drug combinations, and biological networks provide valuable priors on gene relationships. Existing graph-based models commonly use the same network to structure gene representations and mediate intergene interactions, thereby implicitly treating stable associations as perturbation-response pathways. Gene Ontology and control-derived coexpression networks encode relatively stable relationships rather than intervention-specific response directions or magnitudes. We therefore propose GeneGeoFlow, which conditions a control-anchored residual flow on gene-wise geometry derived from biological networks to learn intervention-specific transcriptional responses. GeneGeoFlow derives multi-scale spectral coordinates from Gene Ontology and control-derived coexpression networks. A perturbation-conditioned, gene-wise gating module selects relevant structural scales and network sources, yielding intervention-specific gene geometry. The resulting geometry conditions a control-anchored residual flow without explicitly propagating target-derived signals along the graph. Condition-wise optimal transport couples unpaired control and perturbed populations for training, while a Delta-correlation objective aligns the predicted and observed condition-level expression-shift directions. GeneGeoFlow achieves Pearson Delta scores of 0.8979 on the Norman additive benchmark and 0.9088 on five held-out drug combinations in the fixed ComboSciPlex test split. These results support perturbation-conditioned gene geometry as an effective structural prior for intervention-specific response prediction, without conflating stable gene relationships with response propagation.

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.

Resolving superposition in AI for interpretability and cross-modal alignment in patient-neuronal images

Artificial intelligence is transforming our capability to solve biological challenges. In dimensionality bottleneck regimes exacerbated by high-dimensional biological data, Neural networks force distinct concepts into the lower dimensions known as superposition. Although this superposition is widely known to hinder interpretability, its impact on corrupting the geometry of latent spaces remains critically overlooked. Here, we utilized sparse autoencoders (SAEs) trained on over 100,000 multiplexed images of patient-derived Parkinson's disease and healthy neurons to resolve superposition. This approach bypasses the mathematical non-uniqueness of feature attribution by shifting to interpretable latent representation analysis. We theoretically and empirically demonstrate that superposition contaminates representational metric spaces, and thereby SAEs successfully recover geometric fidelity. By treating these geometrically purified representations as single-cell state vectors, we adapted single-cell RNA sequencing (scRNA-seq) data analysis methodologies directly to the image domain. Finally, we introduce GW-map, utilizing Gromov-Wasserstein optimal transport to align these image representations with authentic scRNA-seq data \emph{de novo}. This coupling reconstructs hierarchical neuronal pathology pathways such as Calcium-AIS scaffold, without reference spatial transcriptomics, establishing a scalable foundation for spatial biology. Code is available at https://github.com/jijihihi/Bio_superposition

One Size does not Fit All: Heterogeneous Latent Space Alignment for Unsupervised Domain Adaptation

Domain shift remains a major obstacle to the reliable deployment of machine learning models in high-stakes environments such as healthcare. While Domain adaptation aims to mitigate these effects, existing approaches suffer from limited expressiveness of latent representations and a reliance on handcrafted, static augmentations. In this work, we address these limitations by proposing a novel deep learning architecture for Unsupervised Domain Adaptation (UDA), specifically optimized for medical image segmentation. Our framework, ADualVUOT, integrates a dual-encoder Variational Autoencoder (VAE) with Continuous Normalizing Flows (CNFs) to increase modeling flexibility and posterior expressiveness. To achieve domain alignment, we leverage Unbalanced Optimal Transport (UOT) through the Gaussian-Gromov-Wasserstein (GGW) distance, which handles structural and topological discrepancies between domains. Furthermore, we incorporate an adversarial augmentation scheme to synthesize worst-case compositions, thus enhancing model robustness. Extensive experiments on medical imaging benchmarks show significant gains over prior OT-based approaches.

Feynman Kac Reweighted Schrödinger Bridge Matching for Surface-Based Tau PET Harmonization

Tau PET imaging is central to tracking Alzheimer's disease progression, but systematic differences between scanners, protocols, and radiotracers across sites introduce nonbiological variability that inflates biomarker variance, reduces sensitivity to disease effects, and can bias downstream clinical assessments. Harmonization methods aim to remove these site-induced shifts while preserving biologically meaningful signal, yet existing approaches struggle when source and target cohorts differ in subgroup composition, risking conflation of site effects with biological variation such as tau-positivity status. We propose the Feynman Kac Reweighted Schröodinger Bridge Matching (FKRSBM) model to address this problem. Rather than routing data through a Gaussian noise prior as in diffusion-based methods, FKRSBM learns a direct stochastic transport process between source and target distributions via entropy-regularized optimal transport. To enforce biologically consistent transport, FKRSBM incorporates a subgroup-aware endpoint proposal derived from a Feynman Kac reweighting of the reference bridge measure, implemented entirely through stratified importance sampling at the data level and requiring no changes to the underlying bridge-matching solver or network architecture. For surface-based neuroimaging, FKRSBM employs a spherical convolutional backbone operating on cortical meshes to perform vertex-level harmonization. We evaluate the method on tau PET SUVR maps, harmonizing PI-2620 data from the HABS-HD cohort into the AV-1451 domain of ADNI. Compared against ComBat, CycleGAN, a diffusion-based method (DF), and unregularized Diffusion Schröodinger Bridge Matching (DSBM), FKRSBM achieves superior distributional alignment, reduced tau-positivity sign mismatch, stronger APOE subgroup alignment, and improved downstream disease classification performance.

Curvature-Guided Geometric Representation for Protein-Ligand Binding Affinity Prediction

Protein-ligand binding affinity (PLA) prediction is critical in drug discovery. Despite the notable advancements in machine learning-based approaches, existing methods struggle to jointly characterize local geometric organization and globally coordinated cross-molecular interactions, limiting their ability to model complex binding mechanisms. Here, we propose RicciBind, a geometric representation framework that integrates curvature-guided hierarchical structure learning with optimal transport (OT)-based cross-domain alignment to model molecular interactions. Specifically, RicciBind leverages Ricci curvature to capture local interaction tightness within molecular structures, enhancing structural awareness and organizing atomic interactions into curvature-aware hierarchical representations. An OT-based cluster matching mechanism then aligns protein and ligand clusters across heterogeneous domains under geometric constraints, enabling globally consistent correspondences and revealing higher-order interaction patterns beyond local neighborhoods. By coupling curvature-guided structure encoding with OT-driven cross-domain alignment, RicciBind effectively models complex interaction semantics and substantially improves both the accuracy and interpretability of binding affinity prediction. Extensive experiments demonstrate that RicciBind achieved superior predictive performance and generalization across PLA benchmarks and virtual screening tasks. Ablation studies further confirmed the essential role of Ricci curvature in enhancing molecular interaction representations.

A Sliced-Wasserstein Framework on Correlation Matrices for EEG Decoding

Electroencephalography (EEG) offers noninvasive, millisecond resolution recordings of neuronal activity and is widely used in neuroscience and healthcare. Many EEG decoding pipelines rely on covariance descriptors for their robustness to noise, but such representations are sensitive to channel-wise scaling. Recent studies have therefore advocated full-rank correlation matrices as a scale-invariant alternative for EEG decoding. In this paper, we study Sliced-Wasserstein (SW) discrepancies for probability distributions on the manifold of full-rank correlation matrices. We adopt the pullback-Euclidean formulation of SW, referred to as Pullback Euclidean Metric Sliced-Wasserstein (PEMSW), and instantiate it under two recently introduced correlation geometries, \textit{i.e.}, the Off-Log Metric (OLM) and Log-Scaled Metric (LSM). This yields two Correlation Sliced-Wasserstein (CorSW) discrepancies with closed-form slicing coordinates and efficient computation through one-dimensional Wasserstein distances. Building on CorSW, we further develop a domain generalization (DG) framework for EEG decoding. Experiments on three EEG datasets demonstrate improved generalization under distribution shifts, with low training overhead and no additional inference cost. The source code is available at github.com/ChenHu-ML/CorSW.

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.

Modeling Temporal scRNA-seq Data with Latent Gaussian Process and Optimal Transport

Single-cell RNA sequencing provides insights into gene expression at single-cell resolution, yet inferring temporal processes from these static snapshot measurements remains a fundamental challenge. Current approaches utilizing neural differential equations and flows are sensitive to overfitting and lack careful considerations of biological variability. In this work, we propose a generative framework that models population trends using a latent heteroscedastic Gaussian process (GP) approximated by Hilbert space methods. To address the absence of genuine cell trajectories, we leverage an optimal transport (OT) objective that aligns generated and observed population distributions. Our method explicitly captures biological heterogeneity by incorporating cell-specific latent time and cell type conditioning to disentangle temporal asynchrony and trajectories to different cell types. We demonstrate state-of-the-art performance on complex interpolation and extrapolation benchmarks and introduce a novel gradient-based strategy for inferring perturbation trajectories.

Learning fMRI activations dictionaries across individual geometries via optimal transport

Dictionary learning is a powerful tool for creating interpretable representations. When applied to functional magnetic resonance imaging (fMRI) data, the resulting patterns of brain activity can be used for various downstream tasks, such as brain state classification or population-level analysis. However, a major challenge is the variability in brain geometry across individuals. This is usually addressed by projecting each individual brain geometry onto a common template, which removes subject-specific information. In this work, we introduce a novel approach to dictionary learning on fMRI data that explicitly accounts for this variability. We use the optimal transport-based Fused Gromov-Wasserstein (FGW) distance to compare graphs with different geometries and features. To address the challenge of computing multiple FGW distances for large graphs such as those arising from fMRI data, we rely on amortized optimization to learn a neural network that predicts an approximation of the optimal transport plans, which substantially reduces the computational cost. Additionally, we learn dictionary atoms that depend on the FGW trade-off parameter, which controls the balance between feature alignment and structural consistency. Numerical experiments on the HCP dataset demonstrate that the proposed approach captures different levels of geometric variability in the data and provides representations that preserve essential information.

RIHA: Report-Image Hierarchical Alignment for Radiology Report Generation

Radiology report generation (RRG) has emerged as a promising approach to alleviate radiologists' workload and reduce human errors by automatically generating diagnostic reports from medical images. A key challenge in RRG is achieving fine-grained alignment between complex visual features and the hierarchical structure of long-form radiology reports. Although recent methods have improved image-text representation learning, they often treat reports as flat sequences, overlooking their structured sections and semantic hierarchies. This simplification hinders precise cross-modal alignment and weakens RRG accuracy. To address this challenge, we propose RIHA (Report-Image Hierarchical Alignment Transformer), a novel end-to-end framework that performs multi-level alignment between radiological images and their corresponding reports across paragraph, sentence, and word levels. This hierarchical alignment enables more precise cross-modal mapping, essential for capturing the nuanced semantics embedded in clinical narratives. Specifically, RIHA introduces a Visual Feature Pyramid (VFP) to extract multi-scale visual features and a Text Feature Pyramid (TFP) to represent multi-granularity textual structures. These components are integrated through a Cross-modal Hierarchical Alignment (CHA) module, leveraging optimal transport to effectively align visual and textual features across various levels. Furthermore, we incorporate Relative Positional Encoding (RPE) into the decoder to model spatial and semantic relationships among tokens, enhancing the token-level alignment between visual features and generated text. Extensive experiments on two benchmark chest X-ray datasets, IU-Xray and MIMIC-CXR, demonstrate that RIHA outperforms existing state-of-the-art models in both natural language generation and clinical efficacy metrics.