LAIOR: a hyperbolic neural ODE variational framework for interpretable single-cell manifold learning and trajectory inference

Single-cell omics data are high-dimensional, sparse, and noisy, and learning embeddings that simultaneously preserve local cell-state structure, global hierarchy, and smooth developmental trajectories remains an open problem. Existing approaches typically achieve only one of these goals: classical methods emphasize either local neighborhoods or global variance; deep generative models cluster cell types well but often fracture trajectory continuity; and hyperbolic embeddings capture hierarchy but are numerically fragile in practice. We present LAIOR (Lorentz attentive interpretable ordinary differential equation (ODE)-regularized variational autoencoder (VAE)), a unified variational framework that combines three complementary inductive biases in a single forward pass: (i) <i>Lorentz geometric regularization</i> encourages tree-like latent hierarchy while remaining numerically stable <i>via</i> tangent-space clamping and exponential-map gating; (ii) a <i>dual-path information bottleneck</i> captures coordinated biological programs rather than forcing latent independence; and (iii) <i>neural ordinary differential equation (ODE) regularization</i> stabilizes latent trajectories through explicit learned dynamics. Across 118 single-cell datasets (53 scRNA-seq and 65 scATAC-seq) benchmarked against 23 baseline methods on 22 complementary metrics, LAIOR improves manifold continuity, trajectory coherence, and embedding fidelity while retaining competitive clustering performance. Ablation and sensitivity analyses show that ODE regularization stabilizes geometric learning and dampens hyperparameter sensitivity. Architecture interpretation experiments on two well-characterized reference systems (human bone marrow and mouse pancreatic endocrinogenesis) demonstrate that LAIOR's encoder and decoder pathways decompose cellular variation into mutually exclusive, biologically coherent latent modules, and biological validation experiments on two previously unseen hematopoietic perturbation cohorts (<i>Dapp1</i> knockout and chemotherapy-induced bone marrow failure) show that the same interpretability contract transfers to perturbed biology. LAIOR generalizes across RNA and chromatin accessibility modalities without architectural changes. Head-to-head comparisons against both dynamical baselines (scTour) and foundation models (scGPT, scFoundation) confirm that explicit geometric and dynamical inductive biases recover trajectory structure that large-scale pretraining alone does not. Together, these results establish LAIOR as a practical, interpretable framework for single-cell manifold and trajectory analysis.