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Learning cell dynamics with neural differential equations

Nature Machine Intelligence volume 7, pages 1969–1984 (2025)Cite this article

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Abstract

Single-cell sequencing measurements facilitate the reconstruction of dynamic biology by capturing snapshot molecular profiles of individual cells. Cell fate decisions in development and disease are orchestrated through an intricate balance of deterministic and stochastic regulatory events. Drift-diffusion equations are effective in modelling single-cell dynamics from high-dimensional single-cell measurements. While existing solutions describe the deterministic dynamics associated with the drift term of these equations at the level of cell state, diffusion is modelled as a constant across cell states. To fully understand the dynamic regulatory logic in development and disease, models explicitly attuned to the balance between deterministic and stochastic biology are required. To address these limitations, we introduce scDiffEq, a generative framework for learning neural stochastic differential equations that approximate biology’s deterministic and stochastic dynamics. Using lineage-traced single-cell data, we demonstrate that scDiffEq offers an improved reconstruction of cell trajectories and prediction of cell fate from multipotent progenitors during haematopoiesis. By imparting in silico perturbations to multipotent progenitor cells, we find that scDiffEq accurately recapitulates the dynamics of CRISPR-perturbed haematopoiesis. We generalize this approach beyond lineage-traced or multi-time-point datasets to model the dynamics of single-cell data from a single time point. Using scDiffEq, we simulate high-resolution developmental cell trajectories, which can model their drift and diffusion, enabling us to study their time-dependent gene-level dynamics.

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Fig. 1: scDiffEq algorithm overview and applications.
Fig. 2: Benchmarking scDiffEq performance using lineage-traced haematopoietic development data.
Fig. 3: In silico perturbation analysis and generalization across datasets.
Fig. 4: Decomposition of cell-specific drift-diffusion dynamics and transcriptional regulation during lineage commitment.

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Data availability

All datasets used in this study were previously published and are publicly available. The LARRY dataset is available via GitHub at https://github.com/AllonKleinLab/paper-data/tree/master/Lineage_tracing_on_transcriptional_landscapes_links_state_to_fate_during_differentiation (commit: af842ce)26. The human haematopoiesis dataset, including the final list of filtered cells was provided by the authors of ref. 9 and is available under the GEO accession number GSE193517 as well as via direct download using the ‘dynamo’ Python package9. The pancreatic endocrinogenesis dataset is available via GitHub at https://github.com/theislab/scvelo_notebooks/raw/master/data/Pancreas/endocrinogenesis_day15.h5ad (commit: 2425e7f)26,43. All data generated in this study are available via Zenodo at https://doi.org/10.5281/zenodo.17238611 (ref. 53).

Code availability

An implementation of scDiffEq is available as a Python package via GitHub at https://github.com/scDiffEq/scDiffEq and https://pypi.org/project/scdiffeq/ (ref. 54). All analysis code, notebooks and scripts to reproduce the results presented in this paper are deposited in a dedicated reproducibility repository via GitHub at https://github.com/scDiffEq/scdiffeq-analyses (ref. 53). Comprehensive documentation is accessible at https://scdiffeq.com.

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Acknowledgements

We graciously thank C. Weinreb for their assistance in using the LARRY dataset. We thank S. Saksena, G. H. Ting Yeo and D. K. Gifford for their assistance in creating a benchmark comparison to PRESCIENT15. We thank X. Qiu for their assistance in comparing scDiffEq with Dynamo and using the human haematopoiesis scNT-seq dataset. We thank the entire Pinello and Getz Laboratories for thoughtful feedback and discussion throughout the preparation of this paper. We thank S. Anand for pivotal discussions that guided us towards the useful implementation of neural differential equations. We thank A. Ravi for useful conceptual design discussions. We thank D. Jia, J. Zhou and H. Levine for useful conversations and assistance in simulating synthetic datasets for proof-of-concept experiments. We thank M. Miller for editorial assistance. While working on this paper, M.E.V. was supported by the National Institutes of Health (NIH) under the Ruth L. Kirschstein National Research Service Award (grant no. 1F31CA257625) from the National Cancer Institute (NCI). L.P. was supported by the National Institutes of Health (NIH) Genomic Innovator Award (grant no. 1R35HG010717-01) and the Rappaport MGH Research Scholar Award 2024-2029. G.G. was partially funded by the Paul C. Zamecnik Chair in Oncology at the Mass General Cancer Center.

Author information

Author notes

  1. These authors jointly supervised this work:Gad Getz, Luca Pinello.

Authors and Affiliations

  1. Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA, USA

    Michael E. Vinyard

  2. Broad Institute of MIT and Harvard, Cambridge, MA, USA

    Michael E. Vinyard, Anders W. Rasmussen, Ruitong Li, Gad Getz & Luca Pinello

  3. Department of Pathology, Massachusetts General Hospital, Boston, MA, USA

    Michael E. Vinyard, Gad Getz & Luca Pinello

  4. Department of Biomedical Informatics, Harvard Medical School, Cambridge, MA, USA

    Ruitong Li

  5. Department of Systems Biology, Harvard Medical School, Boston, MA, USA

    Allon M. Klein

  6. Krantz Family Center for Cancer Research, Massachusetts General Hospital, Boston, MA, USA

    Gad Getz & Luca Pinello

  7. Department of Pathology, Harvard Medical School, Boston, MA, USA

    Gad Getz & Luca Pinello

  8. Department of Biostatistics, Harvard TH Chan School of Public Health, Boston, MA, USA

    Luca Pinello

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Contributions

All authors contributed to the conceptualization of the methodology, experiments and analyses. M.E.V. performed the M.E.V., A.W.R. and R.L. performed the fate-prediction benchmarking experiments. M.E.V. performed the time-point interpolation experiments. M.E.V. performed the gene perturbation experiments. M.E.V. and R.L. performed the investigation of model attributes and gene-level analyses. M.E.V. wrote the software package. All authors wrote the paper. A.M.K. provided guidance and edited the paper. G.G. and L.P. provided supervision and guidance in designing the experimental strategy and funded the research.

Corresponding authors

Correspondence to Gad Getz or Luca Pinello.

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Competing interests

G.G. receives research funds from IBM, Pharmacyclics/Abbvie, Bayer, Genentech, Calico, Ultima Genomics, Inocras, Google, Kite and Novartis and is an inventor on patent applications filed by the Broad Institute related to MSMuTect, MSMutSig, POLYSOLVER, SignatureAnalyzer-GPU, MSEye, MinimuMM-seq and DLBclass. He is a founder and consultant and holds privately held equity in Scorpion Therapeutics; he is also a founder of, and holds privately held equity in, Predicta Biosciences; and holds privately held equity in Antares Therapeutics. M.E.V. is a founder of, and holds privately held equity in, Quintessence Laboratories, Inc. The other authors declare no competing interests.

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Vinyard, M.E., Rasmussen, A.W., Li, R. et al. Learning cell dynamics with neural differential equations. Nat Mach Intell 7, 1969–1984 (2025). https://doi.org/10.1038/s42256-025-01150-3

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