Neural ordinary differential equations, widely called Neural ODEs, are reshaping how researchers model living systems by treating biological processes as continuous flows rather than discrete snapshots. Introduced in 2018, these models blend the flexibility of deep learning with the mathematical language that biologists have used to describe dynamics for centuries. The result is a tool that can learn how cells develop, how drugs move through the body, and how diseases spread, all while respecting the continuous nature of time that traditional neural networks tend to ignore.
Why Continuous Time Matters in Biology
Most deep learning architectures process data in fixed steps. An image classifier sees one frame; a recurrent network advances one tick at a time. But biological processes do not happen in tidy intervals. A patient’s blood pressure changes continuously between the moments a nurse records it. A cell differentiates along a smooth trajectory, not in jumps. Neural ODEs address this mismatch by defining a neural network that describes the rate of change of a system at any point in time, then using a numerical solver to trace the system’s trajectory forward or backward. The approach is essentially a continuous-time version of the residual networks already familiar in deep learning, as multiple research groups have formally established.1NeurIPS Proceedings. Neural Ordinary Differential Equations (Neural ODEs) are the continuous analog of Residual Neural Networks That analogy matters because it means the enormous toolkit built around residual networks can, in principle, be adapted to work in continuous time.
For biology and health specifically, this continuity solves a practical headache. Real-world data in these fields is almost always irregularly sampled. Lab values arrive whenever a doctor orders them, not on a schedule. Single-cell gene expression is measured at a handful of developmental stages, with large gaps between them. Neural ODEs can fill those gaps naturally because they define the system’s dynamics everywhere along the time axis, not just at the observed points. That single property has opened the door to applications across a surprisingly wide range of biomedical problems.
Handling the Chaos of Electronic Health Records
Electronic health records are among the messiest datasets in all of science. A hospitalized patient might have vital signs recorded every hour, lab panels drawn every few days, and imaging done once during the entire stay. Standard time-series models struggle with this irregularity. They either force the data onto a fixed grid, losing information, or require elaborate imputation schemes that can introduce bias.
Neural ODE-based methods sidestep this by learning the underlying continuous dynamics and evaluating them whenever an observation happens to exist. One approach combines gated recurrent units with neural ODEs and Bayesian estimation to impute missing values in high-dimensional EHR time series, using the learned dynamics to fill in the sporadically observed measurements rather than relying on simple interpolation.2BioMedInformatics. A Stochastic Multivariate Irregularly Sampled Time Series Imputation Method for Electronic Health Records Another line of work frames EHR analysis through initial value problem solvers, treating the patient’s clinical trajectory as a differential equation to be solved forward from any starting point.3Proceedings of the AAAI Conference on Artificial Intelligence. IVP-VAE: Modeling EHR Time Series with Initial Value Problem Solvers The practical payoff is more accurate predictions of clinical deterioration or treatment response, even when the raw data has large temporal holes.
Watching Cells Develop in Continuous Time
Single-cell RNA sequencing has given biologists the ability to measure gene expression in individual cells, but each measurement destroys the cell. You get a snapshot of thousands of cells at a given moment, not a movie of one cell changing over time. This is a fundamental limitation: the developmental trajectory of any single cell is never directly observed.
Neural ODEs offer a way around this by learning the continuous dynamics that connect snapshots taken at different developmental stages. A tool called scNODE, for example, trains on gene expression data collected at multiple timepoints across different species and tissues, then predicts what the expression landscape looks like at intermediate times that were never directly measured.4Bioinformatics. scNODE: generative model for temporal single cell transcriptomic data prediction The model essentially reconstructs the movie of cellular development from a handful of still photographs. A broader review of generative models for cell dynamics has highlighted how Neural ODE-based approaches can overcome the inherent destructiveness of single-cell measurements, aiding mechanistic modeling of cellular development and approximating population dynamics.5PubMed Central. Generative models of cell dynamics: from Neural ODEs to flow matching
Mapping Gene Regulatory Networks
Understanding which genes regulate which others is one of the central problems in molecular biology. Traditional approaches either fit rigid mathematical models that cannot capture the full complexity of real regulatory circuits, or use purely data-driven methods that produce predictions hard to interpret biologically. Neural ODEs sit in a sweet spot between these extremes.
A framework called PHOENIX, built on neural ODEs constrained by biochemical kinetics, has been tested across several scales of biological complexity, from synchronized yeast cells with oscillating gene expression to genome-scale regulatory networks in breast cancer samples and B cells treated with the drug Rituximab.6PubMed Central. Biologically informed NeuralODEs for genome-wide regulatory dynamics The key advantage is that the model incorporates prior biological knowledge, such as known transcription factor binding relationships, while still letting the neural network learn regulatory interactions that were not anticipated. Another tool, DynNet, integrates neural ODEs with biophysical models of gene expression to learn the stochastic dynamics of gene regulatory systems involved in cell fate decisions.7Nature Communications. Inferring stochastic dynamics by biophysical Neural ODE using single-cell transcriptomics
A separate framework called Cflows combines neural ODE networks with Granger causality analysis to infer continuous cell state transitions and gene regulatory interactions from static single-cell data.8bioRxiv. Revealing dynamic temporal trajectories and underlying regulatory networks with Cflows And PerturbODE goes a step further by using interpretable neural ODEs to model how cells respond to genetic perturbations, then deriving the underlying causal regulatory network from the learned parameters, which enables simulating the effects of gene knockouts that have never been experimentally tested.9arXiv.org. Interpretable Neural ODEs for Gene Regulatory Network Discovery under Perturbations This capacity to predict the outcomes of unseen interventions is where the approach moves from descriptive science toward something genuinely useful for drug target discovery.
Drug Response and Precision Dosing
Pharmacology has always been built on differential equations. The way a drug is absorbed, distributed, metabolized, and excreted is classically modeled as a system of ODEs with parameters fit to clinical data. Neural ODEs extend this tradition by letting the network learn aspects of drug dynamics that are too complex to capture with a handful of hand-specified compartments.
A neural-pharmacokinetic/pharmacodynamic framework has been developed that combines core pharmacological principles with neural ODEs to predict how individual patients will respond to treatment over time, using only early clinical data as input.10Nature Machine Intelligence. Deep learning prediction of patient response time course from early data via neural-pharmacokinetic/pharmacodynamic modelling The approach preserves the interpretability of classical pharmacology while gaining the flexibility to capture patient-specific quirks that rigid models miss.
A related method, neural controlled differential equations, has been applied to model drug concentration profiles under multiple dosing, flexibly incorporating various types of input data and handling the discontinuous dynamics that occur each time a new dose is administered.11PubMed Central. Neural Controlled Differential Equation and Its Application in Pharmacokinetics and Pharmacodynamics For precision dosing specifically, a latent neural ODE model trained on tacrolimus pharmacokinetic data learned to organize its internal representation so that patients with different genetic profiles and dosing formulations were mapped to distinct regions, even though the model was never explicitly told about those variables.12British Journal of Clinical Pharmacology. Latent Neural Ordinary Differential Equations for Model-Informed Precision Dosing: Overcoming Structural Assumptions in Pharmacokinetics The model independently discovered that a patient’s CYP3A5 genotype and their dosing schedule were the most important factors shaping drug levels. That kind of emergent biological insight, arising without being programmed in, is exactly what makes these methods exciting for personalized medicine.
Epidemiology and Cancer Dynamics
Classical compartmental models of infectious disease, like the susceptible-infected-removed framework, have well-known limitations. They assume constant parameters, struggle with multiple waves of infection, and cannot easily incorporate immune memory or behavioral changes. Researchers have developed a hybrid approach that uses neural ODEs combined with Bayesian optimization to learn the dynamics of a modified model incorporating immune memory, producing continuous multi-wave infection profiles directly from data.13Heliyon. A combined neural ODE-Bayesian optimization approach to resolve dynamics and estimate parameters for a modified SIR model with immune memory A more ambitious model, NeuralCODE, studies how population movements between regions influence outbreak development, using neural ODEs to capture the relationships between geographic mobility and disease spread.14ACM Transactions on Knowledge Discovery from Data. NeuralCODE: Neural Compartmental Ordinary Differential Equations Model with AutoML for Interpretable Epidemic Forecasting
In oncology, the problem shifts to modeling tumor growth and treatment response. Graph Neural ODEs have been developed to model tumor microenvironment dynamics with mathematically guaranteed stability properties, capturing the spatial heterogeneity that bulk ODE models miss while maintaining the continuous-time evolution that discrete graph neural networks handle poorly.15Proceedings of the AAAI Conference on Artificial Intelligence. Graph Neural ODEs with Stability and Conservation Guarantees for Tumor Microenvironment Dynamics For survival prediction, a tumor dynamic neural ODE model has shown that its learned metrics extract enough information from longitudinal tumor size data that adding eleven standard clinical baseline covariates provided no additional predictive value for overall survival.16npj Systems Biology and Applications. Explainable deep learning for tumor dynamic modeling and overall survival prediction using Neural-ODE That is a striking result: it suggests the continuous dynamics of how a tumor responds to treatment carry more prognostic information than the standard clinical variables oncologists typically rely on.
Brain Dynamics and Human Movement
The brain is another system where continuous dynamics matter enormously. Neural activity unfolds on multiple timescales simultaneously: fast electrical spikes on the order of milliseconds, slower oscillations measured by EEG, and even slower hemodynamic changes visible in fMRI. A tool called BrainDyn uses a sheaf neural ODE to model generative brain dynamics, validated on both resting-state fMRI data from a neurodevelopmental cohort and scalp EEG from a seizure corpus, covering both slow and fast neural activity across healthy and clinical populations.17arXiv. BrainDyn: A Sheaf Neural ODE for Generative Brain Dynamics Separately, neural ODEs have been used to analyze effective connectivity between brain regions, tracking how firing rates and local field potentials in premotor and primary motor cortex influence each other during brain-machine interface tasks in primates.18PLoS ONE. Multiscale effective connectivity analysis of brain activity using neural ordinary differential equations
Outside the brain, neural ODEs are being applied to full-body human movement. A method called MP-ODE handles irregularly timed motion capture data by learning the continuous dynamics of human poses, which is useful when sensor readings arrive at uneven intervals.19Pattern Recognition Letters. Neural ordinary differential equation for irregular human motion prediction One striking demonstration involved modeling baseball pitching as a low-dimensional dynamical system. The researchers found that roughly half the variance in the latter portion of a pitching motion could be predicted from just the initial eight percent of the temporal sequence, meaning the rest of the throw unfolded according to ODE-defined dynamics in a learned latent space once the pitcher’s body was set in motion.20arXiv. Data-driven modelling of low-dimensional dynamical structures underlying complex full-body human movement That finding has implications for sports science and injury prevention, where understanding the deterministic component of complex movements could help identify when a motion deviates from its natural trajectory.
Dealing with Biological Noise
One honest limitation of standard Neural ODEs is that they are deterministic: given the same starting point, they always produce the same trajectory. Biology is not like that. Cells with identical genomes in identical environments still behave differently because of inherent molecular randomness. This has driven the development of neural stochastic differential equations, which extend the framework to include noise terms that can capture the randomness of real biological systems.
An extrinsic-noise-driven neural SDE framework has been demonstrated on three different biological systems: circadian rhythms, DNA-binding dynamics, and NF-κB signaling processes, showing that it can reconstruct how cell-to-cell variability modulates reaction dynamics.21PubMed Central. Reconstructing noisy gene regulation dynamics using extrinsic-noise-driven neural stochastic differential equations A physics-informed neural SDE network has been applied to learning cellular dynamics from single-cell RNA-seq data, modeling differentiation as a stochastic process where a neural network learns the energy landscape that cells roll down during development.22Bioinformatics. A physics-informed neural SDE network for learning cellular dynamics from time-series scRNA-seq data These stochastic extensions are essential for making Neural ODE-family methods honest about the limits of prediction in biology. A deterministic model of gene expression will always understate the range of possible outcomes for a cell; a stochastic model can at least try to capture how wide that range is.
Computational Growing Pains
For all their conceptual elegance, Neural ODEs carry a real computational cost. Training requires solving a differential equation at every forward pass, and the numerical solvers used for this can be slow, especially when the underlying dynamics are stiff. Stiffness is common in biology because biochemical systems routinely involve processes happening on very different timescales: a fast enzymatic reaction coupled to a slow transcriptional response, for instance. Standard Neural ODEs and physics-informed variants can require orders of magnitude more iterations to converge on stiff problems, and even then may fail to preserve the oscillatory frequency or amplitude of the true dynamics.23arXiv. Physics-Informed Neural ODEs with Scale-Aware Residuals for Learning Stiff Biophysical Dynamics
Work on specialized solvers is addressing this. The SciML ecosystem, for instance, provides solvers specifically designed for stiff systems that arise in biology.24npj systems biology and applications. Current state and open problems in universal differential equations for systems biology And new architectures are being designed that use trainable numerical schemes to speed up computation without sacrificing accuracy.25PubMed. Enhanced Computational Complexity in Continuous-Depth Models: Neural Ordinary Differential Equations With Trainable Numerical Schemes But the speed problem remains a genuine barrier to deploying these models in real-time clinical settings, where a prediction needs to arrive in seconds rather than minutes.
Interpretability and Trust in Clinical Settings
Speed is one barrier to clinical deployment; trust is another. When a Neural ODE predicts that a patient will deteriorate in six hours, a clinician needs to understand why the model thinks so. Black-box predictions are unacceptable in healthcare. Several of the frameworks discussed above, including PHOENIX, PerturbODE, and the neural-PK/PD models, deliberately build in biological constraints that make their internal workings at least partially transparent. The biophysically motivated neural ODE framework in-CAHOOTTS, for example, decomposes gene expression dynamics into fundamental biophysical processes like mRNA transcription and degradation, so that the learned parameters have direct biological meaning.26bioRxiv. Dynamic Gene Regulatory Network Inference with Interpretable, Biophysically-Motivated Neural ODEs
Uncertainty quantification adds another layer. In clinical and epidemiological applications, knowing how confident a model is in its predictions matters as much as the predictions themselves. A systematic assessment of uncertainty quantification methods for universal differential equations, which include Neural ODEs, found that ensemble-based and Markov chain Monte Carlo methods outperformed variational inference for quantifying uncertainty in both the model parameters and the predicted state trajectories.27PubMed Central. Assessment of uncertainty quantification in universal differential equations This is still an active area of research, and the high dimensionality of Neural ODE parameter spaces makes honest uncertainty estimates computationally expensive. But for any clinical application, these estimates are non-negotiable.
Quantitative MRI and Medical Signal Processing
The irregular-sampling advantage of Neural ODE-family methods extends beyond electronic health records into medical imaging and signal processing. In quantitative MRI, different acquisition protocols produce variable numbers of measurements at non-uniform parameter settings. Neural controlled differential equations have been shown to function as a generic tool for estimating quantitative MRI parameters regardless of sequence length or configuration, handling incomplete and irregularly sampled data from multiple types of MRI protocols including T1-mapping and dynamic contrast-enhanced imaging.28arXiv. Acquisition-Independent Deep Learning for Quantitative MRI Parameter Estimation using Neural Controlled Differential Equations This acquisition-independence is valuable because it means a single trained model could work across different scanners and imaging protocols rather than requiring retraining for each one.
For cardiac monitoring, a GAN framework called ODE-GAN learns the dynamics of heart signals in the form of an ordinary differential equation, generating realistic synthetic ECG data that could be used for data augmentation or privacy-preserving research.29Proceedings of the AAAI Conference on Artificial Intelligence. ECG ODE-GAN: Learning Ordinary Differential Equations of ECG Dynamics via Generative Adversarial Learning And a multimodal approach called RealDiffFusionNet uses neural controlled differential equations with multi-head attention to combine imaging data, time-varying clinical measurements, and static patient characteristics for disease progression modeling, designed specifically for the kind of messy, real-world data that clinical researchers actually work with.30arXiv. RealDiffFusionNet: Neural Controlled Differential Equation Informed Multi-Head Attention Fusion Networks for Disease Progression Modeling Using Real-World Data
Flow Matching and the Generative Frontier
The newest evolution in this space involves connecting Neural ODEs to generative models through a technique called flow matching. Where a standard Neural ODE learns the dynamics of one system, flow matching learns a mapping between entire distributions of data. Infusing optimal transport principles into continuous normalizing flows enables training a Neural ODE that matches distributions of samples across time, minimizing displacement costs while reconstructing how populations evolve.5PubMed Central. Generative models of cell dynamics: from Neural ODEs to flow matching This is particularly well suited for single-cell biology, where you observe populations of cells at each timepoint rather than tracking individual cells. Flow matching has been described as a powerful framework for learning mappings between high-dimensional data distributions, making it applicable to problems across molecular and cell biology.31Nature Machine Intelligence. Flow matching for generative modelling in bioinformatics and computational biology
The practical implications are significant. Rather than trying to infer what happens to individual cells between measurement timepoints, a flow matching model learns how the entire cloud of cells transforms over time. This population-level view aligns naturally with how single-cell experiments actually work, where you sacrifice one batch of cells for each measurement and need to connect the dots between batches. It also opens the door to generating synthetic single-cell datasets for testing hypotheses before running expensive experiments, or for simulating the effects of drugs on cell populations before they are tested in the lab.