Differential equations describe asystem'sbehavior by specifying its instantaneous dynamics.

Neural ordinary differential equations (neural ODEs) are a popular family of continuous-depth deep learning models. In this work, we consider a large family of parameterized ODEs with continuous-in-time parameters, which include time-dependent neural ODEs. We derive a generalization bound for this class by a Lipschitz-based argument. By leveraging the analogy between neural ODEs and deep residual networks, our approach yields in particular a generalization bound for a class of deep residual networks. The bound involves the magnitude of the difference between successive weight matrices. We illustrate numerically how this quantity affects the generalization capability of neural networks.

Neural Ordinary Differential Equations (Neural ODEs) are the continuous analog of Residual Neural Networks (ResNets). We investigate whether the discrete dynamics defined by a ResNet are close to the continuous one of a Neural ODE. We first quantify the distance between the ResNet's hidden state trajectory and the solution of its corresponding Neural ODE. Our bound is tight and, on the negative side, does not go to $0$ with depth $N$ if the residual functions are not smooth with depth. On the positive side, we show that this smoothness is preserved by gradient descent for a ResNet with linear residual functions and small enough initial loss. It ensures an implicit regularization towards a limit Neural ODE at rate $\frac1N$, uniformly with depth and optimization time.

In this paper, we model the subspace of convolutional filters with a neural ordinary differential equation (ODE) to enable gradual changes in generated images. Decomposing convolutional filters over a set of filter atoms allows efficiently modeling and sampling from a subspace of high-dimensional filters. By further modeling filters atoms with a neural ODE, we show both empirically and theoretically that such introduced continuity can be propagated to the generated images, and thus achieves gradually evolved image generation. We support the proposed framework of image generation with continuous filter atoms using various experiments, including image-to-image translation and image generation conditioned on continuous labels. Without auxiliary network components and heavy supervision, the proposed continuous filter atoms allow us to easily manipulate the gradual change of generated images by controlling integration intervals of neural ordinary differential equation. This research sheds the light on using the subspace of network parameters to navigate the diverse appearance of image generation.

We propose the Nesterov neural ordinary differential equations (NesterovNODEs), whose layers solve the second-order ordinary differential equations (ODEs) limit of Nesterov's accelerated gradient (NAG) method, and a generalization called GNesterovNODEs. Taking the advantage of the convergence rate $\mathcal{O}(1/k^{2})$ of the NAG scheme, GNesterovNODEs speed up training and inference by reducing the number of function evaluations (NFEs) needed to solve the ODEs. We also prove that the adjoint state of a GNesterovNODEs also satisfies a GNesterovNODEs, thus accelerating both forward and backward ODE solvers and allowing the model to be scaled up for large-scale tasks. We empirically corroborate the advantage of GNesterovNODEs on a wide range of practical applications, including point cloud separation, image classification, and sequence modeling. Compared to NODEs, GNesterovNODEs require a significantly smaller number of NFEs while achieving better accuracy across our experiments.

The advancement of human healthspan and bioengineering relies heavily on predicting the behavior of complex biological systems. While high-throughput multiomics data is becoming increasingly abundant, converting this data into actionable predictive models remains a bottleneck. High-capacity, datadriven simulation systems are critical in this landscape; unlike classical mechanistic models restricted by prior knowledge, these architectures can infer latent interactions directly from observational data, allowing for the simulation of temporal trajectories and the anticipation of downstream intervention effects in personalized medicine and synthetic biology. To address this challenge, we introduce Neural Ordinary Differential Equations (NODEs) as a dynamic framework for learning the complex interplay between the proteome and metabolome. We applied this framework to time-series data derived from engineered Escherichia coli strains, modeling the continuous dynamics of metabolic pathways. The proposed NODE architecture demonstrates superior performance in capturing system dynamics compared to traditional machine learning pipelines. Our results show a greater than 90% improvement in root mean squared error over baselines across both Limonene (up to 94.38% improvement) and Isopentenol (up to 97.65% improvement) pathway datasets. Furthermore, the NODE models demonstrated a 1000x acceleration in inference time, establishing them as a scalable, high-fidelity tool for the next generation of metabolic engineering and biological discovery.

We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a blackbox differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.

Forecasting system behaviour near and across bifurcations is crucial for identifying potential shifts in dynamical systems. While machine learning has recently been used to learn critical transitions and bifurcation structures from data, most studies remain limited as they exclusively focus on discrete-time methods and local bifurcations. To address these limitations, we use Neural Ordinary Differential Equations which provide a data-driven framework for learning system dynamics. Our results show that Neural Ordinary Differential Equations can recover underlying bifurcation structures directly from time-series data by learning parameter-dependent vector fields. Notably, we demonstrate that Neural Ordinary Differential Equations can forecast bifurcations even beyond the parameter regions represented in the training data. We demonstrate our approach on three test cases: the Lorenz system transitioning from non-chaotic to chaotic behaviour, the Rössler system moving from chaos to period doubling, and a predator-prey model exhibiting collapse via a global bifurcation.

Recent advances in protein structure prediction, such as AlphaFold, have demonstrated the power of deep neural architectures like the Evoformer for capturing complex spatial and evolutionary constraints on protein conformation. However, the depth of the Evoformer, comprising 48 stacked blocks, introduces high computational costs and rigid layerwise discretization. Inspired by Neural Ordinary Differential Equations (Neural ODEs), we propose a continuous-depth formulation of the Evoformer, replacing its 48 discrete blocks with a Neural ODE parameterization that preserves its core attention-based operations. This continuous-time Evoformer achieves constant memory cost (in depth) via the adjoint method, while allowing a principled trade-off between runtime and accuracy through adaptive ODE solvers. Benchmarking on protein structure prediction tasks, we find that the Neural ODE-based Evoformer produces structurally plausible predictions and reliably captures certain secondary structure elements, such as alpha-helices, though it does not fully replicate the accuracy of the original architecture. However, our model achieves this performance using dramatically fewer resources, just 17.5 hours of training on a single GPU, highlighting the promise of continuous-depth models as a lightweight and interpretable alternative for biomolecular modeling. This work opens new directions for efficient and adaptive protein structure prediction frameworks.

Fluid-structure interaction is common in engineering and natural systems, where floating-body motion is governed by added mass, drag, and background flows. Modeling these dissipative dynamics is difficult: black-box neural models regress state derivatives with limited interpretability and unstable long-horizon predictions. We propose Floating-Body Hydrodynamic Neural Networks (FHNN), a physics-structured framework that predicts interpretable hydrodynamic parameters such as directional added masses, drag coefficients, and a streamfunction-based flow, and couples them with analytic equations of motion. This design constrains the hypothesis space, enhances interpretability, and stabilizes integration. On synthetic vortex datasets, FHNN achieves up to an order-of-magnitude lower error than Neural ODEs, recovers physically consistent flow fields. Compared with Hamiltonian and Lagrangian neural networks, FHNN more effectively handles dissipative dynamics while preserving interpretability, which bridges the gap between black-box learning and transparent system identification.