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Neural ordinary differential equations (Neural ODEs) propose the idea that a sequence of layers in a neural network is just a discretisation of an ODE, and thus can instead be directly modelled by a parameterised ODE. This idea has had…

Neural ordinary differential equations (ODEs) are an emerging class of deep learning models for dynamical systems. They are particularly useful for learning an ODE vector field from observed trajectories (i.e., inverse problems). We here…

When deploying a trained machine learning model in the real world, it is inevitable to receive inputs from out-of-distribution (OOD) sources. For instance, in continual learning settings, it is common to encounter OOD samples due to the…

Ordinary differential equations (ODEs) provide a powerful framework for modeling dynamic systems arising in a wide range of scientific domains. However, most existing ODE methods focus on a single system, and do not adequately address the…

A class of neural networks that gained particular interest in the last years are neural ordinary differential equations (neural ODEs). We study input-output relations of neural ODEs using dynamical systems theory and prove several results…

Neural Ordinary Differential Equations (ODEs) are elegant reinterpretations of deep networks where continuous time can replace the discrete notion of depth, ODE solvers perform forward propagation, and the adjoint method enables efficient,…

Background. Commonly, Deep Neural Networks (DNNs) generalize well on samples drawn from a distribution similar to that of the training set. However, DNNs' predictions are brittle and unreliable when the test samples are drawn from a…

Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…

Fractional Differential Equations (FDEs) are essential tools for modelling complex systems in science and engineering. They extend the traditional concepts of differentiation and integration to non-integer orders, enabling a more precise…

In numerous robotics and mechanical engineering applications, among others, data is often constrained on smooth manifolds due to the presence of rotational degrees of freedom. Common datadriven and learning-based methods such as neural…

The order/dimension of models derived on the basis of data is commonly restricted by the number of observations, or in the context of monitored systems, sensing nodes. This is particularly true for structural systems (e.g., civil or…

Irregular sampling intervals and missing values in real-world time series data present challenges for conventional methods that assume consistent intervals and complete data. Neural Ordinary Differential Equations (Neural ODEs) offer an…

Neural ordinary differential equations (NODE) have garnered significant attention for their design of continuous-depth neural networks and the ability to learn data/feature dynamics. However, for high-dimensional systems, estimating…

In this paper, we study a certain class of nonlocal partial differential equations (PDEs). The equations arise from a key problem in network science, i.e., network generation from local interaction rules, which result in a change of the…

Neural ordinary differential equations (neural ODEs) can effectively learn dynamical systems from time series data, but their behavior on graph-structured data remains poorly understood, especially when applied to graphs with different size…

Embedding nonlinear dynamical systems into artificial neural networks is a powerful new formalism for machine learning. By parameterizing ordinary differential equations (ODEs) as neural network layers, these Neural ODEs are…

A key appeal of the recently proposed Neural Ordinary Differential Equation (ODE) framework is that it seems to provide a continuous-time extension of discrete residual neural networks. As we show herein, though, trained Neural ODE models…

We investigate the convergence rates of variational posterior distributions for statistical inverse problems involving nonlinear partial differential equations (PDEs). Departing from exact Bayesian inference, variational inference…

Neural Jump ODEs model the conditional expectation between observations by neural ODEs and jump at arrival of new observations. They have demonstrated effectiveness for fully data-driven online forecasting in settings with irregular and…

Neural Ordinary Differential Equations (NODEs), a framework of continuous-depth neural networks, have been widely applied, showing exceptional efficacy in coping with some representative datasets. Recently, an augmented framework has been…