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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…

This article presents a novel resolution to the problem of spline interpolation versus least-squares fitting on smooth Riemannian manifolds utilizing the method of gradient flows of networks. This approach represents a contribution to both…

Neural controlled differential equations (Neural CDEs) are a continuous-time extension of recurrent neural networks (RNNs), achieving state-of-the-art (SOTA) performance at modelling functions of irregular time series. In order to interpret…

Implicit-depth models such as Deep Equilibrium Networks have recently been shown to match or exceed the performance of traditional deep networks while being much more memory efficient. However, these models suffer from unstable convergence…

Problems involving approximation from scattered data where data is arranged quasi-uniformly have been treated by RBF methods for decades. Treating data with spatially varying density has not been investigated with the same intensity, and is…

Inverse problem or parameter estimation of ordinary differential equations (ODEs), the iterative process of minimizing the mismatch between model-predicted and experimental states by tuning the parameter values within an optimization…

This paper investigates the mean square exponential stabilization problem for a class of coupled PDE-ODE systems with Markov jump parameters. The considered system consists of multiple coupled hyperbolic PDEs and a finite-dimensional ODE,…

Optical coherence tomography (OCT) is a prevalent imaging technique for retina. However, it is affected by multiplicative speckle noise that can degrade the visibility of essential anatomical structures, including blood vessels and tissue…

A neural ordinary differential equation (neural ODE) is a machine learning model that is commonly described as a continuous-depth generalization of a residual network (ResNet) with a single residual block, or conversely, the ResNet can be…

Solving initial value problems and boundary value problems of Linear Ordinary Differential Equations (ODEs) plays an important role in many applications. There are various numerical methods and solvers to obtain approximate solutions…

Due to their dynamic properties such as irregular sampling rate and high-frequency sampling, Continuous Time Series (CTS) are found in many applications. Since CTS with irregular sampling rate are difficult to model with standard Recurrent…

Implicit models separate the definition of a layer from the description of its solution process. While implicit layers allow features such as depth to adapt to new scenarios and inputs automatically, this adaptivity makes its computational…

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…

We present Neural Splines, a technique for 3D surface reconstruction that is based on random feature kernels arising from infinitely-wide shallow ReLU networks. Our method achieves state-of-the-art results, outperforming recent neural…

Neural ordinary differential equations (neural ODEs) have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of their optimal architecture remains elusive. In this work, we examine the…

Numerically solving ordinary differential equations (ODEs) is a naturally serial process and as a result the vast majority of ODE solver software are serial. In this manuscript we developed a set of parallelized ODE solvers using…

In deep learning, often the training process finds an interpolator (a solution with 0 training loss), but the test loss is still low. This phenomenon, known as benign overfitting, is a major mystery that received a lot of recent attention.…

Modelling statistical relationships beyond the conditional mean is crucial in many settings. Conditional density estimation (CDE) aims to learn the full conditional probability density from data. Though highly expressive, neural network…

This paper addresses the optimization problem of minimizing non-convex continuous functions, which is relevant in the context of high-dimensional machine learning applications characterized by over-parametrization. We analyze a randomized…

We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary…