Related papers: Continuous Ordinary Differential Equations and Tra…
Neural Ordinary Differential Equations (NODEs) have proven to be a powerful modeling tool for approximating (interpolation) and forecasting (extrapolation) irregularly sampled time series data. However, their performance degrades…
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…
Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on…
There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of…
In this article, a class of second order differential equations on [0,1], driven by a general H\"older continuous function and with multiplicative noise, is considered. We first show how to solve this equation in a pathwise manner, thanks…
Continuous-depth neural networks, such as the Neural Ordinary Differential Equations (ODEs), have aroused a great deal of interest from the communities of machine learning and data science in recent years, which bridge the connection…
We study ordinary differential equations of the type $u^{(n)}(t)=f(u(t))$ with initial conditions $u(0) = u'(0) =... = u^{(m-1)}(0) = 0 $ and $u^{(m)}(0) \neq 0$ where $m \geq n$, no additional assumption is made on $f$. We establish some…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
In the design of algorithms, the greedy paradigm provides a powerful tool for solving efficiently classical computational problems, within the framework of procedural languages. However, expressing these algorithms within the declarative…
Neural ordinary differential equations (ODEs) have attracted much attention as continuous-time counterparts of deep residual neural networks (NNs), and numerous extensions for recurrent NNs have been proposed. Since the 1980s, ODEs have…
We analyze task orderings in continual learning for linear regression, assuming joint realizability of training data. We focus on orderings that greedily maximize dissimilarity between consecutive tasks, a concept briefly explored in prior…
Regular functions from infinite words to infinite words can be equivalently specified by MSO-transducers, streaming $\omega$-string transducers as well as deterministic two-way transducers with look-ahead. In their one-way restriction, the…
Continuous deep learning models, referred to as Neural Ordinary Differential Equations (Neural ODEs), have received considerable attention over the last several years. Despite their burgeoning impact, there is a lack of formal analysis…
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…
We prove that functions over the reals computable in polynomial time can be characterised using discrete ordinary differential equations (ODE), also known as finite differences. We also provide a characterisation of functions computable in…
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,…
Neural ordinary differential equations describe how values change in time. This is the reason why they gained importance in modeling sequential data, especially when the observations are made at irregular intervals. In this paper we propose…
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…
Feedforward neural networks offer a promising approach for solving differential equations. However, the reliability and accuracy of the approximation still represent delicate issues that are not fully resolved in the current literature.…
Motivated by the use of high speed circuit switches in large scale data centers, we consider the problem of circuit switch scheduling. In this problem we are given demands between pairs of servers and the goal is to schedule at every time…