Related papers: DelayDiffEq: Generating Delay Differential Equatio…
Delays are ubiquitous in applied problems, but often do not arise as the simple constant discrete delays that analysts and numerical analysts like to treat. In this chapter we show how state-dependent delays arise naturally when modeling…
Applications in quantitative finance such as optimal trade execution, risk management of options, and optimal asset allocation involve the solution of high dimensional and nonlinear Partial Differential Equations (PDEs). The connection…
In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then,…
Stiff systems of ordinary differential equations (ODEs) are pervasive in many science and engineering fields, yet standard neural ODE approaches struggle to learn them. This limitation is the main barrier to the widespread adoption of…
Neural ordinary differential equations (ODEs) have been attracting increasing attention in various research domains recently. There have been some works studying optimization issues and approximation capabilities of neural ODEs, but their…
We design and analyse a new numerical method to solve ODE system based on the structural method. We compute approximations of solutions together with its derivatives up to order $K$ by solving an entire block corresponding to $R$ time…
Sampling from Diffusion Models can alternatively be seen as solving differential equations, where there is a challenge in balancing speed and image visual quality. ODE-based samplers offer rapid sampling time but reach a performance limit,…
This papers studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs). It presents a new framework using discrete ODEs as a central tool for computation and provides several implicit characterizations…
A numerical method for coupled 3D-1D problems with discontinuous solutions at the interfaces is derived and discussed. This extends a previous work on the subject where only continuous solutions were considered. Thanks to properly defined…
Since the advent of the ``Neural Ordinary Differential Equation (Neural ODE)'' paper, learning ODEs with deep learning has been applied to system identification, time-series forecasting, and related areas. Exploiting the diffeomorphic…
This paper concerns the stability of analytical and numerical solutions of nonlinear stochastic delay differential equations (SDDEs). We derive sufficient conditions for the stability, contractivity and asymptotic contractivity in mean…
A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a…
Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of…
We present DeepFDM, a differentiable finite-difference framework for learning spatially varying coefficients in time-dependent partial differential equations (PDEs). By embedding a classical forward-Euler discretization into a convolutional…
We introduce the Optimizing a Discrete Loss (ODIL) framework for the numerical solution of Partial Differential Equations (PDE) using machine learning tools. The framework formulates numerical methods as a minimization of discrete residuals…
Ordinary and stochastic differential equations (ODEs and SDEs) are widely used to model continuous-time processes across various scientific fields. While ODEs offer interpretability and simplicity, SDEs incorporate randomness, providing…
The Obreshkov method is a single-step multi-derivative method used in the numerical solution of differential equations and has been used in recent years in efficient circuit simulation. It has been shown that it can be made of arbitrary…
Continuous-time neural processes are performant sequential decision-makers that are built by differential equations (DE). However, their expressive power when they are deployed on computers is bottlenecked by numerical DE solvers. This…
We develop a transformer-based sequence-to-sequence model that recovers scalar ordinary differential equations (ODEs) in symbolic form from irregularly sampled and noisy observations of a single solution trajectory. We demonstrate in…
Performant numerical solving of differential equations is required for large-scale scientific modeling. In this manuscript we focus on two questions: (1) how can researchers empirically verify theoretical advances and consistently compare…