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We introduce an ODE solver for the PyTorch ecosystem that can solve multiple ODEs in parallel independently from each other while achieving significant performance gains. Our implementation tracks each ODE's progress separately and is…

In this work, a novel quantum Fourier ordinary differential equation (ODE) solver is proposed to solve both linear and nonlinear partial differential equations (PDEs). Traditional quantum ODE solvers transform a PDE into an ODE system via…

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…

Learning neural ODEs often requires solving very stiff ODE systems, primarily using explicit adaptive step size ODE solvers. These solvers are computationally expensive, requiring the use of tiny step sizes for numerical stability and…

The existing Neural ODE formulation relies on an explicit knowledge of the termination time. We extend Neural ODEs to implicitly defined termination criteria modeled by neural event functions, which can be chained together and…

Ordinary differential equations (ODEs) describe dynamical systems evolving deterministically in continuous time. Accurate data-driven modeling of systems as ODEs, a central problem across the natural sciences, remains challenging,…

Solving linear ordinary differential equations (ODE) is one of the most promising applications for quantum computers to demonstrate exponential advantages. The challenge of designing a quantum ODE algorithm is how to embed non-unitary…

Calibrating chemical kinetics in a reaction-diffusion system is challenging because of complex dynamics governed by tightly coupled chemistry and transport, while experimental observations are often sparse and noisy. We propose a physics…

Ordinary differential equations (ODE) have been widely used for modeling dynamical complex systems. For high-dimensional ODE models where the number of differential equations is large, it remains challenging to estimate the ODE parameters…

The spatiotemporal resolution of Partial Differential Equations (PDEs) plays important roles in the mathematical description of the world's physical phenomena. In general, scientists and engineers solve PDEs numerically by the use of…

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…

The infinite-depth paradigm pioneered by Neural ODEs has launched a renaissance in the search for novel dynamical system-inspired deep learning primitives; however, their utilization in problems of non-trivial size has often proved…

Operator learning has emerged as a promising paradigm for developing efficient surrogate models to solve partial differential equations (PDEs). However, existing approaches often overlook the domain knowledge inherent in the underlying PDEs…

We consider adaptive finite element methods for solving a multiscale system consisting of a macroscale model comprising a system of reaction-diffusion partial differential equations coupled to a microscale model comprising a system of…

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…

Diffusion models (DMs) create samples from a data distribution by starting from random noise and iteratively solving a reverse-time ordinary differential equation (ODE). Because each step in the iterative solution requires an expensive…

When a system of first order linear ordinary differential equations has eigenvalues of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing…

The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…

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…

Recent research in deep learning has shown that neural networks can learn differential equations governing dynamical systems. In this paper, we adapt this concept to Virtual Analog (VA) modeling to learn the ordinary differential equations…