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We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of…

Outstanding advances in solid-state laser technology, employing the optical parametric chirped-pulse-amplification (OPCPA) technique, have led physicists to focus laser pulses to highly-relativistic intensities which led to novel schemes…

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,…

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…

Deep Implicit Functions (DIFs) represent 3D geometry with continuous signed distance functions learned through deep neural nets. Recently DIFs-based methods have been proposed to handle shape reconstruction and dense point correspondences…

We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show…

Neural-ODE parameterize a differential equation using continuous depth neural network and solve it using numerical ODE-integrator. These models offer a constant memory cost compared to models with discrete sequence of hidden layers in which…

Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently…

By interpreting the forward dynamics of the latent representation of neural networks as an ordinary differential equation, Neural Ordinary Differential Equation (Neural ODE) emerged as an effective framework for modeling a system dynamics…

End-to-end learning of dynamical systems with black-box models, such as neural ordinary differential equations (ODEs), provides a flexible framework for learning dynamics from data without prescribing a mathematical model for the dynamics.…

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…

This paper is concerned with the theory, construction and application of variable-stepsize implicit Peer two-step methods that are super-convergent for variable stepsizes, i.e., preserve their classical order achieved for uniform stepsizes…

There are many methods for finding a particular solution to a nonhomogeneous linear ordinary differential equation (ODE) with constant coefficients. The method of undetermined coefficients, Laplace transform method and differential operator…

The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…

In scientific and engineering applications, solving partial differential equations (PDEs) across various parameters and domains normally relies on resource-intensive numerical methods. Neural operators based on deep learning offered a…

In this paper, we develop a second-order accurate time-stepping scheme for the tempered time-fractional advection-dispersion equation based on a sum-of-exponentials (SOE) approximation to the convolution kernel involved in the fractional…

We consider a strongly nonlinear PDE system describing solid-solid phase transitions in shape memory alloys. The system accounts for the evolution of an order parameter (related to different symmetries of the crystal lattice in the phase…

We present a hybrid a-priori/a-posteriori goal oriented error estimator for a combination of dynamic iteration-based solution of ordinary differential equations discretized by finite elements. Our novel error estimator combines estimates…

Segmented models are widely used to describe non-stationary sequential data with discrete change points. Their estimation usually requires solving a mixed discrete-continuous optimization problem, where the segmentation is the discrete part…

State-of-the-art fully intrinsic networks for non-rigid shape matching often struggle to disambiguate the symmetries of the shapes leading to unstable correspondence predictions. Meanwhile, recent advances in the functional map framework…