Related papers: Proximal Implicit ODE Solvers for Accelerating Lea…

The neural ordinary differential equation (ODE) framework has emerged as a powerful tool for developing accelerated surrogate models of complex physical systems governed by partial differential equations (PDEs). A popular approach for PDE…

The proximal gradient algorithm has been popularly used for convex optimization. Recently, it has also been extended for nonconvex problems, and the current state-of-the-art is the nonmonotone accelerated proximal gradient algorithm.…

We give an algorithm for efficient step size control in numerical integration of non-stiff initial value problems, based on a formula tailormade to methods where the numerical solution is compared with a solution of lower order.

Modern deep learning algorithms use variations of gradient descent as their main learning methods. Gradient descent can be understood as the simplest Ordinary Differential Equation (ODE) solver; namely, the Euler method applied to the…

Neural Ordinary Differential Equations (ODE) are a promising approach to learn dynamic models from time-series data in science and engineering applications. This work aims at learning Neural ODE for stiff systems, which are usually raised…

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 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…

In previous literature, backward error analysis was used to find ordinary differential equations (ODEs) approximating the gradient descent trajectory. It was found that finite step sizes implicitly regularize solutions because terms…

Ordinary differential equations (ODEs) provide a powerful framework for modeling dynamic systems arising in a wide range of scientific domains. However, most existing ODE methods focus on a single system, and do not adequately address the…

Neural ordinary differential equations (Neural ODEs) is a class of machine learning models that approximate the time derivative of hidden states using a neural network. They are powerful tools for modeling continuous-time dynamical systems,…

This paper proposes a novel inverse kinematics (IK) solver of articulated robotic systems for path planning. IK is a traditional but essential problem for robot manipulation. Recently, data-driven methods have been proposed to quickly solve…

Nonlinear PDE solvers require fine space-time discretizations and local linearizations, leading to high memory cost and slow runtimes. Neural operators such as FNOs and DeepONets offer fast single-shot inference by learning…

Incremental gradient and incremental proximal methods are a fundamental class of optimization algorithms used for solving finite sum problems, broadly studied in the literature. Yet, without strong convexity, their convergence guarantees…

Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable…

Recent work in deep learning focuses on solving physical systems in the Ordinary Differential Equation or Partial Differential Equation. This current work proposed a variant of Convolutional Neural Networks (CNNs) that can learn the hidden…

Two-step predictor/corrector methods are provided to solve three classes of problems that present themselves as systems of ordinary differential equations (ODEs). In the first class, velocities are given from which displacements are to be…

Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration…

We propose a data-driven framework for learning reduced-order moment dynamics from PDE-governed systems using Neural ODEs. In contrast to derivative-based methods like SINDy, which necessitate densely sampled data and are sensitive to…

Neural Ordinary Differential Equations (NODEs) are a new class of models that transform data continuously through infinite-depth architectures. The continuous nature of NODEs has made them particularly suitable for learning the dynamics of…

Image registration plays an important role in medical image analysis. Conventional optimization based methods provide an accurate estimation due to the iterative process at the cost of expensive computation. Deep learning methods such as…