Related papers: Operator-learning-inspired Modeling of Neural Ordi…

The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…

This paper introduces an operator-based neural network, the mirror-padded Fourier neural operator (MFNO), designed to learn the dynamics of stochastic systems. MFNO extends the standard Fourier neural operator (FNO) by incorporating mirror…

Driven by rapid advances in artificial intelligence and modern GPU computing capabilities, deep learning methods based on the optimization paradigm have provided new pathways to solve spatiotemporal physical problems, whose mathematical…

Modeling dynamical systems is crucial for a wide range of tasks, but it remains challenging due to complex nonlinear dynamics, limited observations, or lack of prior knowledge. Recently, data-driven approaches such as Neural Ordinary…

Operator learning is a variant of machine learning that is designed to approximate maps between function spaces from data. The Fourier Neural Operator (FNO) is one of the main model architectures used for operator learning. The FNO combines…

Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…

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…

We introduce the Laplace neural operator (LNO), which leverages the Laplace transform to decompose the input space. Unlike the Fourier Neural Operator (FNO), LNO can handle non-periodic signals, account for transient responses, and exhibit…

Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering, and mathematical problems involving functions of several variables, such as the propagation of heat…

This study presents a novel unsupervised convolutional Neural Network (NN) architecture with nonlocal interactions for solving Partial Differential Equations (PDEs). The nonlocal Peridynamic Differential Operator (PDDO) is employed as a…

Numerically solving partial differential equations typically requires fine discretization to resolve necessary spatiotemporal scales, which can be computationally expensive. Recent advances in deep learning have provided a new approach to…

Differential equations are frequently used in engineering domains, such as modeling and control of industrial systems, where safety and performance guarantees are of paramount importance. Traditional physics-based modeling approaches…

This study investigates the application of machine learning, specifically Fourier Neural Operator (FNO) and Convolutional Neural Network (CNN), to learn time-advancement operators for parametric partial differential equations (PDEs). Our…

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training…

The evolution of dynamical systems is generically governed by nonlinear partial differential equations (PDEs), whose solution, in a simulation framework, requires vast amounts of computational resources. In this work, we present a novel…

We propose derivative-informed neural operators (DINOs), a general family of neural networks to approximate operators as infinite-dimensional mappings from input function spaces to output function spaces or quantities of interest. After…

Extrapolation remains a grand challenge in deep neural networks across all application domains. We propose an operator learning method to solve time-dependent partial differential equations (PDEs) continuously and with extrapolation in time…

In computational physics, a longstanding challenge lies in finding numerical solutions to partial differential equations (PDEs). Recently, research attention has increasingly focused on Neural Operator methods, which are notable for their…

Numerical solution of partial differential equations (PDEs) plays a vital role in various fields of science and engineering. In recent years, deep neural networks (DNNs) have emerged as a powerful tool for solving PDEs, leveraging their…

Developing neural operators that accurately predict the behavior of systems governed by partial differential equations (PDEs) across unseen parameter regimes is crucial for robust generalization in scientific and engineering applications.…