相关论文: A review of shape-morphing solutions and evolution…

We propose a Spiking Neural Network (SNN)-based explicit numerical scheme for long time integration of time-dependent Ordinary and Partial Differential Equations (ODEs, PDEs). The core element of the method is a SNN, trained to use…

The numerical solution of time-dependent radiative transfer problems is challenging, both, due to the high dimension as well as the anisotropic structure of the underlying integro-partial differential equation. In this paper we propose a…

Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…

We propose a novel multi-scale message passing neural network algorithm for learning the solutions of time-dependent PDEs. Our algorithm possesses both temporal and spatial multi-scale resolution features by incorporating multi-scale…

The variational multiscale (VMS) formulation formally segregates the evolution of the coarse-scales from the fine-scales. VMS modeling requires the approximation of the impact of the fine scales in terms of the coarse scales. In linear…

We study dynamical Galerkin schemes for evolutionary partial differential equations (PDEs), where the projection operator changes over time. When selecting a subset of basis functions, the projection operator is non-differentiable in time…

One of the main challenges in solving time-dependent partial differential equations is to develop computationally efficient solvers that are accurate and stable. Here, we introduce a graph neural network approach to finding efficient PDE…

Training neural networks sequentially in time to approximate solution fields of time-dependent partial differential equations can be beneficial for preserving causality and other physics properties; however, the sequential-in-time training…

We present a general abstract framework for the systematic numerical approximation of dissipative evolution problems. The approach is based on rewriting the evolution problem in a particular form that complies with an underlying energy or…

The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in science and engineering. It is an integro-differential equation involving time,…

We propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central…

An important class of spatio-temporal models is constructed by leveraging the hierarchical structure of dynamical (or, state-space) models. This paper proposes a new statistical dynamical model for spatio-temporal processes motivated by…

Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology…

The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards…

Friedrichs' systems (FS) are symmetric positive linear systems of first-order partial differential equations (PDEs), which provide a unified framework for describing various elliptic, parabolic and hyperbolic semi-linear PDEs such as the…

In this paper we investigate the numerical solution of stochastic partial differential equations (SPDEs) for a wider class of stochastic equations. We focus on non-diagonal colored noise instead of the usual space-time white noise. By…

In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based…

We propose the *State Space Neural Operator* (SS-NO), a compact architecture for learning solution operators of time-dependent partial differential equations (PDEs). Our formulation extends structured state space models (SSMs) to joint…

We develop fast and scalable methods for computing reduced-order nonlinear solutions (RONS). RONS was recently proposed as a framework for reduced-order modeling of time-dependent partial differential equations (PDEs), where the modes…

Neural network-based solvers for partial differential equations (PDEs) have attracted considerable attention, yet they often face challenges in accuracy and computational efficiency. In this work, we focus on time-dependent PDEs and observe…