Related papers: Convection-Diffusion Equation: A Theoretically Cer…

An abstract framework is developed that enables the analysis of algebraically stabilized discretizations in a unified way. This framework is applied to a discretization of this kind for convection-diffusion-reaction equations. The…

Convolutional layers are one of the basic building blocks of modern deep neural networks. One fundamental assumption is that convolutional kernels should be shared for all examples in a dataset. We propose conditionally parameterized…

Recent studies reveal the connection between GNNs and the diffusion process, which motivates many diffusion-based GNNs to be proposed. However, since these two mechanisms are closely related, one fundamental question naturally arises: Is…

Based on a natural connection between ResNet and transport equation or its characteristic equation, we propose a continuous flow model for both ResNet and plain net. Through this continuous model, a ResNet can be explicitly constructed as a…

Since the seminal work of [9] and their Physics-Informed neural networks (PINNs), many efforts have been conducted towards solving partial differential equations (PDEs) with Deep Learning models. However, some challenges remain, for…

Score-based diffusion models in infinite-dimensional function spaces provide a mathematically principled framework for modelling function-valued data, offering key advantages such as resolution invariance and the ability to handle irregular…

Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these…

Recent studies have demonstrated the success of deep learning in solving forward and inverse problems in engineering and scientific computing domains, such as physics-informed neural networks (PINNs). Source inversion problems under sparse…

This work attempts to interpret modern deep (convolutional) networks from the principles of rate reduction and (shift) invariant classification. We show that the basic iterative gradient ascent scheme for optimizing the rate reduction of…

The layered structure of deep neural networks hinders the use of numerous analysis tools and thus the development of its interpretability. Inspired by the success of functional brain networks, we propose a novel framework for…

A framework of finite-velocity model based Boltzmann equation has been developed for convection-diffusion equations. These velocities are kept flexible and adjusted to control numerical diffusion. A flux difference splitting based kinetic…

We propose a novel method for simulating conditioned diffusion processes (diffusion bridges) in Euclidean spaces. By training a neural network to approximate bridge dynamics, our approach eliminates the need for computationally intensive…

Operational weather forecasting models have advanced for decades on both the explicit numerical solvers and the empirical physical parameterization schemes. However, the involved high computational costs and uncertainties in these existing…

Over the last few years, Neural Processes have become a useful modelling tool in many application areas, such as healthcare and climate sciences, in which data are scarce and prediction uncertainty estimates are indispensable. However, the…

Many complex systems have natural representations as multi-layer networks. While these formulations retain more information than standard single-layer network models, there is not yet a fully developed theory for computing network metrics…

With the growth of deep learning, how to describe deep neural networks unifiedly is becoming an important issue. We first formalize neural networks mathematically with their directed graph representations, and prove a generation theorem…

This paper proposes and studies two extensions of applying hp-variational physics-informed neural networks, more precisely the FastVPINNs framework, to convection-dominated convection-diffusion-reaction problems. First, a term in the spirit…

Diffusion-driven instability is a fundamental mechanism underlying pattern formation in spatially extended systems. In almost all existing works, diffusion across the links of the underlying network is modeled through scalar weights,…

The gradient scheme framework is based on a small number of properties and encompasses a large number of numerical methods for diffusion models. We recall these properties and develop some new generic tools associated with the gradient…

Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays,…