Related papers: Loss Jump During Loss Switch in Solving PDEs with …

Partial differential equations (PDEs) are typically used as models of physical processes but are also of great interest in PDE-based image processing. However, when it comes to their use in imaging, conventional numerical methods for…

Equations governing physico-chemical processes are usually known at microscopic spatial scales, yet one suspects that there exist equations, e.g. in the form of Partial Differential Equations (PDEs), that can explain the system evolution at…

Whilst the partial differential equations that govern the dynamics of our world have been studied in great depth for centuries, solving them for complex, high-dimensional conditions and domains still presents an incredibly large…

This work aims to estimate the drift and diffusion functions in stochastic differential equations (SDEs) driven by a particular class of L\'evy processes with finite jump intensity, using neural networks. We propose a framework that…

In this work we systematically analyze general properties of differential equations used as machine learning models. We demonstrate that the gradient of the loss function with respect to to the hidden state can be considered as a…

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…

Systems modeled by partial differential equations (PDEs) are at least as ubiquitous as systems that are by nature finite-dimensional and modeled by ordinary differential equations (ODEs). And yet, systematic and readily usable…

Physics-informed neural network (PINN) is a data-driven solver for partial and ordinary differential equations(ODEs/PDEs). It provides a unified framework to address both forward and inverse problems. However, the complexity of the…

Spatiotemporal partial differential equations (PDEs) underpin a wide range of scientific and engineering applications. Neural PDE solvers offer a promising alternative to classical numerical methods. However, existing approaches typically…

We use elliptic partial differential equations (PDEs) as examples to show various properties and behaviors when shallow neural networks (SNNs) are used to represent the solutions. In particular, we study the numerical ill-conditioning,…

Designing universal artificial intelligence (AI) solver for partial differential equations (PDEs) is an open-ended problem and a significant challenge in science and engineering. Currently, data-driven solvers have achieved great success,…

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…

Physics-informed neural networks (PINNs) have recently become a powerful tool for solving partial differential equations (PDEs). However, finding a set of neural network parameters that lead to fulfilling a PDE can be challenging and…

It has been demonstrated that deep neural networks outperform traditional machine learning. However, deep networks lack generalisability, that is, they will not perform as good as in a new (testing) set drawn from a different distribution…

There have been growing interests in leveraging experimental measurements to discover the underlying partial differential equations (PDEs) that govern complex physical phenomena. Although past research attempts have achieved great success…

There have been extensive studies on solving differential equations using physics-informed neural networks. While this method has proven advantageous in many cases, a major criticism lies in its lack of analytical error bounds. Therefore,…

The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modeled into the equations as random coefficients. However, very often the variability of…

Many natural systems, such as neurons firing in the brain or basketball teams traversing a court, give rise to time series data with complex, nonlinear dynamics. We can gain insight into these systems by decomposing the data into segments…

A quadratic approximation of neural network loss landscapes has been extensively used to study the optimization process of these networks. Though, it usually holds in a very small neighborhood of the minimum, it cannot explain many…

Ordinary differential equations (ODEs), via their induced flow maps, provide a powerful framework to parameterize invertible transformations for the purpose of representing complex probability distributions. While such models have achieved…