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Motivated by the remarkable success of artificial intelligence (AI) across diverse fields, the application of AI to solve scientific problems, often formulated as partial differential equations (PDEs), has garnered increasing attention.…
Partial differential equation (PDE) models are widely used in engineering and natural sciences to describe spatio-temporal processes. The parameters of the considered processes are often unknown and have to be estimated from experimental…
Efficient and stable solution of partial differential equations (PDEs) is central to scientific and engineering applications, yet existing numerical solvers rely heavily on matrix based discretizations, while learning based methods require…
In recent years, the researches about solving partial differential equations (PDEs) based on artificial neural network have attracted considerable attention. In these researches, the neural network models are usually designed depend on…
In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation…
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin.…
Modeling complex spatiotemporal dynamical systems, such as the reaction-diffusion processes, have largely relied on partial differential equations (PDEs). However, due to insufficient prior knowledge on some under-explored dynamical…
Physics-Informed Neural Networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs). PINNs are based on simple architectures, and learn the behavior of complex…
Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds…
Partial Differential Equations (PDEs) model various physical phenomena, such as electromagnetic fields and fluid mechanics. Methods like Sparse Identification of Nonlinear Dynamics (SINDy) and PDE-Net 2.0 have been developed to identify and…
This book encompasses both traditional and modern methods treating partial differential equation (PDE) of first order and second order. There is a balance in making a selfcontained mathematical text and introducing new subjects. The Lie…
Partial differential equations (PDEs) play a foundational role in modeling physical phenomena. This study addresses the challenging task of determining variable coefficients within PDEs from measurement data. We introduce a novel neural…
Many physics and engineering applications demand Partial Differential Equations (PDE) property evaluations that are traditionally computed with resource-intensive high-fidelity numerical solvers. Data-driven surrogate models provide an…
A unified method for extracting geometric shape features from binary image data using a steady state partial differential equation (PDE) system as a boundary value problem is presented in this paper. The PDE and functions are formulated to…
Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on…
Simulations of complex physical systems are typically realized by discretizing partial differential equations (PDEs) on unstructured meshes. While neural networks have recently been explored for surrogate and reduced order modeling of PDE…
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…
Inpainting-based image compression is emerging as a promising competitor to transform-based compression techniques. Its key idea is to reconstruct image information from only few known regions through inpainting. Specific partial…
Partial differential equations often contain unknown functions that are difficult or impossible to measure directly, hampering our ability to derive predictions from the model. Workflows for recovering scalar PDE parameters from data are…
Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs),…