Related papers: Data-driven discovery of Green's functions
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to derive the relations between the involved physical quantities…
Green's function plays a significant role in both theoretical analysis and numerical computing of partial differential equations (PDEs). However, in most cases, Green's function is difficult to compute. The troubles arise in the following…
In this paper, we introduce PDE-LEARN, a novel deep learning algorithm that can identify governing partial differential equations (PDEs) directly from noisy, limited measurements of a physical system of interest. PDE-LEARN uses a Rational…
Partial differential equations are often used to model various physical phenomena, such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics and image processing, and many analytic approaches or traditional…
Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, and the neural network parameters are calibrated to…
Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge. Insufficient prior knowledge hinders the determination of an accurate candidate library, while noisy observations lead to imprecise…
The data-driven discovery of partial differential equations (PDEs) consistent with spatiotemporal data is experiencing a rebirth in machine learning research. Training deep neural networks to learn such data-driven partial differential…
We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable…
Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and…
Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional…
In this paper, we present a deep surrogate model for learning the Green's function associated with the reaction-diffusion operator in rectangular domain. The U-Net architecture is utilized to effectively capture the mapping from source to…
During the past three decades, the advantageous concept of the Green's function has been extended from linear systems to nonlinear ones. At that, there exist a rigorous and an approximate extensions. The rigorous extension introduces the…
We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic…
Partial differential equations (PDEs) are indispensable for modeling many physical phenomena and also commonly used for solving image processing tasks. In the latter area, PDE-based approaches interpret image data as discretizations of…
Deep learning-based methods have shown remarkable effectiveness in solving PDEs, largely due to their ability to enable fast simulations once trained. However, despite the availability of high-performance computing infrastructure, many…
Motivated by recent research on Physics-Informed Neural Networks (PINNs), we make the first attempt to introduce the PINNs for numerical simulation of the elliptic Partial Differential Equations (PDEs) on 3D manifolds. PINNs are one of the…
Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by…
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…
We present a numerical framework for deep neural network (DNN) modeling of unknown time-dependent partial differential equations (PDE) using their trajectory data. Unlike the recent work of [Wu and Xiu, J. Comput. Phys. 2020], where the…