Related papers: Solving Image PDEs with a Shallow Network

Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…

Fast and accurate solution of time-dependent partial differential equations (PDEs) is of key interest in many research fields including physics, engineering, and biology. Generally, implicit schemes are preferred over the explicit ones for…

In this thesis we develop techniques to efficiently solve numerical Partial Differential Equations (PDEs) using Graphical Processing Units (GPUs). Focus is put on both performance and re--usability of the methods developed, to this end a…

In this work we study the problem about learning a partial differential equation (PDE) from its solution data. PDEs of various types are used as examples to illustrate how much the solution data can reveal the PDE operator depending on the…

Deriving governing equations in Electromagnetic (EM) environment based on first principles can be quite tough when there are some unknown sources of noise and other uncertainties in the system. For nonlinear multiple-physics electromagnetic…

Image deblurring techniques play important roles in many image processing applications. As the blur varies spatially across the image plane, it calls for robust and effective methods to deal with the spatially-variant blur problem. In this…

We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…

Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…

Deep learning has been highly successful in some applications. Nevertheless, its use for solving partial differential equations (PDEs) has only been of recent interest with current state-of-the-art machine learning libraries, e.g.,…

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…

In this paper we establish a connection between non-convex optimization methods for training deep neural networks and nonlinear partial differential equations (PDEs). Relaxation techniques arising in statistical physics which have already…

Model discrepancy, defined as the difference between model predictions and reality, is ubiquitous in computational models for physical systems. It is common to derive partial differential equations (PDEs) from first principles physics, but…

There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…

Fast and accurate solutions of time-dependent partial differential equations (PDEs) are of pivotal interest to many research fields, including physics, engineering, and biology. Generally, implicit/semi-implicit schemes are preferred over…

The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…

Partial differential equations (PDEs) that fit scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects, such as physics and finance. The data-driven discovery of PDEs from…

Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering, and mathematical problems involving functions of several variables, such as the propagation of heat…

Utilizing machine learning to address partial differential equations (PDEs) presents significant challenges due to the diversity of spatial domains and their corresponding state configurations, which complicates the task of encompassing all…

Graph neural networks are increasingly becoming the go-to approach in various fields such as computer vision, computational biology and chemistry, where data are naturally explained by graphs. However, unlike traditional convolutional…

Solutions of partial differential equations (PDEs) on manifolds have provided important applications in different fields in science and engineering. Existing methods are majorly based on discretization of manifolds as implicit functions,…