Artificial neural networks for solving ordinary and partial differential equations [PDF]
IEEE Trans. Neural Networks, 1998 We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts.
Isaac E. Lagaris +5 more
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Learning the solution operator of parametric partial differential equations with physics-informed DeepONets [PDF]
Science Advances, 2021 Enabling the rapid emulation of parametric differential equations with physics-informed deep operator networks.
Sifan Wang, Hanwen Wang, P. Perdikaris
semanticscholar +1 more source
Physics-Informed Neural Operator for Learning Partial Differential Equations [PDF]
ACM / IMS Journal of Data Science, 2021 In this paper, we propose physics-informed neural operators (PINO) that combine training data and physics constraints to learn the solution operator of a given family of parametric Partial Differential Equations (PDE).
Zong-Yi Li +7 more
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Transformer for Partial Differential Equations' Operator Learning [PDF]
Trans. Mach. Learn. Res., 2022 Data-driven learning of partial differential equations' solution operators has recently emerged as a promising paradigm for approximating the underlying solutions.
Zijie Li, Kazem Meidani, A. Farimani
semanticscholar +1 more source
User’s guide to viscosity solutions of second order partial differential equations [PDF]
, 1992 The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be
M. Crandall, H. Ishii, P. Lions
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Three ways to solve partial differential equations with neural networks — A review [PDF]
GAMM-Mitteilungen, 2021 Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and their suitability ...
Jan Blechschmidt, O. Ernst
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Solving high-dimensional partial differential equations using deep learning [PDF]
Proceedings of the National Academy of Sciences of the United States of America, 2017 Significance Partial differential equations (PDEs) are among the most ubiquitous tools used in modeling problems in nature. However, solving high-dimensional PDEs has been notoriously difficult due to the “curse of dimensionality.” This paper introduces ...
Jiequn Han, Arnulf Jentzen, Weinan E
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Data-driven discovery of partial differential equations [PDF]
Science Advances, 2016 Researchers propose sparse regression for identifying governing partial differential equations for spatiotemporal systems. We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by ...
S. Rudy, S. Brunton, J. Proctor, J. Kutz
semanticscholar +1 more source
Stochastic Hyperbolic Systems, Small Perturbations and Pathwise Approximation [PDF]
, 2020 This paper is devoted to the study of hyperbolic systems of linear partial differential equations perturbed by a Brownian motion. The existence and uniqueness of solutions are proved by an energy method.
Aboulalaa, Adnan
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Regular polynomial interpolation and approximation of global solutions of linear partial differential equations [PDF]
, 2007 We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations.
Kampen, Joerg
core +4 more sources