# DGM: A deep learning algorithm for solving partial differential equations

@article{Sirignano2017DGMAD, title={DGM: A deep learning algorithm for solving partial differential equations}, author={Justin A. Sirignano and Konstantinos V. Spiliopoulos}, journal={J. Comput. Phys.}, year={2017}, volume={375}, pages={1339-1364} }

## 1,101 Citations

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### A deep learning approach for solving Poisson’s equations

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A mesh-free deep learning method for solving PDE systems, especially for Poisson’s equations, is studied, which design suitable neural networks that can approximate solutions of a PDE hy formulating it as an optimization problem.

### r-Adaptive Deep Learning Method for Solving Partial Differential Equations

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An r − adaptive algorithm to solve Partial Diﬀerential Equations using a Deep Neural Network and focuses on the latter to solve one- and two-dimensional problems whose solutions are smooth, singular, and/or exhibit strong gradients.

### Solving Nonlinear and High-Dimensional Partial Differential Equations via Deep Learning

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The main goals of this paper are to elucidate the features, capabilities and limitations of DGM by analyzing aspects of its implementation for a number of different PDEs and PDE systems.

### D3M: A Deep Domain Decomposition Method for Partial Differential Equations

- Computer ScienceIEEE Access
- 2020

A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs) and shows that the D3M approximation solution converges to the exact solution of the underlying PDEs.

### Meta-Auto-Decoder for Solving Parametric Partial Differential Equations

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- 2021

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions,…

### The neural network collocation method for solving partial differential equations

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- 2020

This technique could provide a foundation for deep learning methods to be used as preconditioners to traditional methods, where the deep learning method will get close to a solution and traditional solvers can finish the solution.

### A mesh-free method for interface problems using the deep learning approach

- Computer ScienceJ. Comput. Phys.
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### FinNet: Finite Diﬀerence Neural Network for Solving Diﬀerential Equations

- Computer Science
- 2022

This work analyzes potential issues with the well-known Physic Informed Neural Network for diﬀerential equations with little constraints on the boundary and introduces a novel technique called FinNet, for solving di-erential equations by incorporating ﬁnite di-erence into deep learning.

### Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions

- Computer Science, MathematicsArXiv
- 2020

It is shown that deep neural networks are capable of representing solutions of the Poisson equation without incurring the curse of dimension and the proofs are based on a probabilistic representation of the solution to thePoisson equation as well as a suitable sampling method.

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