Difference between revisions of "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations"

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(Introduction)
(Introduction)
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== Introduction ==
 
== Introduction ==
  
In recent years, there has been an enormous growth in the amount of data and computing power available to researchers. Unfortunately, for many real-world scenarios, the cost of data acquisition is simply too high to collect an amount of data sufficient to guarantee robustness or convergence of training algorithms. In such situations, researchers are faced with the challenge of trying to generate results based on partial or incomplete datasets.
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In recent years, there has been an enormous growth in the amount of data and computing power available to researchers. Unfortunately, for many real-world scenarios, the cost of data acquisition is simply too high to collect an amount of data sufficient to guarantee robustness or convergence of training algorithms. In such situations, researchers are faced with the challenge of trying to generate results based on partial or incomplete datasets. Regularization techniques or methods which can artificially inflate the dataset become particularly useful in these situations; however, such techniques are often highly dependent of the specifics of the problem.
  
Luckily, for problems that we have data for, we also have a vast amount of prior-known information. this prior-known information often manifests in the form of mathematical models, specifically partial differential equations (PDEs). The authors provide a technique where a PDE model of a physical system can be used as a regularization agent that constrains the space of admissible solutions to a manageable size. This technique effectively artificially enhances the size of the dataset, allowing the optimization to converge more quickly and more accurately to the solution even when only a few training examples are available.
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Luckily, in important real-world scenarios that we endeavor to analyze, there is often a wealth of existing information from which we can draw. This existing information commonly manifests in the form of a mathematical model, particularly a set of partial differential equations (PDEs). In this paper, the authors provide a technique for incorporating the information of a physical system contained in a PDE into the optimization of a deep neural network. This technique is most useful in situations where established PDE models exist, but where our amount of available data is too small for neural network training. In essence, the accompanying PDE model can be used as a regularization agent, constraining the space of acceptable solutions to help the optimization converge more quickly and more accurately.
  
 
== Another Section ==
 
== Another Section ==
 
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Revision as of 14:19, 13 November 2020

Presented by

Cameron Meaney

Introduction

In recent years, there has been an enormous growth in the amount of data and computing power available to researchers. Unfortunately, for many real-world scenarios, the cost of data acquisition is simply too high to collect an amount of data sufficient to guarantee robustness or convergence of training algorithms. In such situations, researchers are faced with the challenge of trying to generate results based on partial or incomplete datasets. Regularization techniques or methods which can artificially inflate the dataset become particularly useful in these situations; however, such techniques are often highly dependent of the specifics of the problem.

Luckily, in important real-world scenarios that we endeavor to analyze, there is often a wealth of existing information from which we can draw. This existing information commonly manifests in the form of a mathematical model, particularly a set of partial differential equations (PDEs). In this paper, the authors provide a technique for incorporating the information of a physical system contained in a PDE into the optimization of a deep neural network. This technique is most useful in situations where established PDE models exist, but where our amount of available data is too small for neural network training. In essence, the accompanying PDE model can be used as a regularization agent, constraining the space of acceptable solutions to help the optimization converge more quickly and more accurately.

Another Section

Weeee