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arXiv 2007.11133cs.LGstat.ML
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基于生成对抗网络的微分方程解的无监督学习

Unsupervised Learning of Solutions to Differential Equations with Generative Adversarial Networks

IACS(应用计算科学研究所);Harvard University(哈佛大学)

Dylan Randle, Pavlos Protopapas, David Sondak

摘要:

微分方程的解具有重要的科学和工程意义。近年来,人们对用神经网络求解微分方程的兴趣越来越大。本文提出了一种利用无监督神经网络求解微分方程的新方法,该方法将生成对抗网络(GANs)应用于\emph{learn the loss function},以优化神经网络。我们目前的实证结果表明,我们的方法,我们称之为微分方程GAN(DEQ GAN),可以获得多个数量级的均方误差比替代的无监督神经网络方法的基础上(平方)$L_2$,$L_1$,和Huber损失函数。此外,我们表明,DEQGAN实现的解决方案的精度是有竞争力的传统数值方法。最后,我们分析了我们的方法的稳定性,发现它对超参数的选择很敏感,我们在附录中提供了这些参数。代码可在https://github.com/dylanrandle/denn获得。请将任何电子邮件发送至dylanrandle@alumni.harvard.edu。

英文摘要:

Solutions to differential equations are of significant scientific and engineering relevance. Recently, there has been a growing interest in solving differential equations with neural networks. This work develops a novel method for solving differential equations with unsupervised neural networks that applies Generative Adversarial Networks (GANs) to \emph{learn the loss function} for optimizing the neural network. We present empirical results showing that our method, which we call Differential Equation GAN (DEQGAN), can obtain multiple orders of magnitude lower mean squared errors than an alternative unsupervised neural network method based on (squared) $L_2$, $L_1$, and Huber loss functions. Moreover, we show that DEQGAN achieves solution accuracy that is competitive with traditional numerical methods. Finally, we analyze the stability of our approach and find it to be sensitive to the selection of hyperparameters, which we provide in the appendix. Code available at https://github.com/dylanrandle/denn. Please address any electronic correspondence to dylanrandle@alumni.harvard.edu.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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