VarNet: Variational Neural Networks for the Solution of Partial Differential Equations
VarNet: Variational Neural Networks for the Solution of Partial Differential Equations
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发表时间：
2019-12
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通讯作者：
Reza Khodayi-mehr;M. Zavlanos
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作者：
Reza Khodayi-mehr;M. Zavlanos
本文提出了一种新的基于模型的无监督学习方法Varnet，用于求解偏微分方程(PDE)的深度神经网络(NNS)。特别地，我们提出了一种新的损失函数，它依赖于偏微分方程组的变分(积分)形式，而不是文献中常用的微分形式。我们的损失函数是非离散化的，高度可并行化的，并且在捕获偏微分方程组的解方面更有效，因为它使用了低阶导数，并且在测量的非零时空区域上进行训练。在给定这个损失函数的情况下，我们还提出了一种基于偏微分方程残差提供的反馈来优化选择用于训练神经网络的空时样本的方法。用Varnet得到的模型是光滑的，不需要内插。它们也很容易区分，可以直接用于PDE的控制和优化。最后，Varnet可以直接合并参数PDE模型，使其成为PDE模型降阶(MOR)的自然工具。以对流扩散偏微分方程组为例，通过大量的数值实验验证了该方法的有效性。
In this paper we propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks (NNs). Particularly, we propose a novel loss function that relies on the variational (integral) form of PDEs as apposed to their differential form which is commonly used in the literature. Our loss function is discretization-free, highly parallelizable, and more effective in capturing the solution of PDEs since it employs lower-order derivatives and trains over measure non-zero regions of space-time. Given this loss function, we also propose an approach to optimally select the space-time samples, used to train the NN, that is based on the feedback provided from the PDE residual. The models obtained using VarNet are smooth and do not require interpolation. They are also easily differentiable and can directly be used for control and optimization of PDEs. Finally, VarNet can straight-forwardly incorporate parametric PDE models making it a natural tool for model order reduction (MOR) of PDEs. We demonstrate the performance of our method through extensive numerical experiments for the advection-diffusion PDE as an important case-study.