ConvPDE-UQ: Convolutional neural networks with quantified uncertainty for heterogeneous elliptic partial differential equations on varied domains
ConvPDE-UQ: Convolutional neural networks with quantified uncertainty for heterogeneous elliptic partial differential equations on varied domains
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DOI：
10.1016/j.jcp.2019.05.026
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发表时间：
2019-10
期刊：
影响因子：
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通讯作者：
Nick Winovich;K. Ramani;Guang Lin
中科院分区：
文献类型：
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作者：
Nick Winovich;K. Ramani;Guang Lin
在这项工作中，我们介绍了ConvPDE-UQ框架，用于使用卷积神经网络构建偏微分方程（PDE）的轻量级数值求解器。基于绿色函数的存在性和性质，从理论上证明了神经网络在变域上逼近偏微分方程求解器的合理性。这些求解器能够有效地将传统数值方法的计算需求减少到卷积网络的单个前向传递中。该网络架构还被设计为预测逐点高斯后验分布，权重被训练为最小化所观察到的解决方案的相关负对数似然。这种设置有助于同时对网络的解决方案进行训练和不确定性量化，允许求解器为其预测提供逐点不确定性。相关的训练过程避免了其他最先进的不确定性模型所使用的计算昂贵的贝叶斯推理步骤，并允许训练扩展到不同问题域学习所需的大型数据集。该框架的性能表现在三个不同的类的偏微分方程组成的两个线性椭圆问题的设置和非线性泊松问题。在每个类的单个离线训练过程之后，所提出的网络能够准确地预测线性和非线性椭圆问题的解决方案，这些问题具有在任何指定的二维域上定义的异构源项，仅使用卷积神经网络的单个前向传递。此外，预测逐点不确定性的分析与实验证据建立网络的不确定性量化方案的有效性。
In this work, we introduce the ConvPDE-UQ framework for constructing light-weight numerical solvers for partial differential equations (PDEs) using convolutional neural networks. A theoretical justification for the neural network approximation to partial differential equation solvers on varied domains is established based on the existence and properties of Green's functions. These solvers are able to effectively reduce the computational demands of traditional numerical methods into a single forward-pass of a convolutional network. The network architecture is also designed to predict pointwise Gaussian posterior distributions, with weights trained to minimize the associated negative log-likelihood of the observed solutions. This setup facilitates simultaneous training and uncertainty quantification for the network's solutions, allowing the solver to provide pointwise uncertainties for its predictions. The associated training procedure avoids the computationally expensive Bayesian inference steps used by other state-of-the-art uncertainty models and allows training to be scaled to the large data sets required for learning on varied problem domains. The performance of the framework is demonstrated on three distinct classes of PDEs consisting of two linear elliptic problem setups and a nonlinear Poisson problem. After a single offline training procedure for each class, the proposed networks are capable of accurately predicting the solutions to linear and nonlinear elliptic problems with heterogeneous source terms defined on any specified two-dimensional domain using just a single forward-pass of a convolutional neural network. Additionally, an analysis of the predicted pointwise uncertainties is presented with experimental evidence establishing the validity of the network's uncertainty quantification schema.