...
  • 主题
高级搜索  >
历史搜索 清空历史
首页> 外文期刊>Journal of Computational Physics >Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks
【24h】

Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks

机译：使用深神经网络的高维随机椭圆局部微分方程的自由模拟解决方案

获取原文
获取原文并翻译 | 示例
           
页面导航
  • 摘要
  • 著录项
  • 相似文献
  • 相关主题

摘要

Stochastic partial differential equations (SPDEs) are ubiquitous in engineering and computational sciences. The stochasticity arises as a consequence of uncertainty in input parameters, constitutive relations, initial/boundary conditions, etc. Because of these functional uncertainties, the stochastic parameter space is often high-dimensional, requiring hundreds, or even thousands, of parameters to describe it. This poses an insurmountable challenge to response surface modeling since the number of forward model evaluations needed to construct an accurate surrogate grows exponentially with the dimension of the uncertain parameter space; a phenomenon referred to as the curse of dimensionality. State-of-the-art methods for high-dimensional uncertainty propagation seek to alleviate the curse of dimensionality by performing dimensionality reduction in the uncertain parameter space. However, one still needs to perform forward model evaluations that potentially carry a very high computational burden. We propose a novel methodology for high-dimensional uncertainty propagation of elliptic SPDEs which lifts the requirement for a deterministic forward solver. Our approach is as follows. We parameterize the solution of the elliptic SPDE using a deep residual network (ResNet). In a departure from traditional squared residual (SR) based loss function for training the ResNet, we introduce a physics-informed loss function derived from variational principles. Specifically, our loss function is the expectation of the energy functional of the PDE over the stochastic variables. We demonstrate our solver-free approach through various examples where the elliptic SPDE is subjected to different types of high-dimensional input uncertainties. Also, we solve high-dimensional uncertainty propagation and inverse problems. (C) 2019 Elsevier Inc. All rights reserved.
机译：随机偏微分方程（SPDES）在工程和计算科学中无处不在。由于这些功能不确定性，由于这些功能的不确定性，因此，由于输入参数，本构关系，初始/边界条件等的不确定性而产生的随机性。随机参数空间通常是高维，需要数百，甚至数千个参数来描述它。这对响应曲面建模构成了不可逾越的挑战，因为需要构建准确的代理所需的正向模型评估的数量，以不确定参数空间的尺寸呈指数呈指数增长;一种现象称为维度的诅咒。用于高维不确定性传播的最先进方法，寻求通过在不确定参数空间中进行维度降低来缓解维度的诅咒。然而，一个仍然需要执行潜在携带非常高的计算负担的前向模型评估。我们提出了一种新的椭圆形型椭圆形状不确定性传播的方法，该方法升压了确定性正向求解器的要求。我们的方法如下。我们使用深度剩余网络（RESET）来参数化椭圆SPDE的解决方案。在从传统的平方剩余（SR）的损失函数的攻击中，我们介绍了从变分原理得出的物理信息丢失函数。具体地，我们的损失函数是预期PDE在随机变量上的能量功能。我们通过各种示例展示了无求解方法，其中椭圆形SPDE经受不同类型的高维输入不确定性。此外，我们解决了高维不确定性传播和逆问题。 （c）2019 Elsevier Inc.保留所有权利。

著录项

相似文献

  • 外文文献
  • 中文文献
  • 专利
获取原文

客服邮箱：kefu@zhangqiaokeyan.com

京公网安备：11010802029741号 ICP备案号：京ICP备15016152号-6 六维联合信息科技 (北京) 有限公司©版权所有

  • 客服微信

  • 服务号