Many-query problems, arising from uncertainty quantification, Bayesian inversion, Bayesian optimal experimental design, and optimization under uncertainty-require numerous evaluations of a parameter-to-output map. These evaluations become prohibitive if this parametric map is high-dimensional and involves expensive solution of partial differential equations (PDEs). To tackle this challenge, we propose to construct surrogates for high-dimensional PDE-governed parametric maps in the form of projected neural networks that parsimoniously capture the geometry and intrinsic low-dimensionality of these maps. Specifically, we compute Jacobians of these PDE-based maps, and project the high-dimensional parameters onto a low-dimensional derivative-informed active subspace; we also project the possibly high-dimensional outputs onto their principal subspace. This exploits the fact that many high-dimensional PDE-governed parametric maps can be well-approximated in low-dimensional parameter and output subspace. We use the projection basis vectors in the active subspace as well as the principal output subspace to construct the weights for the first and last layers of the neural network, respectively. This frees us to train the weights in only the low-dimensional layers of the neural network. The architecture of the resulting neural network captures to first order, the low-dimensional structure and geometry of the parametric map. We demonstrate that the proposed projected neural network achieves greater generalization accuracy than a full neural network, especially in the limited training data regime afforded by expensive PDE-based parametric maps. Moreover, we show that the number of degrees of freedom of the inner layers of the projected network is independent of the parameter and output dimensions, and high accuracy can be achieved with weight dimension independent of the discretization dimension.
翻译:由不确定性量化、 Bayesian 内向性、 Bayesian 最佳实验设计和在不确定情况下对参数到输出图进行大量评价,这些评价变得令人望而却步。如果这一参数图是高维的,并且涉及部分差异方程(PDEs)的昂贵解决方案。为了应对这一挑战,我们提议以预测神经网络的形式为高维PDE-受控的参数图建造代孕仪,这种预测神经网络网络以低维度参数和输出亚空间为单位。具体地说,我们用这些基于PDE的直径地图的雅各才进行计算,并将高维参数投射到一个低维度的低维值动态子空间;我们利用预测基矢量的矢量矢量矢量数据,我们只能通过直径网络的一级和最后一级精度来测量直径的直径直径直径直值。