论文标题
从线性抛物线偏微分方程控制的系统数据中学到的非侵入性降低模型的概率误差估计
Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations
论文作者
论文摘要
这项工作得出了一个基于残差的后验误差估计器，用于减少模型从非侵入性模型中学到的减少模型，该模型是从具有控制输入的线性抛物线派抛物线偏微分方程控制的高维系统的数据。结果表明，误差估计器所必需的数量可以完全作为最小二乘问题的解决方案以非侵入性的方式从数据，例如初始条件，控制输入和高维解决方案轨迹等数据中获得，或以概率意义界定。计算过程遵循离线/在线分解。在离线（训练）阶段，高维系统以黑盒方式明智地解决，以生成数据并设置误差估计器。在在线阶段，估算器用于限制新初始条件和新的控制输入的减少模型预测的误差，而无需求助于高维系统。数值结果证明了从数据到还原模型再到认证预测的拟议方法的工作流程。
This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. The computational procedure follows an offline/online decomposition. In the offline (training) phase, the high-dimensional system is judiciously solved in a black-box fashion to generate data and to set up the error estimator. In the online phase, the estimator is used to bound the error of the reduced-model predictions for new initial conditions and new control inputs without recourse to the high-dimensional system. Numerical results demonstrate the workflow of the proposed approach from data to reduced models to certified predictions.
