A novel refinement measure for non-intrusive surrogate modelling of partial differential equations (PDEs) with uncertain parameters is proposed. Our approach uses an empirical interpolation procedure, where the proposed refinement measure is based on a PDE residual and probability density function of the uncertain parameters, and excludes parts of the PDE solution that are not used to compute the quantity of interest. The PDE residual used in the refinement measure is computed by using all the partial derivatives that enter the PDE separately. The proposed refinement measure is suited for efficient parametric surrogate construction when the underlying PDE is known, even when the parameter space is non-hypercube, and has no restrictions on the type of the discretisation method. Therefore, we are not restricted to conventional discretisation techniques, e.g., finite elements and finite volumes, and the proposed method is shown to be effective when used in combination with recently introduced neural network PDE solvers. We present several numerical examples with increasing complexity that demonstrate accuracy, efficiency and generality of the method.
Surrogate-to-code verification of a non-intrusive POD-GPR machine-learning emulator of peak thermomechanical fields, with application to a carbon-carbon aircraft brake disc
organization= Independent Researcher , city= Paris , country= France
Reduced operator inference for nonlinear partial differential equations
California Institute of Technology（加州理工学院）；University of Texas at Austin（德克萨斯大学奥斯汀分校）；Oden Institute for Computational Engineering and Sciences（奥登计算工程与科学研究所）
我们提出了一种新的科学机器学习方法，该方法从数据中学习计算成本低廉的代理模型，用于预测由时间依赖的非线性偏微分方程（PDE）控制的系统的演化，PDE是工程环境中使用的许多计算算法的一种使能技术。我们的公式推广到函数空间偏微分方程设置的算子推理方法以前开发的[B。Peherstorfer和K. Willcox，Data-driven operator inference for non-intrusive project-based model reduction，Computer Methods in Applied Mechanics and Engineering，306（2016）]适用于由常微分方程控制的系统。该方法汇集了两个主要要素。首先，从投影为基础的模型简化的想法被用来明确参数化的学习模型的低维多项式算子，反映了已知的形式的管理PDE。其次，监督机器学习工具用于从数据中推断这种物理信息参数化的简化算子。对于其控制偏微分方程包含更一般（非多项式）非线性的系统，学习模型的性能可以通过使用提升变量变换来提高，提升变量变换暴露了偏微分方程中的多项式结构。所提出的方法在两个示例上进行了演示：一个热方程模型问题，证明了函数空间公式在与基本连续真理一致性方面的好处，以及一个具有超过1800万个自由度的三维燃烧模拟，对于该模型，所学习的降阶模型实现了精确的预测，其中维数减少了五个数量…
