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）
Elizabeth Qian, Ionut-Gabriel Farcas, Karen Willcox
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级，模型运行时间减少了高达九个数量级
英文摘要：
We present a new scientific machine learning method that learns from data a computationally inexpensive surrogate model for predicting the evolution of a system governed by a time-dependent nonlinear partial differential equation (PDE), an enabling technology for many computational algorithms used in engineering settings. Our formulation generalizes to the function space PDE setting the Operator Inference method previously developed in [B. Peherstorfer and K. Willcox, Data-driven operator inference for non-intrusive projection-based model reduction, Computer Methods in Applied Mechanics and Engineering, 306 (2016)] for systems governed by ordinary differential equations. The method brings together two main elements. First, ideas from projection-based model reduction are used to explicitly parametrize the learned model by low-dimensional polynomial operators which reflect the known form of the governing PDE. Second, supervised machine learning tools are used to infer from data the reduced operators of this physics-informed parametrization. For systems whose governing PDEs contain more general (non-polynomial) nonlinearities, the learned model performance can be improved through the use of lifting variable transformations, which expose polynomial structure in the PDE. The proposed method is demonstrated on two examples: a heat equation model problem that demonstrates the benefits of the function space formulation in terms of consistency with the underlying continuous truth, and a three-dimensional combustion simulation with over 18 million degrees of freedom, for which the learned reduced models achieve accurate predictions with a dimension reduction of five orders of magnitude and model runtime reduction of up to nine orders of magnitude.
Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
Operator Inference Aware Quadratic Manifolds with Isotropic Reduced Coordinates for Nonintrusive Model Reduction
Courant Institute of Mathematical Sciences, New York University（纽约大学应用数学科学研究所）；Computational Science Center, National Renewable Energy Laboratory（国家可再生能源实验室计算科学中心）
Robust Reduced-Order Model Stabilization for Partial Differential Equations Based on Lyapunov Theory and Extremum Seeking with Application to the 3D Boussinesq Equations
Mitsubishi Electric Research Laboratories（三菱电机研究实验室）；Interdisciplinary Center for Applied Mathematics, Virginia Tech（弗吉尼亚理工大学跨学科应用数学中心）；Department of Aeronautics and Astronautics, Massachusetts Institute of Technology（麻省理工学院航空航天系）
3D-VAR for Parametrized Partial Differential Equations: A Certified Reduced Basis Approach
Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University（亚琛高等研究学院计算工程科学研究所（AICES），亚琛工业大学）；Numerical Mathematics (IGPM), RWTH Aachen University（数值数学（IGPM），亚琛工业大学）；Aachen Institute for Advanced Study in Computational Engineering Science (AICES) and Faculty of Civil Engineering, RWTH Aachen University（亚琛高等研究学院计算工程科学研究所（AICES）和土木工程学院，亚琛工业大学）
