In this study, we present a new framework of physics-informed non-intrusive reduced-order modeling (ROM) of dynamical systems modeled by parametric, partial differential equations (PDEs). Given new time and parameter values of a PDE, the framework utilizes trained physics-informed ML models to quickly estimate high-fidelity solutions while simultaneously observing the constraints and dynamics of the system. In the offline training phase, proper orthogonal decomposition (POD) decomposes a training database of high-fidelity solutions into POD modes and POD coefficients. A feed-forward neural network is trained to map time-parameter values to the few dominant POD coefficients. The loss function is composed of two terms: (1) error between original data and reconstructed data and (2) PDE residuals where each term of the PDE is expressed using Galerkin expansion on the reduced basis composed of the most dominant POD modes. The PDE residuals are not evaluated using POD–Galerkin (reduced-order) equations. The novelty of this work lies in the construction of PDE residual term and an a priori analysis that allows one to select weighting factor (or Lagrange multiplier) ahead of it. It has been found that a physics-informed ROM minimizing the two terms generates new solutions orders-of-magnitude accurate than a vanilla ROM that minimizes only the first error term. Besides estimating reconstruction error on a database, the framework also allows estimation of reconstruction quality of different terms such as advection and diffusion in the PDE. This is expected to promote better integration and interpretation of ML in reduced-order modeling of dynamical systems. During the online prediction phase, given new values of time and parameters, the generalized coordinates are quickly estimated and used in reconstruction. High-fidelity solutions are thus obtained orders-of-magnitude faster than a conventional numerical simulation. The framework is demonstrated on 1D and 2D Burgers’ equations and an incompressible flow over a backward facing step.

中文翻译：

---

参数化动力系统的物理信息非侵入式降阶建模


在这项研究中，我们提出了一种新的物理信息非侵入式降阶建模 （ROM） 框架，该框架由参数偏微分方程 （PDE） 建模。给定 PDE 的新时间和参数值，该框架利用经过训练的物理信息 ML 模型来快速估计高保真解决方案，同时观察系统的约束和动态。在离线训练阶段，适当的正交分解 （POD） 将高保真解的训练数据库分解为 POD 模式和 POD 系数。训练前馈神经网络将时间参数值映射到少数主要的 POD 系数。损失函数由两项组成：（1） 原始数据和重建数据之间的误差，以及 （2） 偏微分方程残差，其中偏微分方程的每一项都是在由最主要的 POD 模式组成的约化基础上使用加辽金展开来表示的。PDE 残差不使用 POD-Galerkin（降阶）方程进行评估。这项工作的新颖之处在于 PDE 残差项的构造和先验分析，允许人们在它之前选择加权因子（或拉格朗日乘数）。已经发现，最小化这两个项的物理信息 ROM 生成的新解比仅最小化第一个误差项的普通 ROM 准确几个数量级。除了在数据库上估计重建误差外，该框架还允许估计不同项的重建质量，例如偏微分方程中的平流和扩散。预计这将促进 ML 在动态系统的降阶建模中更好地集成和解释。 在在线预测阶段，给定新的时间和参数值，快速估计广义坐标并用于重建。因此，获得高保真解的速度比传统数值仿真快几个数量级。该框架在 1D 和 2D Burgers 方程以及面向后的台阶上的不可压缩流上进行了演示。