Numerical solvers of partial differential equations (PDEs) have been widely employed for simulating physical systems. However, the computational cost remains a major bottleneck in various scientific and engineering applications, which has motivated the development of reduced-order models (ROMs). Recently, machine-learning-based ROMs have gained significant popularity and are promising for addressing some limitations of traditional ROM methods, especially for advection dominated systems. In this chapter, we focus on a particular framework known as Latent Space Dynamics Identification (LaSDI), which transforms the high-fidelity data, governed by a PDE, to simpler and low-dimensional latent-space data, governed by ordinary differential equations (ODEs). These ODEs can be learned and subsequently interpolated to make ROM predictions. Each building block of LaSDI can be easily modulated depending on the application, which makes the LaSDI framework highly flexible. In particular, we present strategies to enforce the laws of thermodynamics into LaSDI models (tLaSDI), enhance robustness in the presence of noise through the weak form (WLaSDI), select high-fidelity training data efficiently through active learning (gLaSDI, GPLaSDI), and quantify the ROM prediction uncertainty through Gaussian processes (GPLaSDI). We demonstrate the performance of different LaSDI approaches on Burgers equation, a non-linear heat conduction problem, and a plasma physics problem, showing that LaSDI algorithms can achieve relative errors of less than a few percent and up to thousands of times speed-ups.

翻译：偏微分方程（PDE）数值求解器已被广泛用于物理系统模拟。然而，计算成本仍是各类科学与工程应用中的主要瓶颈，这推动了降阶模型（ROM）的发展。近年来，基于机器学习的ROM方法日益受到关注，有望解决传统ROM方法的部分局限性，尤其是在对流主导系统中。本章聚焦于一种特定框架——潜空间动力学识别（LaSDI），该框架将受PDE控制的高保真数据转化为由常微分方程（ODE）控制的低维潜空间数据。这些常微分方程可被学习并随后进行插值，以实现ROM预测。LaSDI的每个模块均可根据具体应用灵活调整，这使得LaSDI框架具有高度灵活性。特别地，我们介绍了将热力学定律纳入LaSDI模型（tLaSDI）的策略、通过弱形式增强噪声鲁棒性的方法（WLaSDI）、通过主动学习高效选择高保真训练数据的技术（gLaSDI、GPLaSDI），以及利用高斯过程量化ROM预测不确定性的方法（GPLaSDI）。我们通过Burgers方程、非线性热传导问题和等离子体物理问题展示了不同LaSDI方法的性能，结果表明LaSDI算法可实现低于百分之几的相对误差，并达到数千倍的加速比。