The parametric greedy latent space dynamics identification (gLaSDI) framework has demonstrated promising potential for accurate and efficient modeling of high-dimensional nonlinear physical systems. However, it remains challenging to handle noisy data. To enhance robustness against noise, we incorporate the weak-form estimation of nonlinear dynamics (WENDy) into gLaSDI. In the proposed weak-form gLaSDI (WgLaSDI) framework, an autoencoder and WENDy are trained simultaneously to discover intrinsic nonlinear latent-space dynamics of high-dimensional data. Compared to the standard sparse identification of nonlinear dynamics (SINDy) employed in gLaSDI, WENDy enables variance reduction and robust latent space discovery, therefore leading to more accurate and efficient reduced-order modeling. Furthermore, the greedy physics-informed active learning in WgLaSDI enables adaptive sampling of optimal training data on the fly for enhanced modeling accuracy. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including viscous and inviscid Burgers' equations, time-dependent radial advection, and the Vlasov equation for plasma physics. With data that contains 5-10% Gaussian white noise, WgLaSDI outperforms gLaSDI by orders of magnitude, achieving 1-7% relative errors. Compared with the high-fidelity models, WgLaSDI achieves 121 to 1,779x speed-up.

翻译：参数化贪婪潜空间动力学辨识（gLaSDI）框架已展现出在高维非线性物理系统的精确高效建模方面的良好潜力。然而，处理含噪数据仍然具有挑战性。为增强对噪声的鲁棒性，我们将非线性动力学的弱形式估计（WENDy）融入gLaSDI。在所提出的弱形式gLaSDI（WgLaSDI）框架中，自编码器与WENDy被同时训练，以发现高维数据固有的非线性潜空间动力学。与gLaSDI中采用的标准非线性动力学稀疏辨识（SINDy）相比，WENDy能够实现方差缩减和鲁棒的潜空间发现，从而获得更精确高效的低阶建模。此外，WgLaSDI中的贪婪物理信息主动学习能够动态自适应采样最优训练数据，以提升建模精度。通过对多种非线性动力学问题进行建模，包括粘性与无粘性Burgers方程、时间依赖的径向平流以及等离子体物理中的Vlasov方程，验证了所提框架的有效性。在数据包含5-10%高斯白噪声的情况下，WgLaSDI的性能比gLaSDI高出数个数量级，相对误差达到1-7%。与高保真模型相比，WgLaSDI实现了121至1,779倍的加速。