Recent work in data-driven modeling has demonstrated that a weak formulation of model equations enhances the noise robustness of a wide range of computational methods. In this paper, we demonstrate the power of the weak form to enhance the LaSDI (Latent Space Dynamics Identification) algorithm, a recently developed data-driven reduced order modeling technique. We introduce a weak form-based version WLaSDI (Weak-form Latent Space Dynamics Identification). WLaSDI first compresses data, then projects onto the test functions and learns the local latent space models. Notably, WLaSDI demonstrates significantly enhanced robustness to noise. With WLaSDI, the local latent space is obtained using weak-form equation learning techniques. Compared to the standard sparse identification of nonlinear dynamics (SINDy) used in LaSDI, the variance reduction of the weak form guarantees a robust and precise latent space recovery, hence allowing for a fast, robust, and accurate simulation. We demonstrate the efficacy of WLaSDI vs. LaSDI on several common benchmark examples including viscid and inviscid Burgers', radial advection, and heat conduction. For instance, in the case of 1D inviscid Burgers' simulations with the addition of up to 100% Gaussian white noise, the relative error remains consistently below 6% for WLaSDI, while it can exceed 10,000% for LaSDI. Similarly, for radial advection simulations, the relative errors stay below 15% for WLaSDI, in stark contrast to the potential errors of up to 10,000% with LaSDI. Moreover, speedups of several orders of magnitude can be obtained with WLaSDI. For example applying WLaSDI to 1D Burgers' yields a 140X speedup compared to the corresponding full order model. Python code to reproduce the results in this work is available at (https://github.com/MathBioCU/PyWSINDy_ODE) and (https://github.com/MathBioCU/PyWLaSDI).

翻译：近期数据驱动建模研究表明，模型方程的弱形式表述能显著提升多种计算方法的噪声鲁棒性。本文展示了弱形式在增强LaSDI（潜在空间动力学辨识）算法中的强大作用——该算法是最近发展的一种数据驱动降阶建模技术。我们提出了基于弱形式的WLaSDI（弱形式潜在空间动力学辨识）版本。WLaSDI首先压缩数据，随后投影至测试函数并学习局部潜在空间模型。值得注意的是，WLaSDI展现出显著增强的噪声鲁棒性。其通过弱形式方程学习技术获取局部潜在空间，相较于LaSDI中使用的标准稀疏非线性动力学辨识（SINDy），弱形式的方差缩减特性保证了潜在空间恢复的鲁棒性与精度，从而实现快速、稳健且准确的仿真。我们在多个常见基准算例（含粘性与无粘Burgers方程、径向平流及热传导）上对比验证了WLaSDI与LaSDI的性能。例如，在添加高达100%高斯白噪声的一维无粘Burgers仿真中，WLaSDI的相对误差始终低于6%，而LaSDI的误差可能超过10,000%。类似地，径向平流仿真中WLaSDI的相对误差维持在15%以下，而LaSDI的潜在误差可达10,000%。此外，WLaSDI可实现数个数量级的加速，例如应用于一维Burgers方程时，相较对应全阶模型实现了140倍加速。本文结果复现的Python代码见(https://github.com/MathBioCU/PyWSINDy_ODE)和(https://github.com/MathBioCU/PyWLaSDI)。