The simulation of many complex phenomena in engineering and science requires solving expensive, high-dimensional systems of partial differential equations (PDEs). To circumvent this, reduced-order models (ROMs) have been developed to speed up computations. However, when governing equations are unknown or partially known, typically ROMs lack interpretability and reliability of the predicted solutions. In this work we present a data-driven, non-intrusive framework for building ROMs where the latent variables and dynamics are identified in an interpretable manner and uncertainty is quantified. Starting from a limited amount of high-dimensional, noisy data the proposed framework constructs an efficient ROM by leveraging variational autoencoders for dimensionality reduction along with a newly introduced, variational version of sparse identification of nonlinear dynamics (SINDy), which we refer to as Variational Identification of Nonlinear Dynamics (VINDy). In detail, the method consists of Variational Encoding of Noisy Inputs (VENI) to identify the distribution of reduced coordinates. Simultaneously, we learn the distribution of the coefficients of a pre-determined set of candidate functions by VINDy. Once trained offline, the identified model can be queried for new parameter instances and new initial conditions to compute the corresponding full-time solutions. The probabilistic setup enables uncertainty quantification as the online testing consists of Variational Inference naturally providing Certainty Intervals (VICI). In this work we showcase the effectiveness of the newly proposed VINDy method in identifying interpretable and accurate dynamical system for the R\"ossler system with different noise intensities and sources. Then the performance of the overall method - named VENI, VINDy, VICI - is tested on PDE benchmarks including structural mechanics and fluid dynamics.

翻译：在工程与科学领域，模拟许多复杂现象需要求解昂贵的高维偏微分方程（PDE）系统。为规避此问题，降阶模型（ROM）被开发用于加速计算。然而，当控制方程未知或部分已知时，传统ROM通常缺乏预测解的可解释性与可靠性。本文提出一种数据驱动的非侵入式框架，用于构建以可解释方式识别隐变量与动力学并量化不确定性的ROM。该框架从有限的高维含噪数据出发，利用变分自编码器进行降维，并结合新提出的非线性动力学稀疏识别（SINDy）的变分版本——我们称之为非线性动力学的变分识别（VINDy），从而构建高效的ROM。具体而言，该方法通过噪声输入的变分编码（VENI）识别降维坐标的分布，同时利用VINDy学习预设候选函数集系数的分布。离线训练完成后，可针对新参数实例与新初始条件查询已识别模型，计算对应的全时程解。概率化设置使得在线测试能够通过变分推断自然提供置信区间（VICI），从而实现不确定性量化。本研究首先展示了新提出的VINDy方法在不同噪声强度与来源的Rössler系统中识别可解释且精确的动力学系统的有效性。随后，将名为VENI、VINDy、VICI的整体方法在结构力学与流体动力学等PDE基准问题上进行性能测试。