[论文解读] VENI, VINDy, VICI: a generative reduced-order modeling framework with uncertainty quantification

Paolo Conti, Jonas Kneifl+5 more|arXiv (Cornell University)|May 31, 2024

Model Reduction and Neural Networks被引用 5

一句话总结

一个数据驱动、非侵入式的 ROM 框架，结合变分编码（VENI）、变分 SINDY（VINDy）以及带不确定区间的变分推断（VICI），以重建无噪声的高维解并量化不确定性。 在 Rössler 系统和高维偏微分方程基准测试中得到验证。

ABSTRACT

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 Roessler 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.

研究动机与目标

• 解决从受限噪声数据中构建可解释且可靠的 ROM 的挑战。
• 在统一的变分框架中整合降维、系统辨识和不确定性量化。
• 提供离线训练以学习潜在动力学，并进行在线生成并附带预测不确定性。
• 在低维混沌系统和高维 PDE 基准测试上证明有效性。

提出的方法

• 使用 VENI 通过变分编码器和高斯解码器将带噪声的高维数据映射到低维潜在分布。
• 应用 VINDy 学习潜在坐标的概率性、稀疏动力学模型，作为候选函数的线性组合且系数带不确定性。
• 通过在变分目标下联合优化重构、潜在动力学和系数先验进行离线训练。
• 在线时，使用 VICI 为新参数/初始条件生成全场解及不确定区间。
• 为便于求 KL 项的可处理性，假设高斯或拉普拉斯先验，并使用重参数化实现基于梯度的优化。

实验结果

研究问题

• RQ1 VENI-VINDy-VICI 流水线是否能够从噪声较大、维度较高的数据中恢复可解释的潜在动力学？
• RQ2 该框架在新的参数实例上是否提供准确的抗噪重建和可靠的不确定性量化（确定区间）？
• RQ3 该方法在低维混沌系统和高维 PDE 基准（如 MEMS 梁、非稳态反应扩散等）上的表现如何？
• RQ4 离线联合训练在稳定潜在表示和动力学发现中起到何种作用？

主要发现

• VINDy 组件在不同噪声条件下能够识别 Rössler 系统的可解释潜在动力学。
• 集成的 VENI–VINDy–VICI 框架在在线预测时能提供准确的全场 PDE 解并通过 Certainty Intervals 提供不确定性。
• 该方法在包括 MEMS 梁共振器和参数化反应-扩散问题在内的高维 PDE 基准测试中得到验证。
• 公开源码用于 VENI、VINDy、VICI 的代码已在公开代码库中提供。
• 离线训练联合优化重构、潜在动力学和系数先验，以产生具有不确定性量化能力的生成型 ROM。