Use of generative models and deep learning for physics-based systems is currently dominated by the task of emulation. However, the remarkable flexibility offered by data-driven architectures would suggest to extend this representation to other aspects of system synthesis including model inversion and identifiability. We introduce inVAErt (pronounced \emph{invert}) networks, a comprehensive framework for data-driven analysis and synthesis of parametric physical systems which uses a deterministic encoder and decoder to represent the forward and inverse solution maps, normalizing flow to capture the probabilistic distribution of system outputs, and a variational encoder designed to learn a compact latent representation for the lack of bijectivity between inputs and outputs. We formally investigate the selection of penalty coefficients in the loss function and strategies for latent space sampling, since we find that these significantly affect both training and testing performance. We validate our framework through extensive numerical examples, including simple linear, nonlinear, and periodic maps, dynamical systems, and spatio-temporal PDEs.

翻译：基于生成模型与深度学习对物理系统进行建模的研究目前主要集中于仿真任务。然而，数据驱动架构所提供的卓越灵活性表明，其表征能力可扩展至系统综合的其他环节，包括模型反演与可辨识性分析。我们提出inVAErt（发音为\emph{invert}）网络——一个用于参数化物理系统数据驱动分析与综合的综合性框架。该框架采用确定性编码器与解码器分别表示正问题与反问题的解映射，利用归一化流捕捉系统输出的概率分布，并设计了变分编码器以学习输入与输出间非双射关系的紧凑隐空间表征。我们系统研究了损失函数中惩罚系数的选择策略及隐空间采样方法，发现这些因素对训练与测试性能均有显著影响。通过涵盖简单线性映射、非线性映射、周期映射、动力系统及时空偏微分方程的大量数值算例，我们验证了该框架的有效性。