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 "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, a 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：在基于物理系统的生成模型与深度学习中，目前主要应用集中于仿真模拟任务。然而，数据驱动架构所提供的显著灵活性表明，应将其应用范围扩展至系统合成的其他方面，包括模型反演与可辨识性分析。本文提出inVAErt网络——一种用于参数化物理系统数据驱动分析与合成的综合框架。该框架采用确定性编码器与解码器分别表征正向与逆解映射，通过归一化流捕捉系统输出的概率分布，并设计变分编码器以学习输入与输出间缺乏双射性时的紧凑潜在表示。我们系统研究了损失函数中惩罚系数的选取策略及潜在空间采样方法，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在内的大量数值算例，验证了该框架的有效性。