Emulation, Inverse Problem and Probabilistic Modeling of Physics-Based Systems

Owing to their strong expressive power and great flexibility, deep neural network-based surrogate models have tremendously advanced the development of digital twins across a wide spectrum of physics-based systems. Apart from traditional architectures, modern artificial intelligence techniques such as normalizing flows and variational auto-encoder further equip the data-driven modeling with probabilistic perspectives and generative capabilities. However, among diverse aspects of studying a physical system, emulating the forward problem has been a primary interest, while other crucial components, like inverse problem and parameter identifiability analysis receive relatively less attention. In this work, we propose InVAErt networks, a data-driven framework that comprehensively synthesizes physics-based systems, including input-to-output forward emulation, probabilistic modeling of outputs, solving inverse problem in an amortized fashion, as well as addressing input parameter non-identifiability. In particular, the augmentation of a latent space constructed through a variational network facilitates the discovery of the structurally non-identifiable manifold embedded in the input space that maps to a common output. In addition, several approaches of dealing with practical identifiability induced by missing observations in the outputs, measurement noise and mis-specification error are also proposed, including a physics-based missing data imputation method and artificial noise injection during network training. For validation, a series of numerical experiments are carried out, starting from simple maps, to nonlinear dynamical systems, space-time PDEs and large scale hemodynamic models used for real-time inference of physiological states from real electronic health records (EHR).

得益于强大的表达能力与出色的灵活性，基于深度神经网络的代理模型（surrogate models）极大推动了跨多类物理系统的数字孪生（digital twins）发展。除传统架构外，现代人工智能技术如归一化流（normalizing flows）与变分自编码器（variational auto-encoder）进一步为数据驱动建模赋予了概率视角与生成能力。然而，在物理系统研究的诸多维度中，正问题（forward problem）仿真始终是核心研究方向，而反问题（inverse problem）与参数可辨识性分析（parameter identifiability analysis）等关键环节却相对受关注不足。本研究提出了InVAErt网络，这是一种可全面适配物理系统的数据驱动框架，能够实现输入到输出的正问题仿真、输出的概率建模、以摊销方式求解反问题，以及解决输入参数不可辨识问题。具体而言，通过变分网络构建的隐空间（latent space）增强模块，有助于发现嵌入于输入空间中、映射至同一输出的结构不可辨识流形。此外，本研究还针对由输出缺失观测、测量噪声与模型设定误差所引发的实际可辨识性问题，提出了多种解决方案，包括基于物理的缺失数据填充方法，以及网络训练阶段的人工噪声注入策略。为验证所提框架的有效性，本研究开展了一系列数值实验，实验覆盖从简单映射、非线性动力学系统、时空偏微分方程（PDEs），到可基于真实电子健康记录（EHR）实时推理生理状态的大规模血流动力学模型等多种场景。