We introduce a physics-driven deep latent variable model (PDDLVM) to learn simultaneously parameter-to-solution (forward) and solution-to-parameter (inverse) maps of parametric partial differential equations (PDEs). Our formulation leverages conventional PDE discretization techniques, deep neural networks, probabilistic modelling, and variational inference to assemble a fully probabilistic coherent framework. In the posited probabilistic model, both the forward and inverse maps are approximated as Gaussian distributions with a mean and covariance parameterized by deep neural networks. The PDE residual is assumed to be an observed random vector of value zero, hence we model it as a random vector with a zero mean and a user-prescribed covariance. The model is trained by maximizing the probability, that is the evidence or marginal likelihood, of observing a residual of zero by maximizing the evidence lower bound (ELBO). Consequently, the proposed methodology does not require any independent PDE solves and is physics-informed at training time, allowing the real-time solution of PDE forward and inverse problems after training. The proposed framework can be easily extended to seamlessly integrate observed data to solve inverse problems and to build generative models. We demonstrate the efficiency and robustness of our method on finite element discretized parametric PDE problems such as linear and nonlinear Poisson problems, elastic shells with complex 3D geometries, and time-dependent nonlinear and inhomogeneous PDEs using a physics-informed neural network (PINN) discretization. We achieve up to three orders of magnitude speed-up after training compared to traditional finite element method (FEM), while outputting coherent uncertainty estimates.

翻译：我们提出一种物理驱动的深度潜变量模型（PDDLVM），用于同时学习参数化偏微分方程（PDEs）的参数-解（正向）映射与解-参数（逆）映射。该公式结合传统PDE离散化技术、深度神经网络、概率建模和变分推断，构建了一个完全概率化的统一框架。在所提出的概率模型中，正向和逆映射均被近似为高斯分布，其均值和协方差由深度神经网络参数化。PDE残差被假设为零观测值的随机向量，因此我们将其建模为具有零均值和用户预设协方差的随机向量。模型通过最大化观测到零残差的概率（即证据或边际似然）进行训练，通过最大化证据下界（ELBO）实现。因此，所提方法无需任何独立PDE求解，且在训练阶段即融入物理信息，使得训练后能够实时求解PDE正反问题。该框架可轻易扩展以无缝整合观测数据，从而解决逆问题并构建生成模型。我们通过有限元离散的参数化PDE问题（如线性和非线性泊松问题、复杂三维几何弹性壳体问题、以及使用物理信息神经网络（PINN）离散的时变非线性非齐次PDE）展示了方法的效率与鲁棒性。与传统有限元法（FEM）相比，训练后我们实现了高达三个数量级的加速，同时输出具有一致性的不确定性估计。