We introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning probability measures to the spatial domain, which allows us to treat collocation grids probabilistically as random variables to be marginalised out. Adapting this spatial statistics view, we solve forward and inverse problems for parametric PDEs in a way that leads to the construction of Gaussian process models of solution fields. The implementation of these random grids poses a unique set of challenges for inverse physics informed deep learning frameworks and we propose a new architecture called Grid Invariant Convolutional Networks (GICNets) to overcome these challenges. We further show how to incorporate noisy data in a principled manner into our physics informed model to improve predictions for problems where data may be available but whose measurement location does not coincide with any fixed mesh or grid. The proposed method is tested on a nonlinear Poisson problem, Burgers equation, and Navier-Stokes equations, and we provide extensive numerical comparisons. We demonstrate significant computational advantages over current physics informed neural learning methods for parametric PDEs while improving the predictive capabilities and flexibility of these models.

翻译：我们提出了一类新的空间随机物理与数据驱动的深度潜变量模型，用于参数化偏微分方程（PDEs），这些模型通过可扩展的变分神经过程实现。通过将概率测度赋予空间域，我们能够将配点网格视为需要边缘化的随机变量。采用这种空间统计视角，我们以一种构建解场高斯过程模型的方式解决参数化PDEs的正向和逆向问题。随机网格的实现对逆物理信息深度学习框架提出了独特挑战，为此我们提出了一种名为网格不变卷积网络（GICNets）的新架构来克服这些挑战。我们进一步展示了如何将含噪数据以原则性方式融入物理信息模型，从而在可能存在数据但测量位置与任何固定网格或网络不重合的问题中改进预测能力。所提方法在非线性泊松问题、伯格斯方程和纳维-斯托克斯方程上进行了测试，并提供了广泛的数值比较。我们证明了该方法相比当前用于参数化PDEs的物理信息神经学习方法具有显著的计算优势，同时提升了这些模型的预测能力和灵活性。