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，用于处理参数化偏微分方程（PDEs），该模型通过可扩展的变分神经过程实现。我们通过向空间域赋予概率测度来实现这一目标，从而允许将配点网格视为待边际化的随机变量。采用这种空间统计学视角，我们以构建解场的高斯过程模型的方式，求解参数化PDEs的正向与逆向问题。这些随机网格的实现为逆向物理驱动深度学习框架带来了一系列独特挑战，为此我们提出了一种名为网格不变卷积网络（GICNets）的新架构来解决这些挑战。我们还展示了如何以原理性方式将含噪数据纳入物理驱动模型，从而改进在可能有数据但测量位置与固定网格不重合的场景中的预测性能。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。结果表明，与现有用于参数化PDEs的物理驱动神经学习方法相比，我们的方法在计算效率上具有显著优势，同时提升了模型的预测能力和灵活性。