We propose Physics-Aware Neural Implicit Solvers (PANIS), a novel, data-driven framework for learning surrogates for parametrized Partial Differential Equations (PDEs). It consists of a probabilistic, learning objective in which weighted residuals are used to probe the PDE and provide a source of {\em virtual} data i.e. the actual PDE never needs to be solved. This is combined with a physics-aware implicit solver that consists of a much coarser, discretized version of the original PDE, which provides the requisite information bottleneck for high-dimensional problems and enables generalization in out-of-distribution settings (e.g. different boundary conditions). We demonstrate its capability in the context of random heterogeneous materials where the input parameters represent the material microstructure. We extend the framework to multiscale problems and show that a surrogate can be learned for the effective (homogenized) solution without ever solving the reference problem. We further demonstrate how the proposed framework can accommodate and generalize several existing learning objectives and architectures while yielding probabilistic surrogates that can quantify predictive uncertainty.

提出了物理感知神经隐式求解器（PANIS），这是一种新颖的数据驱动框架，用于学习参数化的偏微分方程（PDE）的代理解。它由一个概率性的学习目标组成，在此目标中，使用加权残差来探测PDE并提供源数据，而实际的PDE无需解决。与之结合的是一种物理感知隐式求解器，它由原始PDE的一个粗糙、离散化版本组成，为高维问题提供必要的信息瓶颈，并在分布不同的情况下实现泛化（例如，不同的边界条件）。演示了在随机异质材料的背景下其能力，在该背景下输入参数表示材料的微观结构。将该框架扩展到多尺度问题，并展示了可以学习到有效（均质化）解的代理且无需解决原问题。此外，还演示了如何适应和泛化几种现有的学习目标和架构，同时产生能够量化预测不确定性的概率性代理解。

物理感知的神经隐式求解器：在异质介质中应用的多尺度、参数化的偏微分方程