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.

翻译：我们提出了一种新颖的数据驱动框架——物理感知神经隐式求解器（PANIS），用于学习参数化偏微分方程（PDEs）的代理模型。该框架包含一个概率性学习目标，其中通过加权残差来探测偏微分方程并提供**虚拟**数据源（即无需实际求解原始偏微分方程）。该方法与一个物理感知隐式求解器相结合，该求解器由原偏微分方程经大幅粗化离散后的版本构成，为高维问题提供了必要的信息瓶颈，并能在分布外场景（例如不同边界条件）中实现泛化。我们在随机异质材料背景下展示了其能力，其中输入参数代表材料微观结构。我们将该框架扩展至多尺度问题，并证明无需求解参考问题即可学习有效（均匀化）解的代理模型。进一步，我们展示了所提框架如何兼容并泛化多种现有学习目标与架构，同时生成能够量化预测不确定性的概率代理模型。