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 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.

中文翻译：

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物理感知神经隐式求解器，适用于异构介质中的多尺度参数 PDE


我们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，用于学习参数化偏微分方程 （PDE） 的代理。它由一个概率学习目标组成，其中加权残差用于探测 PDE 并提供虚拟数据源，即永远不需要解决实际的 PDE。这与物理感知隐式求解器相结合，该求解器由原始 PDE 的更粗略的离散化版本组成，为高维问题提供了必要的信息瓶颈，并支持在分布外设置（例如不同的边界条件）中泛化。我们在随机异质材料的背景下展示了它的能力，其中输入参数代表材料的微观结构。我们将框架扩展到多尺度问题，并表明可以在不解决参考问题的情况下为有效（同质化）解决方案学习代理物。我们进一步展示了所提出的框架如何适应和推广几个现有的学习目标和架构，同时产生可以量化预测不确定性的概率代理。