We present a new framework for computing fine-scale solutions of multiscale Partial Differential Equations (PDEs) using operator learning tools. Obtaining fine-scale solutions of multiscale PDEs can be challenging, but there are many inexpensive computational methods for obtaining coarse-scale solutions. Additionally, in many real-world applications, fine-scale solutions can only be observed at a limited number of locations. In order to obtain approximations or predictions of fine-scale solutions over general regions of interest, we propose to learn the operator mapping from coarse-scale solutions to fine-scale solutions using a limited number (and possibly noisy) observations of the fine-scale solutions. The approach is to train multi-fidelity homogenization maps using mathematically motivated neural operators. The operator learning framework can efficiently obtain the solution of multiscale PDEs at any arbitrary point, making our proposed framework a mesh-free solver. We verify our results on multiple numerical examples showing that our approach is an efficient mesh-free solver for multiscale PDEs.

翻译：我们提出了一种新框架，利用算子学习工具计算多尺度偏微分方程的细尺度解。获取多尺度偏微分方程的细尺度解具有挑战性，但存在许多廉价的计算方法用于获取粗尺度解。此外，在诸多实际应用中，细尺度解仅能在有限位置观测到。为了在感兴趣的一般区域上获得细尺度解的近似或预测，我们提出利用有限（且可能含噪声）的细尺度解观测数据，学习从粗尺度解到细尺度解的算子映射。该方法是采用数学驱动的神经算子训练多保真度均质化映射。该算子学习框架能高效获取多尺度偏微分方程在任意点的解，使所提框架成为无网格求解器。我们通过多个数值算例验证了结果，证明该方法作为多尺度偏微分方程的高效无网格求解器的有效性。