In the present work, we introduce a novel approach to enhance the precision of reduced order models by exploiting a multi-fidelity perspective and DeepONets. Reduced models provide a real-time numerical approximation by simplifying the original model. The error introduced by the such operation is usually neglected and sacrificed in order to reach a fast computation. We propose to couple the model reduction to a machine learning residual learning, such that the above-mentioned error can be learned by a neural network and inferred for new predictions. We emphasize that the framework maximizes the exploitation of high-fidelity information, using it for building the reduced order model and for learning the residual. In this work, we explore the integration of proper orthogonal decomposition (POD), and gappy POD for sensors data, with the recent DeepONet architecture. Numerical investigations for a parametric benchmark function and a nonlinear parametric Navier-Stokes problem are presented.

翻译：在本文中，我们提出一种新颖方法，通过利用多保真视角和DeepONet来提升降阶模型的精度。降阶模型通过简化原始模型提供实时数值近似，但通常为达到快速计算而忽略或牺牲该操作引入的误差。我们提出将模型降阶与机器学习残差学习相结合，使得上述误差可通过神经网络学习，并用于新预测的推断。需强调的是，该框架最大化地利用高保真信息，既用于构建降阶模型，也用于学习残差。本文探索将本征正交分解（POD）及面向传感器数据的稀疏POD与最新DeepONet架构相结合。针对参数化基准函数和非线性参数化Navier-Stokes问题进行了数值研究。