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：本文提出了一种新方法，通过利用多保真视角和DeepONet来提高降阶模型的精度。降阶模型通过简化原始模型提供实时数值近似，但该操作引入的误差通常被忽略或牺牲以换取快速计算。我们建议将模型降阶与机器学习残差学习相结合，使得上述误差能够被神经网络学习并用于新预测的推断。我们强调该框架最大化了高保真信息的利用，既用于构建降阶模型，也用于残差学习。本研究探讨了将本征正交分解（POD）和用于传感器数据的间隙POD与最新DeepONet架构的集成，并针对参数化基准函数和非线性参数化纳维-斯托克斯问题进行了数值研究。