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 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 learnt by a neural network and inferred for new predictions. We emphasize that the framework maximizes the exploitation of the 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：本文提出了一种新颖方法，通过利用多保真度视角和DeepONet来提升降阶模型的精度。降阶模型通过简化原始模型实现实时数值近似，但此类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，使得上述误差能被神经网络学习并用于新预测。我们强调，该框架最大化利用高保真信息，既用于构建降阶模型，也用于学习残差。本文探索了本征正交分解（POD）及用于传感器数据的残缺POD与最新DeepONet架构的集成，并展示了参数化基准函数和非线性参数化Navier-Stokes问题的数值研究。