We present a deep learning framework for correcting existing dynamical system models utilizing only a scarce high-fidelity data set. In many practical situations, one has a low-fidelity model that can capture the dynamics reasonably well but lacks high resolution, due to the inherent limitation of the model and the complexity of the underlying physics. When high resolution data become available, it is natural to seek model correction to improve the resolution of the model predictions. We focus on the case when the amount of high-fidelity data is so small that most of the existing data driven modeling methods cannot be applied. In this paper, we address these challenges with a model-correction method which only requires a scarce high-fidelity data set. Our method first seeks a deep neural network (DNN) model to approximate the existing low-fidelity model. By using the scarce high-fidelity data, the method then corrects the DNN model via transfer learning (TL). After TL, an improved DNN model with high prediction accuracy to the underlying dynamics is obtained. One distinct feature of the propose method is that it does not assume a specific form of the model correction terms. Instead, it offers an inherent correction to the low-fidelity model via TL. A set of numerical examples are presented to demonstrate the effectiveness of the proposed method.

翻译：本文提出了一种深度学习框架，用于在仅有稀缺高保真数据集的条件下修正现有动力学系统模型。在许多实际场景中，我们可能拥有一个能够较好捕捉动态特性但分辨率不足的低保真模型，这源于模型本身的固有局限性和底层物理过程的复杂性。当获得高分辨率数据时，自然希望通过模型修正来提高模型预测的分辨率。我们重点关注高保真数据量极其有限、致使大多数现有数据驱动建模方法无法适用的情况。本文通过一种仅需稀缺高保真数据集的模型修正方法来应对这些挑战。该方法首先构建深度神经网络（DNN）模型来逼近现有低保真模型，随后利用稀缺高保真数据通过迁移学习（TL）对该DNN模型进行修正。经过迁移学习后，可获得对底层动态具有高预测精度的改进型DNN模型。所提方法的一个显著特点是无需假设模型修正项的具体形式，而是通过迁移学习对低保真模型进行本质性修正。文中通过一系列数值算例验证了该方法的有效性。