The Kolmogorov $n$-width of the solution manifolds of transport-dominated problems can decay slowly. As a result, it can be challenging to design efficient and accurate reduced order models (ROMs) for such problems. To address this issue, we propose a new learning-based projection method to construct nonlinear adaptive ROMs for transport problems. The construction follows the offline-online decomposition. In the offline stage, we train a neural network to construct adaptive reduced basis dependent on time and model parameters. In the online stage, we project the solution to the learned reduced manifold. Inheriting the merits from both deep learning and the projection method, the proposed method is more efficient than the conventional linear projection-based methods, and may reduce the generalization error of a solely learning-based ROM. Unlike some learning-based projection methods, the proposed method does not need to take derivatives of the neural network in the online stage.
翻译:解决运输问题的办法的科尔莫戈罗夫(Kolmogorov $n-with)可能缓慢地衰减,因此,为这些问题设计高效和准确的减少订单模型(ROMs)可能具有挑战性。为了解决这一问题,我们提议了一种新的基于学习的预测方法,用于为运输问题建造非线性适应型的ROM。建设是在离线线分解之后进行的。在离线阶段,我们培训神经网络,以根据时间和模型参数来构建适应性减少的基础。在在线阶段,我们预测了学习减少的多功能。根据深层次的学习和预测方法,拟议的方法比传统的线性投影法都更有效率,并可能减少仅以学习为基础的ROM的普遍错误。与某些基于学习的预测方法不同,拟议的方法不需要在在线阶段取取神经网络衍生物。