论文标题
通过具有记忆的深层神经网络学习减少系统
Learning reduced systems via deep neural networks with memory
论文作者
论文摘要
当仅在状态变量的子集上的数据上，我们提出了一种用于构建未知动态系统管理方程的一般数值方法。因此，这些观察到的变量的未知方程是整个状态变量集的减少系统。减少的系统具有基于众所周知的Mori-Zwanzig（MZ）配方的记忆积分。我们恢复还原系统的数值策略首先通过制定MZ公式中存储器积分的离散近似。从涉及有限数量的过去历史数据的意义上讲，所得未知的近似MZ方程是有限维度的。然后，我们提出了一个深层的神经网络结构，该结构直接结合了历史术语以在网络中产生内存。该方法适用于任何有限记忆长度的实际系统。然后，我们使用一组数值示例来证明我们方法的有效性。
We present a general numerical approach for constructing governing equations for unknown dynamical systems when only data on a subset of the state variables are available. The unknown equations for these observed variables are thus a reduced system of the complete set of state variables. Reduced systems possess memory integrals, based on the well known Mori-Zwanzig (MZ) formulism. Our numerical strategy to recover the reduced system starts by formulating a discrete approximation of the memory integral in the MZ formulation. The resulting unknown approximate MZ equations are of finite dimensional, in the sense that a finite number of past history data are involved. We then present a deep neural network structure that directly incorporates the history terms to produce memory in the network. The approach is suitable for any practical systems with finite memory length. We then use a set of numerical examples to demonstrate the effectiveness of our method.
