Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems
Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems
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发表时间：
2018-01
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通讯作者：
M. Raissi;P. Perdikaris;G. Karniadakis
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作者：
M. Raissi;P. Perdikaris;G. Karniadakis
将观测到的数据转化为物理世界的预测数学模型的过程在科学和工程中一直是至关重要的。虽然目前收集数据的速度越来越快，但以自动化的方式从这些观察中设计出有意义的模型仍然是一个悬而未决的问题。在这项工作中，我们提出了一种从数据中识别非线性动力系统的机器学习方法。具体来说，我们将数值分析中的经典工具，即多步时间步方案，与强大的非线性函数逼近器，即深度神经网络相结合，以提取控制给定数据集演变的机制。我们测试了我们的方法在几个涉及识别复杂、非线性和混沌动力学的基准问题上的有效性，并展示了这如何使我们能够准确地学习动力学、预测未来状态和识别吸引力盆地。特别地，我们研究了洛伦兹系统、圆柱后流体流动、Hopf分岔和糖醇振荡模型作为生物系统复杂非线性动力学的典型例子。
The process of transforming observed data into predictive mathematical models of the physical world has always been paramount in science and engineering. Although data is currently being collected at an ever-increasing pace, devising meaningful models out of such observations in an automated fashion still remains an open problem. In this work, we put forth a machine learning approach for identifying nonlinear dynamical systems from data. Specifically, we blend classical tools from numerical analysis, namely the multi-step time-stepping schemes, with powerful nonlinear function approximators, namely deep neural networks, to distill the mechanisms that govern the evolution of a given data-set. We test the effectiveness of our approach for several benchmark problems involving the identification of complex, nonlinear and chaotic dynamics, and we demonstrate how this allows us to accurately learn the dynamics, forecast future states, and identify basins of attraction. In particular, we study the Lorenz system, the fluid flow behind a cylinder, the Hopf bifurcation, and the Glycoltic oscillator model as an example of complicated nonlinear dynamics typical of biological systems.