Stochastic Adversarial Koopman Model for Dynamical Systems
Stochastic Adversarial Koopman Model for Dynamical Systems
复制标题
动力系统的随机对抗库普曼模型
DOI：
--
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
D. Upadhyay
中科院分区：
文献类型：
--
作者：
K. Balakrishnan;D. Upadhyay
动力系统是普遍存在的，并且通常使用控制方程的非线性系统来建模。许多动力系统的数值求解过程已经存在了几十年，但由于动力系统的高维状态空间，可能会很慢。因此，基于深度学习的降阶模型（ROM）是令人感兴趣的，并且沿着这些路线的一个这样的算法家族沿着是基于Koopman理论的。本文将最近开发的对抗Koopman模型（Balakrishnan &Upadhyay，arXiv：2006.05547）扩展到随机空间，其中Koopman算子适用于编码器的潜在编码的概率分布。具体地说，系统的潜在编码被建模为高斯，并通过使用输出两个Koopman矩阵$K_{mu}$和$K_{sigma}$的辅助神经网络在时间上提前。对抗和梯度损失的使用，这被发现，以降低预测误差。一个减少的Koopman制定也进行了假设Koopman矩阵具有三对角结构，这产生的预测与基线模型与完整的Koopman矩阵。随机Koopman模型的功效表现在不同的测试问题，在混沌，流体动力学，燃烧和反应扩散模型。所提出的模型也适用于设置中的Koopman矩阵的条件下，其他输入参数的泛化，这是应用于模拟锂离子电池的状态的时间。在这项研究中讨论的Koopman模型是非常有前途的广泛考虑的问题。
Dynamical systems are ubiquitous and are often modeled using a non-linear system of governing equations. Numerical solution procedures for many dynamical systems have existed for several decades, but can be slow due to high-dimensional state space of the dynamical system. Thus, deep learning-based reduced order models (ROMs) are of interest and one such family of algorithms along these lines are based on the Koopman theory. This paper extends a recently developed adversarial Koopman model (Balakrishnan &Upadhyay, arXiv:2006.05547) to stochastic space, where the Koopman operator applies on the probability distribution of the latent encoding of an encoder. Specifically, the latent encoding of the system is modeled as a Gaussian, and is advanced in time by using an auxiliary neural network that outputs two Koopman matrices $K_{mu}$ and $K_{sigma}$. Adversarial and gradient losses are used and this is found to lower the prediction errors. A reduced Koopman formulation is also undertaken where the Koopman matrices are assumed to have a tridiagonal structure, and this yields predictions comparable to the baseline model with full Koopman matrices. The efficacy of the stochastic Koopman model is demonstrated on different test problems in chaos, fluid dynamics, combustion, and reaction-diffusion models. The proposed model is also applied in a setting where the Koopman matrices are conditioned on other input parameters for generalization and this is applied to simulate the state of a Lithium-ion battery in time. The Koopman models discussed in this study are very promising for the wide range of problems considered.