# Stochastic Adversarial Koopman Model for Dynamical Systems

arXiv: https://arxiv.org/abs/2109.05095 · HTML: https://mlsanity.com/paper/2109.05095/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 55 references that are in the map.
- First posted on arXiv 10 Sep 2021.

## Builds on (most cited first, 21 of 21)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [World Models](https://mlsanity.com/paper/1803.10122/) (2018)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Autoencoding beyond pixels using a learned similarity metric](https://mlsanity.com/paper/1512.09300/) (2015)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Deep multi-scale video prediction beyond mean square error](https://mlsanity.com/paper/1511.05440/) (2015)
- [Stochastic Latent Actor-Critic: Deep Reinforcement Learning with a Latent Variable Model](https://mlsanity.com/paper/1907.00953/) (2019)
- [A guide to convolution arithmetic for deep learning](https://mlsanity.com/paper/1603.07285/) (2016)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Near-Optimal Representation Learning for Hierarchical Reinforcement Learning](https://mlsanity.com/paper/1810.01257/) (2018)
- [Deep Dynamical Modeling and Control of Unsteady Fluid Flows](https://mlsanity.com/paper/1805.07472/) (2018)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability](https://mlsanity.com/paper/1906.03663/) (2019)
- [Deep Variational Koopman Models: Inferring Koopman Observations for Uncertainty-Aware Dynamics Modeling and Control](https://mlsanity.com/paper/1902.09742/) (2019)
- [Deep Adversarial Koopman Model for Reaction-Diffusion systems](https://mlsanity.com/paper/2006.05547/) (2020)

## Built on by (most cited first, 4 of 4)

- [NLP Inspired Training Mechanics For Modeling Transient Dynamics](https://mlsanity.com/paper/2211.02716/) (2022)
- [Stochastic Deep Koopman Model for Quality Propagation Analysis in Multistage Manufacturing Systems](https://mlsanity.com/paper/2309.10193/) (2023)
- [Deep Koopman-based Control of Quality Variation in Multistage Manufacturing Systems](https://mlsanity.com/paper/2407.16933/) (2024)
- [AI-driven Transformer Model for Fault Prediction in Non-Linear Dynamic Automotive System](https://mlsanity.com/paper/2408.12638/) (2024)

## Abstract (arXiv)

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 Kμ and Kσ. 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.