# Learning Nonautonomous Systems via Dynamic Mode Decomposition

arXiv: https://arxiv.org/abs/2306.15618 · HTML: https://mlsanity.com/paper/2306.15618/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 52 references that are in the map.
- First posted on arXiv 27 Jun 2023.

## Builds on (most cited first, 11 of 11)

- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Generalizing Koopman Theory to allow for inputs and control](https://mlsanity.com/paper/1602.07647/) (2016)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [Online dynamic mode decomposition for time-varying systems](https://mlsanity.com/paper/1707.02876/) (2017)
- [Data-Driven Discovery of Coarse-Grained Equations](https://mlsanity.com/paper/2002.00790/) (2020)
- [Data-driven learning of non-autonomous systems](https://mlsanity.com/paper/2006.02392/) (2020)

## Built on by (most cited first, 1 of 1)

- [Koopman Learning with Episodic Memory](https://mlsanity.com/paper/2311.12615/) (2023)

## Abstract (arXiv)

We present a data-driven learning approach for unknown nonautonomous dynamical systems with time-dependent inputs based on dynamic mode decomposition (DMD). To circumvent the difficulty of approximating the time-dependent Koopman operators for nonautonomous systems, a modified system derived from local parameterization of the external time-dependent inputs is employed as an approximation to the original nonautonomous system. The modified system comprises a sequence of local parametric systems, which can be well approximated by a parametric surrogate model using our previously proposed framework for dimension reduction and interpolation in parameter space (DRIPS). The offline step of DRIPS relies on DMD to build a linear surrogate model, endowed with reduced-order bases (ROBs), for the observables mapped from training data. Then the offline step constructs a sequence of iterative parametric surrogate models from interpolations on suitable manifolds, where the target/test parameter points are specified by the local parameterization of the test external time-dependent inputs. We present a number of numerical examples to demonstrate the robustness of our method and compare its performance with deep neural networks in the same settings.