Learning Nonautonomous Systems via Dynamic Mode Decomposition

TL;DR

This paper introduces a data-driven method combining dynamic mode decomposition and parameter interpolation to learn unknown nonautonomous dynamical systems with time-dependent inputs, offering an alternative to neural networks.

Contribution

The paper develops a novel approach that approximates nonautonomous systems using local parameterization and DMD-based surrogate models, enhancing robustness and interpretability.

Findings

Method outperforms deep neural networks in numerical tests.

Efficiently constructs surrogate models for time-dependent inputs.

Demonstrates robustness across various numerical examples.

Abstract

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.