Learning Nonautonomous Systems via Dynamic Mode Decomposition
This work addresses a specific challenge in dynamical systems modeling for researchers in computational mathematics and engineering, representing an incremental improvement over existing methods.
The authors tackled the problem of learning unknown nonautonomous dynamical systems with time-dependent inputs by developing a data-driven approach based on dynamic mode decomposition (DMD). They demonstrated the robustness of their method through numerical examples, showing competitive performance compared to deep neural networks in the same settings.
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.