Learning Nonautonomous Systems via Dynamic Mode Decomposition
Hannah Lu Daniel M. Tartakovsky
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.