Learning dynamical systems with hit-and-run random feature maps
This provides an efficient forecasting method for chaotic systems, reducing hyperparameter tuning compared to alternatives like reservoir computers.
The paper tackles forecasting chaotic dynamical systems by introducing modified random feature maps with data-driven weight selection, skip connections, and localization, achieving state-of-the-art skill with up to 512 dimensions and requiring only one hyperparameter.
We show how random feature maps can be used to forecast dynamical systems with excellent forecasting skill. We consider the tanh activation function and judiciously choose the internal weights in a data-driven manner such that the resulting features explore the nonlinear, non-saturated regions of the activation function. We introduce skip connections and construct a deep variant of random feature maps by combining several units. To mitigate the curse of dimensionality, we introduce localization where we learn local maps, employing conditional independence. Our modified random feature maps provide excellent forecasting skill for both single trajectory forecasts as well as long-time estimates of statistical properties, for a range of chaotic dynamical systems with dimensions up to 512. In contrast to other methods such as reservoir computers which require extensive hyperparameter tuning, we effectively need to tune only a single hyperparameter, and are able to achieve state-of-the-art forecast skill with much smaller networks.