  Modern design, control, and optimization often requires simulation of highly
nonlinear models, leading to prohibitive computational costs. These costs can
be amortized by evaluating a cheap surrogate of the full model. Here we present
a general data-driven method, the continuous-time echo state network (CTESN),
for generating surrogates of nonlinear ordinary differential equations with
dynamics at widely separated timescales. We empirically demonstrate
near-constant time performance using our CTESNs on a physically motivated
scalable model of a heating system whose full execution time increases
exponentially, while maintaining relative error of within 0.2 %. We also show
that our model captures fast transients as well as slow dynamics effectively,
while other techniques such as physics informed neural networks have
difficulties trying to train and predict the highly nonlinear behavior of these
models.
