  Computational models have become a powerful tool in the quantitative sciences
to understand the behaviour of complex systems that evolve in time. However,
they often contain a potentially large number of free parameters whose values
cannot be obtained from theory but need to be inferred from data. This is
especially the case for models in the social sciences, economics, or
computational epidemiology. Yet many current parameter estimation methods are
mathematically involved and computationally slow to run. In this paper we
present a computationally simple and fast method to retrieve accurate
probability densities for model parameters using neural differential equations.
We present a pipeline comprising multi-agent models acting as forward solvers
for systems of ordinary or stochastic differential equations, and a neural
network to then extract parameters from the data generated by the model. The
two combined create a powerful tool that can quickly estimate densities on
model parameters, even for very large systems. We demonstrate the method on
synthetic time series data of the SIR model of the spread of infection, and
perform an in-depth analysis of the Harris-Wilson model of economic activity on
a network, representing a non-convex problem. For the latter, we apply our
method both to synthetic data and to data of economic activity across Greater
London. We find that our method calibrates the model orders of magnitude more
accurately than a previous study of the same dataset using classical
techniques, while running between 195 and 390 times faster.
