  Normalizing flows are an established approach for modelling complex
probability densities through invertible transformations from a base
distribution. However, the accuracy with which the target distribution can be
captured by the normalizing flow is strongly influenced by the topology of the
base distribution. A mismatch between the topology of the target and the base
can result in a poor performance, as is typically the case for multi-modal
problems. A number of different works have attempted to modify the topology of
the base distribution to better match the target, either through the use of
Gaussian Mixture Models (Izmailov et al., 2020; Ardizzone et al., 2020;
Hagemann & Neumayer, 2021) or learned accept/reject sampling (Stimper et al.,
2022). We introduce piecewise normalizing flows which divide the target
distribution into clusters, with topologies that better match the standard
normal base distribution, and train a series of flows to model complex
multi-modal targets. We demonstrate the performance of the piecewise flows
using some standard benchmarks and compare the accuracy of the flows to the
approach taken in Stimper et al. (2022) for modelling multi-modal
distributions. We find that our approach consistently outperforms the approach
in Stimper et al. (2022) with a higher emulation accuracy on the standard
benchmarks.
