  We use a combination of unsupervised clustering and sparsity-promoting
inference algorithms to learn locally dominant force balances that explain
macroscopic pattern formation in self-organized active particle systems. The
self-organized emergence of macroscopic patterns from microscopic interactions
between self-propelled particles can be widely observed nature. Although
hydrodynamic theories help us better understand the physical basis of this
phenomenon, identifying a sufficient set of local interactions that shape,
regulate, and sustain self-organized structures in active particle systems
remains challenging. We investigate a classic hydrodynamic model of
self-propelled particles that produces a wide variety of patterns, like asters
and moving density bands. Our data-driven analysis shows that propagating bands
are formed by local alignment interactions driven by density gradients, while
steady-state asters are shaped by a mechanism of splay-induced negative
compressibility arising from strong particle interactions. Our method also
reveals analogous physical principles of pattern formation in a system where
the speed of the particle is influenced by local density. This demonstrates the
ability of our method to reveal physical commonalities across models. The
physical mechanisms inferred from the data are in excellent agreement with
analytical scaling arguments and experimental observations.
