  Integrable partial differential equation (PDE) systems are of great interest
in natural science, but are exceedingly rare and difficult to discover. To
solve this, we introduce OptPDE, a first-of-its-kind machine learning approach
that Optimizes PDEs' coefficients to maximize their number of conserved
quantities, $n_{\rm CQ}$, and thus discover new integrable systems. We discover
four families of integrable PDEs, one of which was previously known, and three
of which have at least one conserved quantity but are new to the literature to
the best of our knowledge. We investigate more deeply the properties of one of
these novel PDE families, $u_t = (u_x+a^2u_{xxx})^3$. Our paper offers a
promising schema of AI-human collaboration for integrable system discovery:
machine learning generates interpretable hypotheses for possible integrable
systems, which human scientists can verify and analyze, to truly close the
discovery loop.
