Discovering candidates for integrable systems via backpropagation

Integrable partial differential equation (PDE) systems are of great interest in natural science, but are exceedingly rare and difficult to discover. To this end, we introduce OptPDE, a numerical approach that Optimizes PDEs' coefficients via automatic differentiation to maximize their number of conserved quantities, n_{CQ}, and thus discover candidates for integrable systems. The algorithm discovers four families of PDEs with at least one nontrivial conserved quantity. We analytically study one new candidate u_{t}=u_{x}^{3}, revealing its interesting properties. It is worth mentioning that our approach aims to discover systems with as many conserved quantities as possible, which does not guarantee integrability. Despite this limitation, our humble effort suggests that numerical algorithms that leverage automatic differentiation are potentially useful for discovering candidates for integrable systems.