OptPDE: Discovering Novel Integrable Systems via AI-Human Collaboration
This addresses the challenge of finding integrable systems in natural science, which are difficult to identify, by proposing a collaborative AI-human framework for discovery.
The authors tackled the problem of discovering rare integrable partial differential equation (PDE) systems by introducing OptPDE, a machine learning approach that optimizes PDE coefficients to maximize conserved quantities, resulting in the discovery of three new families of integrable PDEs.
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.