OptPDE: Discovering Novel Integrable Systems via AI-Human Collaboration

TL;DR

This paper presents OptPDE, a machine learning method that optimizes PDE coefficients to discover new integrable systems, demonstrating AI-human collaboration in scientific discovery.

Contribution

Introduces OptPDE, a novel AI-driven approach to identify and analyze previously unknown integrable PDE systems through optimization of conserved quantities.

Findings

Discovered four families of integrable PDEs, including three new to literature.

Identified a new PDE family: u_t = (u_x + a^2 u_{xxx})^3.

Showed AI-human collaboration effectively advances integrable system discovery.

Abstract

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.