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.

翻译：可积偏微分方程系统在自然科学中具有重要价值，但其极为罕见且难以发现。为解决这一问题，我们提出OptPDE——一种首创的机器学习方法，通过优化PDE的系数以最大化其守恒量数目$n_{\rm CQ}$，从而发现新的可积系统。我们发现了四个可积PDE族系，其中一族此前已知，其余三族据我们所知至少含有一个守恒量且为文献中首次报道。我们对其中一个新型PDE族系$u_t = (u_x+a^2u_{xxx})^3$的性质进行了深入研究。本文提出了一种理想的可积系统发现人机协作范式：机器学习生成关于潜在可积系统的可解释假设，由人类科学家进行验证与分析，从而真正实现发现闭环。