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.

OptPDE 是一种机器学习方法，通过优化偏微分方程 (PDE) 的系数以最大化其守恒量数量 n_CQ，从而发现新的可积系统，并发现了四个可积 PDE 族群，其中一个已知，三个是文献中新的具有至少一个守恒量的可积 PDE。我们深入研究了其中一个新的 PDE 族群的特性，即 $u_t = (u_x+a^2u_{xxx})^3$，这一研究为 AI 与人类合作推动可积系统的发现提供了有前景的框架：机器学习为可能的可积系统生成可解释的假设，而人类科学家则对其进行验证和分析，从而真正闭合发现的循环。