Closed-form differential equations, including partial differential equations and higher-order ordinary differential equations, are one of the most important tools used by scientists to model and better understand natural phenomena. Discovering these equations directly from data is challenging because it requires modeling relationships between various derivatives that are not observed in the data (equation-data mismatch) and it involves searching across a huge space of possible equations. Current approaches make strong assumptions about the form of the equation and thus fail to discover many well-known systems. Moreover, many of them resolve the equation-data mismatch by estimating the derivatives, which makes them inadequate for noisy and infrequently sampled systems. To this end, we propose D-CIPHER, which is robust to measurement artifacts and can uncover a new and very general class of differential equations. We further design a novel optimization procedure, CoLLie, to help D-CIPHER search through this class efficiently. Finally, we demonstrate empirically that it can discover many well-known equations that are beyond the capabilities of current methods.

翻译：闭式微分方程（包括偏微分方程和高阶常微分方程）是科学家用于建模并深入理解自然现象的最重要工具之一。直接从数据中发现这些方程具有挑战性，原因在于：它需要建模数据中未观测到的各阶导数之间的关系（方程-数据失配），同时涉及在巨大的可能方程空间中搜索。当前方法对方程形式施加了强假设，因此无法发现许多已知的系统。此外，许多方法通过估计导数来解决方程-数据失配问题，这使得它们难以适用于噪声大且采样频率低的系统。为此，我们提出D-CIPHER方法，该方法对测量伪影具有鲁棒性，并能揭示一类全新且非常通用的微分方程。我们进一步设计了一种新颖的优化程序CoLLie，以帮助D-CIPHER高效搜索此类方程。最后，我们通过实验证明，该方法能够发现许多当前方法无法处理的已知方程。