The discovery of underlying surface partial differential equation (PDE) from observational data has significant implications across various fields, bridging the gap between theory and observation, enhancing our understanding of complex systems, and providing valuable tools and insights for applications. In this paper, we propose a novel approach, termed physical-informed sparse optimization (PIS), for learning surface PDEs. Our approach incorporates both $L_2$ physical-informed model loss and $L_1$ regularization penalty terms in the loss function, enabling the identification of specific physical terms within the surface PDEs. The unknown function and the differential operators on surfaces are approximated by some extrinsic meshless methods. We provide practical demonstrations of the algorithms including linear and nonlinear systems. The numerical experiments on spheres and various other surfaces demonstrate the effectiveness of the proposed approach in simultaneously achieving precise solution prediction and identification of unknown PDEs.

翻译：从观测数据中发现底层曲面偏微分方程具有重要的跨学科意义，它能够弥合理论与观测之间的差距，增强对复杂系统的理解，并为实际应用提供有价值的工具和见解。本文提出一种名为物理信息稀疏优化的新型方法，用于学习曲面偏微分方程。该方法在损失函数中同时引入$L_2$物理信息模型损失和$L_1$正则化惩罚项，从而能够识别曲面偏微分方程中的特定物理项。未知函数及曲面上的微分算子通过若干外在无网格方法进行逼近。我们展示了包括线性和非线性系统在内的算法实际演示。在球面及其他多种曲面上的数值实验表明，该方法能在精确预测解的同时有效识别未知偏微分方程。