Many real-world dynamical systems are associated with first integrals (a.k.a. invariant quantities), which are quantities that remain unchanged over time. The discovery and understanding of first integrals are fundamental and important topics both in the natural sciences and in industrial applications. First integrals arise from the conservation laws of system energy, momentum, and mass, and from constraints on states; these are typically related to specific geometric structures of the governing equations. Existing neural networks designed to ensure such first integrals have shown excellent accuracy in modeling from data. However, these models incorporate the underlying structures, and in most situations where neural networks learn unknown systems, these structures are also unknown. This limitation needs to be overcome for scientific discovery and modeling of unknown systems. To this end, we propose first integral-preserving neural differential equation (FINDE). By leveraging the projection method and the discrete gradient method, FINDE finds and preserves first integrals from data, even in the absence of prior knowledge about underlying structures. Experimental results demonstrate that FINDE can predict future states of target systems much longer and find various quantities consistent with well-known first integrals in a unified manner.

翻译：许多现实世界的动力系统都伴随着首次积分（也称为不变量），即随时间保持不变的量。首次积分的发现与理解是自然科学和工业应用中基础且重要的课题。首次积分源于系统能量、动量和质量的守恒定律以及状态约束，这些通常与支配方程的特定几何结构相关。现有旨在确保此类首次积分的神经网络在基于数据建模方面展现了卓越的精度。然而，这些模型嵌入了底层结构，而在神经网络学习未知系统的大多数情况下，这些结构本身也是未知的。这一局限性需被克服，以推动未知系统的科学发现与建模。为此，我们提出了保持首次积分的神经微分方程（FINDE）。通过利用投影法和离散梯度法，FINDE能从数据中发现并保持首次积分，即使缺乏关于底层结构的先验知识。实验结果表明，FINDE能更长时间地预测目标系统的未来状态，并以统一方式发现与已知首次积分一致的各种量。