  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.
