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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 andimportant 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 tospecific geometric structures of the governing equations. Existing neuralnetworks designed to ensure such first integrals have shown excellent accuracyin modeling from data. However, these models incorporate the underlyingstructures, and in most situations where neural networks learn unknown systems,these structures are also unknown. This limitation needs to be overcome forscientific discovery and modeling of unknown systems. To this end, we proposefirst integral-preserving neural differential equation (FINDE). By leveragingthe projection method and the discrete gradient method, FINDE finds andpreserves first integrals from data, even in the absence of prior knowledgeabout underlying structures. Experimental results demonstrate that FINDE canpredict future states of target systems much longer and find various quantitiesconsistent with well-known first integrals in a unified manner.

FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantities