FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantities

Takashi Matsubara, Takaharu Yaguchi

Keywords: neural ordinary differential equations, first integral, conservetaion law

TL;DR: Most real-world dynamical systems are associated with invariant quantities, such as energy, momenta, and mass. Even without prior knowledge, the proposed neural network finds and preserves such quantities from data by leveraging projection methods.

Abstract: Neural networks have shown promise for modeling dynamical systems from data. Recent models, such as Hamiltonian neural networks, have been designed to ensure known geometric structures of target systems and have shown excellent modeling accuracy. However, in most situations where neural networks learn unknown systems, their underlying structures are also unknown. Even in such cases, one can expect that target systems are associated with first integrals (a.k.a. invariant quantities), which are quantities remaining unchanged over time. First integrals come from the conservation laws of system energy, momentum, and mass, from constraints on states, and from other features of governing equations. By leveraging projection methods and discrete gradient methods, we propose first integral-preserving neural differential equations (FINDE). The proposed FINDE finds and preserves first integrals from data, even in the absence of prior knowledge about the underlying structures. Experimental results demonstrate that the proposed FINDE is able to predict future states of given systems much longer and find various quantities consistent with well-known first integrals of the systems in a unified manner.

Community Implementations: [ 2 code implementations](https://www.catalyzex.com/paper/finde-neural-differential-equations-for/code)