FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantities

Many real-world dynamical systems are associated with first integrals (a.k.a. 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. Modeling and predicting real-world systems are fundamental aspects of understanding the world in natural science and improving computer simulations in industry. Target systems include chemical dynamics for discovering new drugs (Raff et al., 2012), climate dynamics for climate change prediction and weather forecasting (Rasp et al., 2020; Trigo & Palutikof, 1999), and physical dynamics of vehicles and robots for optimal control (Nelles, 2001). In addition to image processing and natural language processing (Devlin et al., 2018; He et al., 2016), neural networks have been actively studied for modeling dynamical systems (Nelles, 2001). Their history dates back to at least the 1990s (see Chen et al. (1990); Clouse et al. (1997); Levin & Narendra (1995); Narendra & Parthasarathy (1990); Sjöberg et al. (1994); Wang & Lin (1998) for examples). Recently, two notable but distinct families have been proposed. Physics-informed neural networks (PINNs) directly solve partial differential equations (PDEs) given as symbolic equations (Raissi et al., 2019). Neural ordinary differential equations (NODEs) learn ordinary differential equations (ODEs) from observed data and solve them using numerical integrators (Chen et al., 2018). Our focus this time is on NODEs. Most real-world systems are associated with first integrals (a.k.a. First integrals arise from intrinsic geometric structures of systems and are sometimes more important than superficial dynamics in understanding systems (see Appendix A for details).