Projected Neural Differential Equations for Learning Constrained Dynamics
Authors: Alistair White, Anna Büttner, Maximilian Gelbrecht, Valentin Duruisseaux, Niki Kilbertus, Frank Hellmann, Niklas Boers
Organizations: Munich Climate Center and Earth System Modelling Group, Department of Aerospace and Geodesy, School of Engineering and Design, Technical University of Munich, Munich, Germany · Potsdam Institute for Climate Impact Research, Potsdam, Germany · California Institute of Technology, Pasadena, California, USA · Technical University of Munich, Munich, Germany · Helmholtz Munich, Neuherberg, Germany · Munich Center for Machine Learning (MCML), Munich, Germany · University of Exeter, Exeter, United Kingdom
Neural differential equations offer a powerful approach for learning dynamical systems from data. However, they do not inherently respect known constraints, such as conservation laws, that should be obeyed by the learned dynamics. It is well known that enforcing constraints in data-driven models can enhance their generalizability and numerical stability. In this paper, we introduce projected neural differential equations (PNDEs), a method for constraining neural differential equations based on projection of the predicted velocities onto the tangent space of the manifold that fulfills the constraint. In tests on several examples from different fields, including chaotic dynamical systems and power grid models, PNDEs outperform existing methods for constraining learned dynamics, require fewer hyperparameters, and are computationally more efficient. The proposed approach demonstrates potential for enhancing the modeling of constrained dynamical systems, particularly in domains where accuracy and reliability are essential.