Uncertainty and Structure in Neural Ordinary Differential Equations

Neural ordinary differential equations (ODEs) are an emerging class of deep learning models for dynamical systems. They are particularly useful for learning an ODE vector field from observed trajectories (i.e., inverse problems). We here consider aspects of these models relevant for their application in science and engineering. Scientific predictions generally require structured uncertainty estimates. As a first contribution, we show that basic and lightweight Bayesian deep learning techniques like the Laplace approximation can be applied to neural ODEs to yield structured and meaningful uncertainty quantification. But, in the scientific domain, available information often goes beyond raw trajectories, and also includes mechanistic knowledge, e.g., in the form of conservation laws. We explore how mechanistic knowledge and uncertainty quantification interact on two recently proposed neural ODE frameworks - symplectic neural ODEs and physical models augmented with neural ODEs. In particular, uncertainty reflects the effect of mechanistic information more directly than the predictive power of the trained model could. And vice versa, structure can improve the extrapolation abilities of neural ODEs, a fact that can be best assessed in practice through uncertainty estimates. Our experimental analysis demonstrates the effectiveness of the Laplace approach on both low dimensional ODE problems and a high dimensional partial differential equation.