Supplementary Material Spectrum Gaussian Processes Learning from Noisy and Sparse Data A Derivation of the spectral representation

The ELBO is derived from Jensen's inequality as follows: log p ( Y) ZZZ q ( X, f, w) log p ( Y, X, f, w) q ( X, f, w) d w d f d X (31) = ZZZ p ( f | w) q ( w) The inference procedure of SSGP is shown in Algorithm 1. In the experiments, we set the integration time window =1 . Update the parameters by maximizing the ELBO (13) evaluated using D . In this appendix, we describe baseline models for the experiments in Section 6. D-SymODEN can also apply to the dissipative systems. SympGPR can estimate conservative vector fields from derivative observations by considering Hamiltonian mechanics; we used finite differences for training.