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A promising approach for scalable Gaussian processes (GPs) is theKarhunen-Lo\`eve (KL) decomposition, in which the GP kernel is represented by aset of basis functions which are the eigenfunctions of the kernel operator.Such decomposed kernels have the potential to be very fast, and do not dependon the selection of a reduced set of inducing points. However KL decompositionslead to high dimensionality, and variable selection becomes paramount. Thispaper reports a new method of forward variable selection, enabled by theordered nature of the basis functions in the KL expansion of the BayesianSmoothing Spline ANOVA kernel (BSS-ANOVA), coupled with fast Gibbs sampling ina fully Bayesian approach. It quickly and effectively limits the number ofterms, yielding a method with competitive accuracies, training and inferencetimes for tabular datasets of low feature set dimensionality. The inferencespeed and accuracy makes the method especially useful for dynamic systemsidentification, by modeling the dynamics in the tangent space as a staticproblem, then integrating the learned dynamics using a high-order scheme. Themethods are demonstrated on two dynamic datasets: a `Susceptible, Infected,Recovered' (SIR) toy problem, with the transmissibility used as forcingfunction, along with the experimental `Cascaded Tanks' benchmark dataset.Comparisons on the static prediction of time derivatives are made with a randomforest (RF), a residual neural network (ResNet), and the Orthogonal AdditiveKernel (OAK) inducing points scalable GP, while for the timeseries predictioncomparisons are made with LSTM and GRU recurrent neural networks (RNNs) alongwith the SINDy package.

Forward variable selection enables fast and accurate dynamic system identification with Karhunen-Lo\`eve decomposed Gaussian processes