Simple, low-cost and accurate data-driven geophysical forecasting with learned kernels

Abstract

Modelling geophysical processes as low-dimensional dynamical systems and regressing their vector field from data is a promising approach for learning emulators of such systems. We show that when the kernel of these emulators is also learned from data (using kernel flows, a variant of cross-validation), then the resulting data-driven models are not only faster than equation-based models but are easier to train than neural networks such as the long short-term memory neural network. In addition, they are also more accurate and predictive than the latter. When trained on geophysical observational data, for example the weekly averaged global sea-surface temperature, considerable gains are also observed by the proposed technique in comparison with classical partial differential equation-based models in terms of forecast computational cost and accuracy. When trained on publicly available re-analysis data for the daily temperature of the North American continent, we see significant improvements over classical baselines such as climatology and persistence-based forecast techniques. Although our experiments concern specific examples, the proposed approach is general, and our results support the viability of kernel methods (with learned kernels) for interpretable and computationally efficient geophysical forecasting for a large diversity of processes.

Computational graph completion

“…We have focused the manuscript on the situation where the kernels of the underlying GPs are given/pre-determined. Using data-driven kernels can improve the accuracy of kernel methods by orders of magnitude [34,5,16,15,10,21,35]. There are essentially three categories of methods for learning kernel from data: variants of cross-validation (such as Kernel Flows [34]), maximum likelihood estimation (see [5] for a comparison between Kernel Flows and MLE), and maximum a posteriori estimation [30].…”

“…Remark 5.1. Although this (classical) approach may appear simple, it can be highly effective when the underlying kernel is also learned from data [16,15]. Considering the extrapolation of time series obtained from satellite data as an example, this simple dataadapted kernel perspective outperforms (both in complexity and accuracy) PDE-based and ANN-based methods [15].…”

Deep learning for surrogate modeling of two-dimensional mantle convection

“…(1) One could use the partial differential equations and boundary conditions governing mantle convection to place soft and/or hard constraints on the optimization problem at hand [33,[69][70][71]. (2) Although POD bases are not appropriate for this problem because they do not generalize well among simulations with different parameters, the idea of finding a set of basis functions, that one needs only to learn the coefficients to, remains an attractive one [72][73][74].…”

Gaussian process hydrodynamics

“…which establishes (40). The identity P g(h(•), h(x)) = g(•, h(x)) employed in (41) follows from observing that the identity g(•, h(x))…”

“…When the underlying physics is unknown, the kernel can be learned from data via crossvalidation/maximum likelihood estimation in a given (possibly non-parametric) family of kernels [25,[37][38] . The kernel flow (a variant of cross-validation) approach [37] has been shown to be efficient for learning (possibly stochastic) dynamical systems [39][40][41][42][43] and designing surrogate models [44][45][46] . In particular, this approach has been shown to compare favorably to ANN-based methods (in terms of both complexity and accuracy) for weather/climate prediction using actual satellite data [40] .…”