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Abstract This study presents a data‐driven approach to quantify uncertainties in the ionosphere‐thermosphere (IT) system due to varying solar wind parameters (drivers) during quiet conditions (Kp < 4) and fixed solar radiation and lower atmospheric conditions representative of 16 March 2013. Ensemble simulations of the coupled Whole Atmosphere Model with Ionosphere Plasmasphere Electrodynamics (WAM‐IPE) driven by synthetic solar wind drivers generated through a multi‐channel variational autoencoder (MCVAE) model are obtained. Applying the polynomial chaos expansion (PCE) technique, it is possible to estimate the means and variances of the QoIs as well as the sensitivities of the QoIs with regard to the drivers. Our results highlight unique features of the IT system's uncertainty: (a) the uncertainty of the IT system is larger during nighttime; (b) the spatial distributions of the uncertainty for electron density and zonal drift at fixed local times present 4 peaks in the evening sector, which are associated with the low‐density regions of longitude structure of electron density; (c) the uncertainty of the equatorial electron density is highly correlated with the uncertainty of the zonal drift, especially in the evening sector, while it is weakly correlated with the vertical drift. A variance‐based global sensitivity analysis suggests that the IMF Bz plays a dominant role in the uncertainty of electron density. A further discussion shows that the uncertainty of the IT system is determined by the magnitudes and universal time variations of solar wind drivers. Its temporal and spatial distribution can be modulated by the average state of the IT system. 

We present a stochastic descent algorithm for unconstrained optimization that is particularly efficient when the objective function is slow to evaluate and gradients are not easily obtained, as in some PDE-constrained optimization and machine learning problems. The algorithm maps the gradient onto a low-dimensional ran- dom subspace of dimension at each iteration, similar to coordinate descent but without restricting directional derivatives to be along the axes. Without requiring a full gradient, this mapping can be performed by computing directional deriva- tives (e.g., via forward-mode automatic differentiation). We give proofs for conver- gence in expectation under various convexity assumptions as well as probabilistic convergence results under strong-convexity. Our method provides a novel extension to the well-known Gaussian smoothing technique to descent in subspaces of dimen- sion greater than one, opening the doors to new analysis of Gaussian smoothing when more than one directional derivative is used at each iteration. We also provide a finite-dimensional variant of a special case of the Johnson–Lindenstrauss lemma. Experimentally, we show that our method compares favorably to coordinate descent, Gaussian smoothing, gradient descent and BFGS (when gradients are calculated via forward-mode automatic differentiation) on problems from the machine learning and shape optimization literature.