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We derive upper bounds on the Wasserstein distance ($W_1$), with respect to$\sup$-norm, between any continuous $\mathbb{R}^d$ valued random field indexedby the $n$-sphere and the Gaussian, based on Stein's method. We develop a novelGaussian smoothing technique that allows us to transfer a bound in a smoothermetric to the $W_1$ distance. The smoothing is based on covariance functionsconstructed using powers of Laplacian operators, designed so that theassociated Gaussian process has a tractable Cameron-Martin or ReproducingKernel Hilbert Space. This feature enables us to move beyond one dimensionalinterval-based index sets that were previously considered in the literature.Specializing our general result, we obtain the first bounds on the Gaussianrandom field approximation of wide random neural networks of any depth andLipschitz activation functions at the random field level. Our bounds areexplicitly expressed in terms of the widths of the network and moments of therandom weights. We also obtain tighter bounds when the activation function hasthree bounded derivatives.

Gaussian random field approximation via Stein's method with applications to wide random neural networks