# Deep ReLU neural network approximation of parametric and stochastic elliptic PDEs with lognormal inputs

We investigate non-adaptive methods of deep ReLU neural network approximation of the solution u to parametric and stochastic elliptic PDEs with lognormal inputs on non-compact set ℝ^∞. The approximation error is measured in the norm of the Bochner space L_2(ℝ^∞, V, γ), where γ is the tensor product standard Gaussian probability on ℝ^∞ and V is the energy space. The approximation is based on an m-term truncation of the Hermite generalized polynomial chaos expansion (gpc) of u. Under a certain assumption on ℓ_q-summability condition for lognormal inputs (0< q <∞), we proved that for every integer n > 1, one can construct a non-adaptive compactly supported deep ReLU neural network ϕ_n of size not greater than n on ℝ^m with m = 𝒪 (n/log n), having m outputs so that the summation constituted by replacing polynomials in the m-term truncation of Hermite gpc expansion by these m outputs approximates u with an error bound 𝒪((n/log n)^-1/q). This error bound is comparable to the error bound of the best approximation of u by n-term truncations of Hermite gpc expansion which is 𝒪(n^-1/q). We also obtained some results on similar problems for parametric and stochastic elliptic PDEs with affine inputs, based on the Jacobi and Taylor gpc expansions.

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