Learning Green's functions associated with parabolic partial differential equations
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Given input-output pairs from a parabolic partial differential equation (PDE) in any spatial dimension n≥ 1, we derive the first theoretically rigorous scheme for learning the associated Green's function G. Until now, rigorously learning Green's functions associated with parabolic operators has been a major challenge in the field of scientific machine learning because G may not be square-integrable when n>1, and time-dependent PDEs have transient dynamics. By combining the hierarchical low-rank structure of G together with the randomized singular value decomposition, we construct an approximant to G that achieves a relative error of 𝒪(Γ_ϵ^-1/2ϵ) in the L^1-norm with high probability by using at most 𝒪(ϵ^-n+2/2log(1/ϵ)) input-output training pairs, where Γ_ϵ is a measure of the quality of the training dataset for learning G, and ϵ>0 is sufficiently small. Along the way, we extend the low-rank theory of Bebendorf and Hackbusch from elliptic PDEs in dimension 1≤ n≤ 3 to parabolic PDEs in any dimensions, which shows that Green's functions associated with parabolic PDEs admit a low-rank structure on well-separated domains.
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