Emulating the Darcy Solution Operator by Heat-Semigroup Functional Calculus and Pointwise Products
Abstract
We approximate the solution operator (a,f)↦ u of −div(a∇ u)=f by operator networks whose only primitives are heat-semigroup functions of a fixed reference operator and pointwise products. The main result is an emulation theorem on strongly local Dirichlet spaces with a spectral gap and a Bakry--Émery bound BE(−K,∞): an explicit scheme reaches Ḣ1-error ε uniformly over coefficients with bounded carré du champ and bounded generator image and over L2 data, with parameters depending only on the coefficient bounds, the accuracy and K; on the torus this gives a C2-type coefficient class with constants independent of the dimension. The torus version refines this to Wiener classes of fractional regularity A1+θ, with an explicit rate attained by the scheme's product-defect block. The scheme unrolls the Laplacian-preconditioned Richardson iteration, controls each inexact step in the operator norm of Ḣ1 without Galerkin truncation, and synthesises the coupling ∇ a·∇ u from the product defect of one heat layer. All parameters are explicit; the results are approximation theorems for continuum operators.
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BibTeX
@misc{hanrahan2026emulating,
title = {{Emulating the Darcy Solution Operator by Heat-Semigroup Functional Calculus and Pointwise Products}},
author = {Dawid Hanrahan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20106},
eprint = {2610.20106},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20106},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Emulating the Darcy Solution Operator by Heat-Semigroup Functional Calculus and Pointwise Products
AU - Dawid Hanrahan
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20106
UR - https://arxiv.org/abs/2610.20106
AB - We approximate the solution operator (a,f)↦ u of −div(a∇ u)=f by operator networks whose only primitives are heat-semigroup functions of a fixed reference operator and pointwise products. The main result is an emulation theorem on strongly local Dirichlet spaces with a spectral gap and a Bakry--Émery bound BE(−K,∞): an explicit scheme reaches Ḣ^1-error ε uniformly over coefficients with bounded carré du champ and bounded generator image and over L^2 data, with parameters depending only on the coefficient bounds, the accuracy and K; on the torus this gives a C^2-type coefficient class with constants independent of the dimension. The torus version refines this to Wiener classes of fractional regularity A^1+θ, with an explicit rate attained by the scheme's product-defect block. The scheme unrolls the Laplacian-preconditioned Richardson iteration, controls each inexact step in the operator norm of Ḣ^1 without Galerkin truncation, and synthesises the coupling ∇ a·∇ u from the product defect of one heat layer. All parameters are explicit; the results are approximation theorems for continuum operators.
ER -