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Neural General Operator Networks via Banach Fixed Point Iterations

by M. Feischl and Ch. Schwab and F. Zehetgruber

(Report number 2025-13)

Abstract
Banach's fixed point (BFP) iterations are at the root of constructive proofs for several major existence results in nonlinear functional analysis; in particular, for nonlinear differential and integral equations in function spaces. Examples are the Cauchy-Lipschitz theorem, the Lax-Milgram theorem, the Newton-Kantorovich theorem and the implicit function theorem. We develop a framework based on "unrolled" BFP-iterations to construct families of finite-parametric, generalized neural operator networks ("GONs") which approximate numerical data-to-solution maps for nonlinear operator equations in Banach- and metric spaces.

We prove that GONs obtained by finitely termined, unrolled BFP-iterations converge exponentially in terms of network depth: they furnish approximate solutions with target accuracy \(\varepsilon>0\) at logarithmic (with respect to \(\varepsilon\)) network depth, assuming at hand an \(\varepsilon\)-consistent iterator network. An elementary example is the (well-known) exponential rate of deep NN emulation of rough, self-similar function systems. Similarly, BFP-iterations can, subject to available parsimonious neural emulations of the BFP iterator maps, alleviate or eliminate smoothness assumptions on solutions of nonlinear operator equations in NN expression rate bounds due to ensuring (root) exponential consistency with respect to the depth of the network.

Keywords:

@Techreport{FSZ25_1134,
  author = {M. Feischl and Ch. Schwab and F. Zehetgruber},
  title = {Neural General Operator Networks via Banach Fixed Point Iterations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-13},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-13.pdf },
  year = {2025}
}

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