Fixed and Adaptive Topological DeepONets: Functional Measurements on Hausdorff Locally Convex Spaces

2026-08-05

AI Summary

This work addresses the limitations of conventional DeepONets, which rely on fixed-grid sampling and struggle with function inputs in non-normable locally convex spaces while lacking discretization invariance. The authors introduce Topological DeepONet, the first framework to encode input functions via continuous dual functionals over Hausdorff locally convex spaces. They propose both fixed and adaptive functional measurement mechanisms, integrated with a two-stage coefficient-space strategy, a dedicated decoder trained end-to-end, and an error decomposition theory featuring refined Barron-type rates. The resulting method yields interpretable, compact, and grid-transferable operator learning: it achieves approximately 5.5% resolution-independent error in heterogeneous Darcy flow, reduces average error below 1.2% in control operator tasks using adaptive measurements, and attains a mean relative L² error of 1.685% in steady-state Navier–Stokes vorticity prediction—significantly outperforming comparably sized Fourier Neural Operators in computational efficiency.

Abstract

Deep Operator Networks (DeepONets; arXiv:1910.03193) typically encode an input function through point values on a fixed discretization. Building on the Topological DeepONet framework of Ismailov (arXiv:2603.11972), we replace point samples by continuous linear functionals drawn from the continuous dual of a Hausdorff locally convex space $({V},\{p_α\}_{α\in A})$, whose topology is generated by a point-separating family of seminorms rather than a single norm, and develop fixed and adaptive functional measurement systems. Measurements are combined with the coefficient-space Two-Step procedure of Lee and Shin (arXiv:2309.01020), while a training-only decoder and regularization stabilize the adaptive coordinates. We derive a discrete error decomposition separating measurement, output-basis, and neural-approximation errors, together with a Barron-rate refinement. The framework is evaluated on the antiderivative operator, a non-normable locally convex input space, heterogeneous Darcy flow, a controlled operator, and fixed-time and time-evolving Navier-Stokes vorticity operators. In the heterogeneous Darcy problem, the functional models retain nearly resolution-independent errors of 5.5-5.6% on unseen grids, while in the controlled problem adaptive measurements reduce the mean error below 1.2%. For the fixed-time Navier-Stokes problem, the Adaptive Topological DeepONet is the most accurate DeepONet-based model, attaining a mean relative $L^2$ error of 1.685% +/- 0.017% using 128 functional coordinates. A comparably sized Fourier neural operator (FNO; arXiv:2010.08895) achieves the lower error 0.832% +/- 0.172%, but requires the full 64x64 input field, twice the training time, and 10.7x greater peak GPU memory. The formulation provides compact, interpretable, and discretization-portable coordinates in the continuous dual $V'$, including for non-normable input spaces.

Problem

• Deep Operator Networks
• Hausdorff locally convex spaces
• functional measurements
• non-normable spaces
• operator learning

Innovation

• Topological DeepONet
• locally convex space
• functional measurements
• adaptive coordinates
• operator learning