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This content will become publicly available on November 1, 2027

Title: Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimation
Minimizing partial differential equation (PDE)-residual losses is a common strategy to promote physical consistency in neural operators. However, standard formulations often lack variational correctness, meaning that small residuals do not guarantee small solution errors due to the use of non-compliant norms or ad hoc penalty terms for boundary conditions. This work develops a variationally correct operator learning framework by constructing first-order system least- squares (FOSLS) objectives whose values are provably equivalent to the solution errors in PDE-compliant norms. We demonstrate this framework on the stationary diffusion and linear elasticity equations, incorporating mixed Dirichlet-Neumann boundary conditions via variational lifts to preserve norm equivalence without inconsistent penalties. To ensure the function space conformity required by the FOSLS loss, we propose a Reduced Basis Neural Operator (RBNO). The RBNO predicts coefficients for a pre-computed, conforming reduced basis, thereby ensuring variational stability by design while enabling efficient training. We provide a rigorous convergence analysis that bounds the total error by the sum of finite element discretization error, reduced basis projection error, neural network approximation error, statistical estimation error, and optimization error. Numerical benchmarks validate these theoretical bounds and demonstrate that the proposed approach achieves superior accuracy in PDE-compliant norms compared to standard baselines, while the residual loss serves as a reliable and computable a posteriori error estimator.  more » « less
Award ID(s):
2245097 2513112 2038080 2245111
PAR ID:
10706341
Author(s) / Creator(s):
; ;
Publisher / Repository:
Elsevier
Date Published:
Journal Name:
Computer Methods in Applied Mechanics and Engineering
Volume:
461
Issue:
PB
ISSN:
0045-7825
Page Range / eLocation ID:
119228
Subject(s) / Keyword(s):
Parametric partial differential equations Neural operator Reduced basis methods First-order system least squares Variational correctness A posteriori error estimate
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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