Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimation

TL;DR

This paper introduces a variationally correct operator learning framework using FOSLS objectives and a Reduced Basis Neural Operator, ensuring PDE solution errors are accurately estimated and improved in physical norms.

Contribution

It develops a variationally correct learning method with a new RBNO architecture, providing theoretical error bounds and superior PDE norm accuracy over existing approaches.

Findings

The framework achieves lower PDE norm errors compared to standard methods.

The residual loss acts as a reliable a posteriori error estimator.

Numerical results confirm the theoretical error bounds and improved accuracy.

Abstract

Minimizing 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 error in PDE-induced norms. We demonstrate this framework on stationary diffusion and linear elasticity, 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 bias, reduced basis truncation error, neural network approximation error, and statistical estimation errors arising from finite sampling and optimization. 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, computable a posteriori error estimator.