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

2025-12-24

AI Summary

Physics-informed neural operators lack variational consistency in PDE solving—small residual does not guarantee small solution error. Method: We propose a variationally rigorous framework based on the First-Order System Least Squares (FOSLS) formulation, ensuring the loss function is equivalent to the solution error measured in the PDE-induced norm. Specifically: (1) we design the first variationally correct training objective for neural operators; (2) we introduce the provably convergent Reduced Basis Neural Operator (RBNO) architecture; and (3) we develop a variational enhancement technique that uniformly handles mixed boundary conditions and rigorously interprets the residual loss as a reliable, computable a posteriori error estimator. Results: On steady-state diffusion and linear elasticity problems, our method significantly outperforms standard baselines in PDE-compatible norms. Numerical experiments strictly validate the theoretical error bounds, and the residual provides an efficient, computationally tractable guide for adaptive solution refinement.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Develops a variationally correct operator learning framework for PDEs
• Ensures variational stability via Reduced Basis Neural Operator
• Provides rigorous error bounds and reliable a posteriori estimation

Innovation

• Variationally correct operator learning framework
• Reduced Basis Neural Operator with conforming reduced basis
• First-order system least-squares objectives for error equivalence