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

Why it matters (2 min)

• 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…
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Context

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,…

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Receipts

1. Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimation (arXiv cs.LG)