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

Yuan Qiu, Wolfgang Dahmen, Peng Chen

TL;DR

This paper advances operator learning for parametric PDEs by enforcing variational correctness through First-Order System Least-Squares losses, enabling PDE-compliant residuals that tightly bound solution errors. It introduces a Reduced Basis Neural Operator (RBNO) that learns coefficients in a conforming POD basis, ensuring stability by construction while maintaining training efficiency. A rigorous error decomposition shows total error bounds comprising FE discretization bias, RB projection error, neural approximation error, and sampling/optimization effects, with the residual serving as a computable a posteriori estimator. Numerical experiments on diffusion and linear elasticity demonstrate superior PDE-norm accuracy and effective error estimation, while RBNO outperforms data-driven baselines like PCA-Net and FNO in variationally meaningful metrics.

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.

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

TL;DR

This paper advances operator learning for parametric PDEs by enforcing variational correctness through First-Order System Least-Squares losses, enabling PDE-compliant residuals that tightly bound solution errors. It introduces a Reduced Basis Neural Operator (RBNO) that learns coefficients in a conforming POD basis, ensuring stability by construction while maintaining training efficiency. A rigorous error decomposition shows total error bounds comprising FE discretization bias, RB projection error, neural approximation error, and sampling/optimization effects, with the residual serving as a computable a posteriori estimator. Numerical experiments on diffusion and linear elasticity demonstrate superior PDE-norm accuracy and effective error estimation, while RBNO outperforms data-driven baselines like PCA-Net and FNO in variationally meaningful metrics.

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.
Paper Structure (43 sections, 10 theorems, 182 equations, 16 figures, 8 tables)

This paper contains 43 sections, 10 theorems, 182 equations, 16 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Conceptual background and orientation
  3. First-order system least-squares loss formulation
  4. Stationary diffusion equation with mixed boundary conditions
  5. Second order formulation
  6. First-order system least-squares formulation
  7. Linear elasticity equation
  8. Conforming finite element approximations
  9. Conforming finite element approximation of the diffusion problem
  10. Conforming finite element approximation of the elasticity problem
  11. Reduced basis neural operators
  12. Reduced basis approximations
  13. Loss evaluation
  14. Learning the reduced basis neural operator
  15. Numerical experiments
  16. ...and 28 more sections

Key Result

Theorem 1

Assume $\Omega$ is Lipschitz and $\mathfrak{p}\in L^\infty(\Omega)$ is uniformly bounded with $0<\alpha\le \mathfrak{p}(x)\le \beta<\infty$ independent of $\mathfrak{p}\in\mathfrak{P}$. Let $\Sigma_h=\mathrm{RT}^\circ_k$ and $\mathbb{U}_h=\mathrm{CG}^\circ_m$ on a shape-regular mesh of size $h$ with with equivalence constants depending only on $\alpha,\beta$ and the domain. Moreover, if $\sc\in H^

Figures (16)

  • Figure 1: Visualization of parameter-to-solution map $\mathfrak{p}_h \mapsto [u_h^{\circ}(\mathfrak{p}_h), \sigma_h^{\circ}(\mathfrak{p}_h)]$ at a random parameter sample $\mathfrak{p}_h$ (left) for the heat conduction (first row), Darcy flow (second row), and linear elasticity setup (third and fourth rows).
  • Figure 2: Auxiliary variables $w$ and $z=(z_1,z_2)$ for heat conduction (top left three plots), $w$ for Darcy flow (top right) with $z=(0,0)$, obtained by solving auxiliary problems \ref{['eq:DiriLiftPoisson']} and \ref{['harm']}. Bottom: $\stackunder[0.5pt]{ \stackunder[2pt]{z}{} }{} = (z_{11}, z_{12}; z_{21}, z_{21})$ by solving \ref{['aux-elast']} for linear elasticity with $\stackunder[2pt]{w}{} = (0, 0)$. These variables encode the inhomogeneous Dirichlet and Neumann boundary data.
  • Figure 3: Convergence of the FE loss with respect to mesh size ($h$) and FE order ($k=0,1,2$) for a representative parameter sample. Solid lines show the measured losses and dashed lines the reference rates ($O(h^{2(k+1)})$); Darcy flow follows the predicted asymptotic behavior, while convergence in heat conduction and linear elasticity is limited by reduced solution regularity due to discontinuous coefficients and boundary-induced corner singularities, confirming the analysis in Theorem \ref{['thm:FELSPoisson']} and \ref{['thm:FELSElasticity']}.
  • Figure 4: Comparison between the empirical mean squared error $\mathbb{E}_{\mathfrak{p}\sim\mu}[||s_r(\mathfrak{p}) - s_h(\mathfrak{p})||^2_{\mathbb{H}}]$ and the empirical mean loss difference $\mathbb{E}_{\mathfrak{p}\sim\mu}[\mathcal{L}(s_r(\mathfrak{p});\mathfrak{p}) - \mathcal{L}(s_h(\mathfrak{p}); \mathfrak{p})]$. Both quantities are estimated over $500$ random samples (using RT$_1\times$CG$_2$ elements for $s_h$), confirming the RB loss decomposition \ref{['eq:RB-fiber-equivalence-t_r']} in \ref{['prop:reduced_loss_error_equivalence']}.
  • Figure 5: Comparison between the empirical mean squared error $\mathbb{E}_{\mathfrak{p}\sim\mu}[||s_r(\mathfrak{p}) - s_h(\mathfrak{p})||^2_{\mathbb{H}}]$ of the RB solution $s_r$ (approximated over 500 samples with respect to the FE solution $s_h$ using RT$_1\times$CG$_2$ elements), the square of the $X_h$-projection error of $s_h$ onto the RB space $\mathbb{H}_r$, and the eigenvalue-based error estimate \ref{['eq:low_rank_approximation_via_trailing_eigenvalues']}. These results illustrate the quasi-optimality \ref{['eq:RB-best-approx-t_r']} of the RB approximation in Theorem \ref{['prop:reduced_loss_error_equivalence']} and the tightness of the error estimate.
  • ...and 11 more figures

Theorems & Definitions (23)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Corollary 1
  • proof
  • Proposition 1
  • proof
  • ...and 13 more