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Reduced-Basis Deep Operator Learning for Parametric PDEs with Independently Varying Boundary and Source Data

Yueqi Wang, Guang Lin

TL;DR

RB--DeepONet combines a fixed reduced-basis trunk with a label-free branch network that predicts RB coefficients, yielding an offline–online split for parametric PDE operators. The framework supports independent variation of boundary and source data via boundary and source modal encodings and uses residual-based training to converge to RB–Galerkin solutions. Convergence theory separates RB discretization error from learning error, and numerical experiments show competitive accuracy with substantially fewer trainable parameters and significant online speedups compared to POD--DeepONet and FEONet. The approach offers a stable, interpretable, and data-efficient pathway for large-scale parametric PDEs in design, digital twins, and real-time simulation contexts.

Abstract

Parametric PDEs power modern simulation, design, and digital-twin systems, yet their many-query workloads still hinge on repeatedly solving large finite-element systems. Existing operator-learning approaches accelerate this process but often rely on opaque learned trunks, require extensive labeled data, or break down when boundary and source data vary independently from physical parameters. We introduce RB-DeepONet, a hybrid operator-learning framework that fuses reduced-basis (RB) numerical structure with the branch-trunk architecture of DeepONet. The trunk is fixed to a rigorously constructed RB space generated offline via Greedy selection, granting physical interpretability, stability, and certified error control. The branch network predicts only RB coefficients and is trained label-free using a projected variational residual that targets the RB-Galerkin solution. For problems with independently varying loads or boundary conditions, we develop boundary and source modal encodings that compress exogenous data into low-dimensional coordinates while preserving accuracy. Combined with affine or empirical interpolation decompositions, RB-DeepONet achieves a strict offline-online split: all heavy lifting occurs offline, and online evaluation scales only with the RB dimension rather than the full mesh. We provide convergence guarantees separating RB approximation error from statistical learning error, and numerical experiments show that RB-DeepONet attains accuracy competitive with intrusive RB-Galerkin, POD-DeepONet, and FEONet while using dramatically fewer trainable parameters and achieving significant speedups. This establishes RB-DeepONet as an efficient, stable, and interpretable operator learner for large-scale parametric PDEs.

Reduced-Basis Deep Operator Learning for Parametric PDEs with Independently Varying Boundary and Source Data

TL;DR

RB--DeepONet combines a fixed reduced-basis trunk with a label-free branch network that predicts RB coefficients, yielding an offline–online split for parametric PDE operators. The framework supports independent variation of boundary and source data via boundary and source modal encodings and uses residual-based training to converge to RB–Galerkin solutions. Convergence theory separates RB discretization error from learning error, and numerical experiments show competitive accuracy with substantially fewer trainable parameters and significant online speedups compared to POD--DeepONet and FEONet. The approach offers a stable, interpretable, and data-efficient pathway for large-scale parametric PDEs in design, digital twins, and real-time simulation contexts.

Abstract

Parametric PDEs power modern simulation, design, and digital-twin systems, yet their many-query workloads still hinge on repeatedly solving large finite-element systems. Existing operator-learning approaches accelerate this process but often rely on opaque learned trunks, require extensive labeled data, or break down when boundary and source data vary independently from physical parameters. We introduce RB-DeepONet, a hybrid operator-learning framework that fuses reduced-basis (RB) numerical structure with the branch-trunk architecture of DeepONet. The trunk is fixed to a rigorously constructed RB space generated offline via Greedy selection, granting physical interpretability, stability, and certified error control. The branch network predicts only RB coefficients and is trained label-free using a projected variational residual that targets the RB-Galerkin solution. For problems with independently varying loads or boundary conditions, we develop boundary and source modal encodings that compress exogenous data into low-dimensional coordinates while preserving accuracy. Combined with affine or empirical interpolation decompositions, RB-DeepONet achieves a strict offline-online split: all heavy lifting occurs offline, and online evaluation scales only with the RB dimension rather than the full mesh. We provide convergence guarantees separating RB approximation error from statistical learning error, and numerical experiments show that RB-DeepONet attains accuracy competitive with intrusive RB-Galerkin, POD-DeepONet, and FEONet while using dramatically fewer trainable parameters and achieving significant speedups. This establishes RB-DeepONet as an efficient, stable, and interpretable operator learner for large-scale parametric PDEs.

Paper Structure

This paper contains 18 sections, 8 theorems, 136 equations, 8 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. Parametric PDEs
  4. Finite Element approximation
  5. RB--DeepONet framework
  6. RB trunk construction
  7. Offline reduced operators
  8. Training and inference
  9. Extension to independent boundary and source data
  10. Boundary and source modes construction
  11. Modal encoding and RB right-hand-side assembly
  12. Summary and comparison
  13. Convergence analysis of RB--DeepONet
  14. Numerical experiments
  15. Example 1: Fully-parameterized steady heat conduction
  16. ...and 3 more sections

Key Result

Proposition 4.1

Assume Assumption ass:wellposed holds. Furthermore, we assume the maps $\mathbf{k}\mapsto a(\cdot,\cdot;\mathbf{k})$ and $\mathbf{k}\mapsto \ell(\cdot;\mathbf{k})$ are continuous on $\mathcal{D}$. Then, for a fixed RB basis $\Psi=[\psi_1,\ldots,\psi_N]\subset V_0$ as in Section sec:rb-construction a Consequently, the RB system $\mathbf A_{\rm rb}(\mathbf{k})\,\mathbf c_N(\mathbf{k})=\mathbf F_{\rm

Figures (8)

  • Figure 1: RB--DeepONet workflow: offline preparation (Greedy RB basis, affine precomputation, and boundary/source modes), residual-based training with a fixed trunk and a branch predicting $\mathbf c_\theta$, and online prediction with inputs $\mathbf k$ (Case I) or $[\mathbf k,\mathbf a,\mathbf b]$ (Case II). In the middle column, $\bullet=k$ (Case I), $\bullet=k_{\mathrm{aug}}$ (Case II).
  • Figure 2: Geometrical set-up (left) of the heat conductivity problem and mesh on $\Omega$ (right).
  • Figure 3: Example \ref{['sec:ex1']}: Representative-parameter comparison at $\mathbf{k}=(k_1,k_2)=(10,\,0.5)$ of (top–left) FEM reference, (top–right) FEONet, (bottom–left) POD--DeepONet, and (bottom–right) RB--DeepONet.
  • Figure 4: Example \ref{['sec:ex1']}: Training/validation loss.
  • Figure 5: Example \ref{['sec:ex2']}: Training/validation loss for FEONet and POD--, RB--DeepONet with $(N,r_f,r_g)=(209,128,16)$.
  • ...and 3 more figures

Theorems & Definitions (20)

  • Remark 3.1
  • Remark 3.2: Projection error for unseen data
  • Remark 3.3
  • Remark 3.4: Generality of RB--DeepONet
  • Proposition 4.1
  • Proposition 4.2
  • proof
  • Definition 4.1: Pseudo-dimension anthony2009neuralbartlett2019nearlyshalev2014understanding
  • Theorem 4.1
  • Remark 4.2: On existence of minimizers
  • ...and 10 more