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

TL;DR

RB-DeepONet combines reduced-basis methods with DeepONet architecture to efficiently and accurately solve parametric PDEs with independently varying boundary and source data, offering interpretability and certified error control.

Contribution

It introduces a hybrid framework that fuses reduced-basis numerical structure with DeepONet, enabling label-free training and efficient online evaluation for complex parametric PDEs.

Findings

Achieves accuracy comparable to traditional methods with fewer parameters.

Provides convergence guarantees separating approximation and learning errors.

Offers significant computational speedups over existing operator-learning approaches.

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.