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

arXiv: https://arxiv.org/abs/2511.18260 · HTML: https://mlsanity.com/paper/2511.18260/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 49 references that are in the map.
- First posted on arXiv 23 Nov 2025.

## Builds on (most cited first, 13 of 13)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [A physics-informed operator regression framework for extracting data-driven continuum models](https://mlsanity.com/paper/2009.11992/) (2020)
- [Data-driven reduced-order models via regularized operator inference for a single-injector combustion process](https://mlsanity.com/paper/2008.02862/) (2020)
- [Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference](https://mlsanity.com/paper/2110.07653/) (2021)
- [Data driven approximation of parametrized PDEs by Reduced Basis and Neural Networks](https://mlsanity.com/paper/1904.01514/) (2019)
- [Learning High-Dimensional Parametric Maps via Reduced Basis Adaptive Residual Networks](https://mlsanity.com/paper/2112.07096/) (2021)
- [Finite Element Operator Network for Solving Elliptic-type parametric PDEs](https://mlsanity.com/paper/2308.04690/) (2023)
- [Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEs](https://mlsanity.com/paper/2211.08900/) (2022)
- [A general reduced-order neural operator for spatio-temporal predictive learning on complex spatial domains](https://mlsanity.com/paper/2409.05508/) (2024)
- [ReBaNO: Reduced Basis Neural Operator Mitigating Generalization Gaps and Achieving Discretization Invariance](https://mlsanity.com/paper/2509.09611/) (2025)

## Built on by (most cited first, 1 of 1)

- [Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimation](https://mlsanity.com/paper/2512.21319/) (2025)

## Abstract (arXiv)

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.