# Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation

arXiv: https://arxiv.org/abs/2512.09165 · HTML: https://mlsanity.com/paper/2512.09165/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 53 references that are in the map.
- First posted on arXiv 9 Dec 2025.

## Builds on (most cited first, 22 of 22)

- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs](https://mlsanity.com/paper/2310.00120/) (2023)
- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [Mitigating spectral bias for the multiscale operator learning](https://mlsanity.com/paper/2210.10890/) (2022)
- [PDE Generalization of In-Context Operator Networks: A Study on 1D Scalar Nonlinear Conservation Laws](https://mlsanity.com/paper/2401.07364/) (2024)
- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [Physics-Informed Geometry-Aware Neural Operator](https://mlsanity.com/paper/2408.01600/) (2024)
- [Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responses](https://mlsanity.com/paper/2401.06070/) (2024)
- [Koopman operator learning using invertible neural networks](https://mlsanity.com/paper/2306.17396/) (2023)
- [Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator](https://mlsanity.com/paper/2408.02965/) (2024)
- [A Multifidelity deep operator network approach to closure for multiscale systems](https://mlsanity.com/paper/2303.08893/) (2023)
- [Local neural operator for solving transient partial differential equations on varied domains](https://mlsanity.com/paper/2203.08145/) (2022)
- [Feature-adjacent multi-fidelity physics-informed machine learning for partial differential equations](https://mlsanity.com/paper/2303.11577/) (2023)
- [FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning](https://mlsanity.com/paper/2509.12344/) (2025)
- [Sobolev Training for Operator Learning](https://mlsanity.com/paper/2402.09084/) (2024)
- [Resolution invariant deep operator network for PDEs with complex geometries](https://mlsanity.com/paper/2402.00825/) (2024)
- [Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equations](https://mlsanity.com/paper/2404.15618/) (2024)
- [Chebyshev Feature Neural Network for Accurate Function Approximation](https://mlsanity.com/paper/2409.19135/) (2024)
- [Chebyshev Spectral Neural Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2407.03347/) (2024)

## Built on by (most cited first, 6 of 6)

- [WLNO: Wavelet-Laplace Neural Operator for Solving Partial Differential Equations](https://mlsanity.com/paper/2605.24658/) (2026)
- [The impact of observation density on Bayesian inversion of latent dynamics in shock-dominated flows](https://mlsanity.com/paper/2605.19076/) (2026)
- [Rationally Enriched Chebyshev Trunk Bases for DeepONet Surrogates of High P'eclet Entrance Transport](https://mlsanity.com/paper/2608.19658/) (2026)
- [Quantum SEDONet: Spectrally-Embedded Quantum Deep Operator Networks for Partial Differential Equations](https://mlsanity.com/paper/2608.27626/) (2026)
- [Spectral-Embedded Operator Learning for Three-Phase Interfacial Flow: A Ternary Cahn-Hilliard-Navier-Stokes Benchmark](https://mlsanity.com/paper/2608.29069/) (2026)
- [Extreme-Scale Super-Resolution of Homogeneous Turbulence from a Single Realization](https://mlsanity.com/paper/2610.20305/) (2026)

## Abstract (arXiv)

Deep Operator Networks (DeepONets) have emerged as a powerful framework for data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk network, which operates directly on raw spatial or spatiotemporal coordinates through fully connected layers, often struggles to represent sharp gradients, boundary layers, and other non-periodic solution structures on bounded domains. To address these limitations, we introduce the Spectral-Embedded Deep Operator Network (SEDONet), a novel DeepONet architecture in which the trunk is driven by a fixed Chebyshev spectral dictionary instead of coordinate inputs. This non-periodic spectral embedding provides a principled inductive bias for bounded domains, enabling the learned operator to capture fine-scale features that are difficult for Fourier-based or MLP-only trunks to represent. SEDONet is evaluated on the 2-D Poisson equation, 1-D Burgers' equation, 1-D advection-diffusion equation, Allen-Cahn equation, Lorenz-96 chaotic system, and Darcy flow, covering elliptic, hyperbolic, parabolic, chaotic, and multiscale problems. Across all benchmarks, SEDONet consistently achieves the lowest or statistically comparable relative L2 errors among DeepONet, FEDONet, and SEDONet, with improvements of up to 54% over the baseline DeepONet and consistent gains over Fourier-embedded variants on bounded, non-periodic problems. Energy spectrum analyses further demonstrate that SEDONet more accurately preserves intermediate- and high-frequency solution structures. The proposed framework provides a simple, parameter-neutral modification to DeepONets, offering a robust and computationally efficient spectral approach for surrogate modeling of nonlinear operators in scientific computing.