Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation

TL;DR

The paper tackles the challenge of learning operators for PDEs on bounded, non-periodic domains where standard DeepONet trunks struggle with boundary layers and sharp gradients. It introduces Spectral-Embedded DeepONet (SEDONet), which replaces the trunk input with a fixed tensor-product Chebyshev embedding, preserving the branch while adding a non-periodic spectral prior. Across five benchmarks (2-D Poisson, Burgers, Advection-Diffusion, Lorenz-96, Allen-Cahn), SEDONet achieves lower relative $L^2$ errors and better spectral fidelity than both DeepONet and Fourier-embedded variants, especially on non-periodic problems. The approach is simple, plug-and-play, and builds a bridge between classical Chebyshev spectral methods and modern neural operator architectures, improving accuracy, stability, and boundary-resolution in surrogate PDE models.

Abstract

Deep Operator Networks (DeepONets) have become a central tool in data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk design based on fully connected layers acting on raw spatial or spatiotemporal coordinates struggles to represent sharp gradients, boundary layers, and non-periodic structures commonly found in PDEs posed on bounded domains with Dirichlet or Neumann boundary conditions. To address these limitations, we introduce the Spectral-Embedded DeepONet (SEDONet), a new DeepONet variant in which the trunk is driven by a fixed Chebyshev spectral dictionary rather than coordinate inputs. This non-periodic spectral embedding provides a principled inductive bias tailored to bounded domains, enabling the learned operator to capture fine-scale non-periodic features that are difficult for Fourier or MLP trunks to represent. SEDONet is evaluated on a suite of PDE benchmarks including 2D Poisson, 1D Burgers, 1D advection-diffusion, Allen-Cahn dynamics, and the Lorenz-96 chaotic system, covering elliptic, parabolic, advective, and multiscale temporal phenomena, all of which can be viewed as canonical problems in computational mechanics. Across all datasets, SEDONet consistently achieves the lowest relative L2 errors among DeepONet, FEDONet, and SEDONet, with average improvements of about 30-40% over the baseline DeepONet and meaningful gains over Fourier-embedded variants on non-periodic geometries. Spectral analyses further show that SEDONet more accurately preserves high-frequency and boundary-localized features, demonstrating the value of Chebyshev embeddings in non-periodic operator learning. The proposed architecture offers a simple, parameter-neutral modification to DeepONets, delivering a robust and efficient spectral framework for surrogate modeling of PDEs on bounded domains.

Figures (12)

• Figure 1: SEDONet architecture: The branch network maps discrete samples of the input function $u_0$ to latent coefficients $b_k$, while the trunk network maps Chebyshev spectral features $\Phi(x,t)$ to basis channels $t_k(x,t)$. Their inner product yields the operator evaluation $\widehat{\mathcal{G}}(u_0)(x,t)$.
• Figure 2: Comparison of DeepONet, FEDONet, and SEDONet on a representative test example from the 2D Poisson dataset. The left panel shows the forcing field; the remaining panels show the exact solution, model predictions, and the corresponding pointwise residual fields.
• Figure 3: Relative $\ell_2$ error (mean $\pm$ std) across 1000 unseen Poisson test set for all three architectures. SEDONet achieves both the lowest mean error and the lowest variance.
• Figure 4: Relative $\ell_2$ error across 128 unseen Burgers' test samples for all three architectures (DeepONet, FEDONet, and SEDONet).
• Figure 5: Best-performing Burgers' test sample: Ground truth, DeepONet, FEDONet, and SEDONet predictions with error maps.