Rationally Enriched Chebyshev Trunk Bases for DeepONet Surrogates of High Péclet Entrance Transport

Mingeun Choi; Satish Kumar

cs.LGAug 20, 2026

Rationally Enriched Chebyshev Trunk Bases for DeepONet Surrogates of High Péclet Entrance Transport

Authors: Mingeun Choi, Satish Kumar

Organizations: George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, 30332, GA, USA

Abstract

This study demonstrates a rationally enriched Chebyshev (REC) trunk for deep operator network (DeepONet) surrogate models of singularly perturbed and high-Péclet transport problems whose solution profiles are characterized by thin localized boundary or wall layers. The REC trunk combines Chebyshev polynomial dictionary elements with rational dictionary elements constructed using the adaptive Antoulas-Anderson (AAA) algorithm. Over five independent training runs, the resulting REC-trunk DeepONet is evaluated against a vanilla DeepONet and a Chebyshev-trunk DeepONet whose prescribed dictionary consists only of Chebyshev polynomials across three problems whose singular perturbation parameters are diffusion-to-advection ratios: a singularly perturbed scalar boundary-value problem (BVP), the thermal entrance problem with a prescribed wall temperature, and the concentration entrance problem with an absorbing wall. Across the held-out test profiles, the REC-trunk DeepONet improves over the vanilla DeepONet and remains comparable to the Chebyshev-trunk DeepONet in predicting the scalar profile, with its clearest advantage over the Chebyshev-trunk DeepONet appearing when the perturbation parameter lies between $1.00\times10^{-4}$ and $1.78\times10^{-4}$ , where it reduces the profile-error metrics by up to $19.5\,\%$ relative to the Chebyshev-trunk DeepONet. In predicting the wall-normal temperature and concentration profiles, the REC-trunk DeepONet reduces the profile-error metrics by up to $60.2\,\%$ and $32.2\,\%$ relative to the vanilla and Chebyshev-trunk DeepONets, respectively, while suppressing artificial near-wall oscillations as the Péclet or mass-transfer Péclet number ranges from $10^{2}$ to $10^{4}$ .

Figures & tables

Figure 1: Computational domains and boundary conditions for (a) Problem 1 and (b) Problems 2 and 3. The full channel shown in (b) is the symmetric extension of the half-channel computational domain

y\in[0\mathord{\mathchar 59\relax}1]

Figure 2: Ratios of

E_{2}^{\mathrm{rel}}

E_{\infty}

E_{\max}^{\mathrm{layer}}

, and

E_{\mathrm{LA}}

over

\varepsilon\in[10^{-4}\mathord{\mathchar 59\relax}10^{-2}]

for the singularly perturbed scalar BVP at

N_{\mathrm{out}}=16

Metric ratio	Full test set	First three parameter bins	Seeds with lower REC error	Profiles with lower REC error
$\left.E_{2}^{\mathrm{rel}}\right\|_{\mathrm{REC}}/\left.E_{2}^{\mathrm{rel}}\right\|_{\mathrm{Vanilla}}$	$0.107$	$0.110$	$5/5$	$100\%$
$\left.E_{\infty}\right\|_{\mathrm{REC}}/\left.E_{\infty}\right\|_{\mathrm{Vanilla}}$	$0.0976$	$0.0821$	$5/5$	$100\%$
$\left.E_{\max}^{\mathrm{layer}}\right\|_{\mathrm{REC}}/\left.E_{\max}^{\mathrm{layer}}\right\|_{\mathrm{Vanilla}}$	$0.0977$	$0.0820$	$5/5$	$99.8\%$
$\left.E_{\mathrm{LA}}\right\|_{\mathrm{REC}}/\left.E_{\mathrm{LA}}\right\|_{\mathrm{Vanilla}}$	$0.335$	$0.146$	$5/5$	$92.1\%$
$\left.E_{2}^{\mathrm{rel}}\right\|_{\mathrm{REC}}/\left.E_{2}^{\mathrm{rel}}\right\|_{\mathrm{Chebyshev}}$	$0.962$	$0.905$	$5/5$	$52.6\%$
$\left.E_{\infty}\right\|_{\mathrm{REC}}/\left.E_{\infty}\right\|_{\mathrm{Chebyshev}}$	$1.06$	$0.970$	$0/5$	$43.9\%$

Table 1: Comparison of the REC model with the Vanilla and Chebyshev models for the singularly perturbed scalar BVP at

N_{\mathrm{out}}=16

Figure 3: Ratios of

E_{2}^{\mathrm{rel}}

E_{\infty}

E_{\max}^{\mathrm{layer}}

, and

E_{\mathrm{LA}}

over

\mathrm{Pe}^{-1}\in[10^{-4}\mathord{\mathchar 59\relax}10^{-2}]

for the thermal entrance problem at

N_{\mathrm{out}}=16

Metric ratio	Full test set	First three parameter bins	Seeds with lower REC error	Profiles with lower REC error
$\left.E_{2}^{\mathrm{rel}}\right\|_{\mathrm{REC}}/\left.E_{2}^{\mathrm{rel}}\right\|_{\mathrm{Vanilla}}$	$0.398$	$0.316$	$5/5$	$98.5\%$
$\left.E_{\infty}\right\|_{\mathrm{REC}}/\left.E_{\infty}\right\|_{\mathrm{Vanilla}}$	$0.461$	$0.375$	$5/5$	$93.7\%$
$\left.E_{\max}^{\mathrm{layer}}\right\|_{\mathrm{REC}}/\left.E_{\max}^{\mathrm{layer}}\right\|_{\mathrm{Vanilla}}$	$0.558$	$0.435$	$5/5$	$80.6\%$
$\left.E_{\mathrm{LA}}\right\|_{\mathrm{REC}}/\left.E_{\mathrm{LA}}\right\|_{\mathrm{Vanilla}}$	$0.426$	$0.318$	$5/5$	$85.9\%$
$\left.E_{2}^{\mathrm{rel}}\right\|_{\mathrm{REC}}/\left.E_{2}^{\mathrm{rel}}\right\|_{\mathrm{Chebyshev}}$	$0.747$	$0.773$	$5/5$	$90.6\%$
$\left.E_{\infty}\right\|_{\mathrm{REC}}/\left.E_{\infty}\right\|_{\mathrm{Chebyshev}}$	$0.699$	$0.746$	$5/5$	$90.9\%$

Table 2: Comparison of the REC model with the Vanilla and Chebyshev models for the thermal entrance problem at

N_{\mathrm{out}}=16

Figure 4: Ratios of

E_{2}^{\mathrm{rel}}

E_{\infty}

E_{\max}^{\mathrm{layer}}

, and

E_{\mathrm{LA}}

over

\mathrm{Pe}_{m}^{-1}\in[10^{-4}\mathord{\mathchar 59\relax}10^{-2}]

for the concentration entrance problem at

N_{\mathrm{out}}=16

and

\mathrm{Da}=0.1

1.0

, and

10.0

Metric ratio	Full test set	First three parameter bins	Seeds with lower REC error	Profiles with lower REC error
Problem 3 ( $\mathrm{Da}=0.1$ )
$\left.E_{2}^{\mathrm{rel}}\right\|_{\mathrm{REC}}/\left.E_{2}^{\mathrm{rel}}\right\|_{\mathrm{Vanilla}}$	$0.453$	$0.291$	$5/5$	$94.9\%$
$\left.E_{\infty}\right\|_{\mathrm{REC}}/\left.E_{\infty}\right\|_{\mathrm{Vanilla}}$	$0.476$	$0.329$	$5/5$	$90.9\%$
$\left.E_{\max}^{\mathrm{layer}}\right\|_{\mathrm{REC}}/\left.E_{\max}^{\mathrm{layer}}\right\|_{\mathrm{Vanilla}}$	$0.799$	$0.585$	$5/5$	$64.2\%$
$\left.E_{\mathrm{LA}}\right\|_{\mathrm{REC}}/\left.E_{\mathrm{LA}}\right\|_{\mathrm{Vanilla}}$	$0.827$	$0.578$	$4/5$	$63.8\%$
$\left.E_{2}^{\mathrm{rel}}\right\|_{\mathrm{REC}}/\left.E_{2}^{\mathrm{rel}}\right\|_{\mathrm{Chebyshev}}$	$0.761$	$0.683$	$5/5$	$87.7\%$

Table 3: Comparison of the REC model with the Vanilla and Chebyshev models for the concentration entrance problem at

N_{\mathrm{out}}=16

Figure 5: Full representative solution profiles for the three problems at

N_{\mathrm{out}}=16

. (a) Scalar profile

u(x)

x\in(0\mathord{\mathchar 59\relax}1)

for

\varepsilon=1.01\times 10^{-4}

. (b) Temperature profile

\theta(y)

y\in(0\mathord{\mathchar 59\relax}1)

for

\mathrm{Pe}^{-1}=1.24\times 10^{-4}

and

x=6.70\times 10^{-2}

. (c)–(e) Concentration profiles

c(y)

y\in(0\mathord{\mathchar 59\relax}1)

for

\mathrm{Da}=0.1

1.0

, and

10.0

, where the

\mathrm{Da}=0.1

case uses

\mathrm{Pe}_{m}^{-1}=1.24\times 10^{-4}

and

x=6.70\times 10^{-2}

, while the

\mathrm{Da}=1.0

and

10.0

cases use

\mathrm{Pe}_{m}^{-1}=1.15\times 10^{-4}

and

x=8.32\times 10^{-2}

Figure 6: Near-boundary and near-wall solution profiles for the three problems at

N_{\mathrm{out}}=16

. (a) Scalar profile

u(x)

near the right endpoint for

\varepsilon=1.01\times 10^{-4}

. (b) Temperature profile

\theta(y)

near the wall for

\mathrm{Pe}^{-1}=1.24\times 10^{-4}

and

x=6.70\times 10^{-2}

. (c)–(e) Concentration profiles

c(y)

near the absorbing wall for

\mathrm{Da}=0.1

1.0

, and

10.0

, where the

\mathrm{Da}=0.1

case uses

\mathrm{Pe}_{m}^{-1}=1.24\times 10^{-4}

and

x=6.70\times 10^{-2}

, while the

\mathrm{Da}=1.0

and

10.0

cases use

\mathrm{Pe}_{m}^{-1}=1.15\times 10^{-4}

and

x=8.32\times 10^{-2}

. Each comparison includes the numerical reference, Vanilla, Chebyshev, and REC.

Problem	Parameter	$\left.W_{2}\right\|_{\mathrm{Reference}}$	$\left.W_{2}\right\|_{\mathrm{Chebyshev}}$	$\left.W_{2}\right\|_{\mathrm{REC}}$
Problem 1	$\varepsilon=1.01\times 10^{-4}$	$1.87\times 10^{-1}$	$9.72\times 10^{-1}$	$2.07\times 10^{-1}$
Problem 2	$\mathrm{Pe}^{-1}=1.24\times 10^{-4}$ $x=6.70\times 10^{-2}$	$1.29\times 10^{-1}$	$8.23\times 10^{-1}$	$1.35\times 10^{-1}$
Problem 3 ( $\mathrm{Da}=0.1$ )	$\mathrm{Pe}_{m}^{-1}=1.24\times 10^{-4}$ $x=6.70\times 10^{-2}$	$5.15\times 10^{-2}$	$8.71\times 10^{-1}$	$6.78\times 10^{-2}$
Problem 3 ( $\mathrm{Da}=1.0$ )	$\mathrm{Pe}_{m}^{-1}=1.15\times 10^{-4}$ $x=8.32\times 10^{-2}$	$6.39\times 10^{-2}$	$1.43\times 10^{0}$	$9.26\times 10^{-2}$
Problem 3 ( $\mathrm{Da}=10.0$ )	$\mathrm{Pe}_{m}^{-1}=1.15\times 10^{-4}$ $x=8.32\times 10^{-2}$	$9.22\times 10^{-2}$	$1.63\times 10^{0}$	$9.63\times 10^{-2}$

Table 4:

W_{2}

values for the numerical reference and for the predictions from the Chebyshev and REC models in the five representative cases shown in Fig. 6 , computed over

x\in(0\mathord{\mathchar 59\relax}1)

for Problem 1 and over

y\in(0\mathord{\mathchar 59\relax}1)

for Problems 2 and 3.

Figure 7: Ratios of

E_{2}^{\mathrm{rel}}

E_{\infty}

E_{\max}^{\mathrm{layer}}

, and

E_{\mathrm{LA}}

from the REC model to the corresponding errors from the Chebyshev model for

N_{\mathrm{out}}=16

33

65

, and

97

Figure 8: Ratios of

E_{2}^{\mathrm{rel}}

E_{\infty}

E_{\max}^{\mathrm{layer}}

, and

E_{\mathrm{LA}}

from the REC model to the corresponding errors from the Vanilla model for

N_{\mathrm{out}}=16

33

65

, and

97

Figure 9: Averaged values of

E_{2}^{\mathrm{rel}}

E_{\infty}

E_{\max}^{\mathrm{layer}}

, and

E_{\mathrm{LA}}

for the Vanilla , Chebyshev , and REC models at

N_{\mathrm{out}}=16

PDE Surrogate Modeling

PDE surrogate modeling builds fast approximations of partial differential equation solutions for simulation, prediction, and design.

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Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation

Muhammad Abid, Omer San

Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville, TN 37996, USA.

PDE Surrogate Modeling PDE Operator Learning Spectral Methods Deep Operator Networks

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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

L^2

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.

Abstract

PDE Surrogate Modeling

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