Theory Guided and Interpretable Neural Operator Design for Partial Differential Equation Learning

Zeyuan Song; Zheyu Jiang

cs.LGSep 27, 2026

Theory Guided and Interpretable Neural Operator Design for Partial Differential Equation Learning

Authors: Zeyuan Song, Zheyu Jiang

Organizations: Oklahoma State University Stillwater, OK 74078

Abstract

Accurate numerical solutions of partial differential equations (PDEs) are crucial in numerous science and engineering applications. In this work, we introduce a novel neural PDE solver named AFDONet, which incorporates neural operator learning and adaptive Fourier decomposition (AFD) theory for the first time into a specifically designed variational autoencoder (VAE) structure, to solve a general class of nonlinear PDEs on smooth manifolds. AFDONet is the first neural PDE solver whose architectural and component design is fully guided by an established mathematical framework (in this case, AFD theory), turning neural operator design from an art to a science. Thus, AFDONet also exhibits exceptional mathematical explainability and groundness, and enjoys several desired properties. Furthermore, AFDONet achieves outstanding solution accuracy and competitive computational efficiency in several benchmark problems. In particular, thanks to its deep connections with AFD theory, AFDONet shows superior performance in solving PDEs on i) arbitrary (Riemannian) manifolds, and ii) datasets with sharp gradients. Overall, this work presents a new paradigm for designing explainable neural operator frameworks.

Figures & tables

Figure 1: Our proposed AFDONet framework, which adopts VAE as the backbone, introduces a latent-to-RKHS network and a dynamic CKN decoder to reproduce the AFD setting and operation.

Figure 2: Average $\mathrm{MAE}$ , relative $L^{2}$ error, and total computational time comparisons with respect to dataset size (averaged over five random seeds) for Navier-Stokes equation (static task) (top row), Helmholtz equation (middle row), and Poisson equation (bottom row).

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Average $\mathrm{MAE}$ and total computational time (in seconds) of FNO solver with respect to number of Fourier modes (averaged over five random seeds) for solving the Helmholtz equation 12 .

Latent dimension	MAE	Relative $L^{2}$ error	Time (sec)
16	6.40E-04 $\pm$ 9.90E-05	1.11E-03 $\pm$ 1.91E-04	1058.39 $\pm$ 19.30
20	5.35E-04 $\pm$ 1.36E-04	1.40E-03 $\pm$ 1.03E-03	1190.61 $\pm$ 15.67
32	3.77E-04 $\pm$ 1.28E-04	9.60E-04 $\pm$ 8.03E-04	1110.57 $\pm$ 18.38
64	4.62E-04 $\pm$ 1.35E-04	1.22E-03 $\pm$ 8.92E-04	1173.40 $\pm$ 17.22
100	4.05E-04 $\pm$ 1.09E-04	1.06E-03 $\pm$ 9.94E-04	1365.03 $\pm$ 21.89
128	3.89E-04 $\pm$ 1.26E-04	9.99E-04 $\pm$ 8.48E-04	1406.05 $\pm$ 23.98

Appendix

Table 4: Average

\mathrm{MAE}

, relative

L^{2}

error, and computational time (in seconds) of AFDONet (averaged over five random seeds) for solving Navier-Stokes equation 13 (autoregressive task) under different latent space dimensions.

Parameter	Value
Training epochs	$100$
Loss weights ( $\omega$ )	$10^{-5}$
Loss weights ( $w_{i}$ )	$10^{-8}$
Optimizer	Adam
Learning rate	$10^{-3}$
Batch size	$16$

Appendix

Table 5: Specifications of loss function and training for AFDONet solver.

Figure 4: Ground truth and predicted solutions $(u,v)$ of the Navier-Stokes equation (static task) on the torus and heat map.

Figure 5: Ground truth and predicted solutions $u(x,y)$ of the Helmholtz equation on the planar manifold.

Figure 6: Ground truth and predicted solutions $u(\phi,z)$ of the Poisson equation on the quarter-cylindrical surface.

Figure 7: Ground truth and predicted fields ( $u$ , $v$ ) of the Navier-Stokes equation (for static task) on both the torus $\mathbb{T}^{2}$ and the heatmap for various solvers. Here, the dataset is generated from Gaussian-based randomized vortex fields (dataset size is $5000$ ) [ Pedergnana et al., 2020 ] . Average $\mathrm{MAE}$ and relative $L^{2}$ errors and their standard deviations obtained using five random seeds are also reported.

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Jul 20, 2026, cs.LGJ/K move · Enter open · S save

Adaptive Mamba Neural Operators

Zeyuan Song, Zheyu Jiang

School of Chemical Engineering Oklahoma State University Stillwater, OK 74078, USA

Neural Partial Differential Equation Solvers Neural Operators Partial Differential Equations Fluid Dynamics Fourier

Open paper AskPDF arXiv

Accurately solving partial differential equations (PDEs) on arbitrary geometries and a variety of meshes is an important task in science and engineering applications. In this paper, we propose Adaptive Mamba Neural Operators (AMO), which integrates reproducing kernels for state-space models (SSMs) rather than the kernel integral formulation of SSMs. This is achieved by constructing Takenaka-Malmquist systems for the PDEs. AMO offers new representations that align well with the adaptive Fourier decomposition (AFD) theory and can approximate the solution manifold of PDEs on a wide range of geometries and meshes. In several challenging benchmark PDE problems in the fields of fluid physics, solid physics, and finance on point clouds, structured meshes, regular grids, and irregular domains, AMO consistently outperforms state-of-the-art solvers in terms of relative

L^2

error. Overall, this work presents a new paradigm for designing explainable neural operator frameworks.

Abstract

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