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

arXiv 2609.33715 ↗First posted 27 Sep 2026cs.LG cs.CEcited by 0

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Abstract (arXiv)

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.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 30 of its 76 references that are in the map. First posted on arXiv 27 Sep 2026.

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References · 30

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
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  12. 2023Spherical Fourier Neural Operators: Learning Stable Dynamics on the Spherecited by 233
18 more
  1. 2017Sobolev Training for Neural Networkscited by 143
  2. 2022Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modelingcited by 65
  3. 2021Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modelingcited by 59
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  5. 2024Geom-DeepONet: A Point-cloud-based Deep Operator Network for Field Predictions on 3D Parameterized Geometriescited by 47
  6. 2023Domain Agnostic Fourier Neural Operatorscited by 38
  7. 2023Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loadscited by 33
  8. 2021Approximation and Learning with Deep Convolutional Models: a Kernel Perspectivecited by 31
  9. 2023Phase-Field DeepONet: Physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionalscited by 26
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  13. 2023Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equationscited by 11
  14. 2020Variational Autoencoding of PDE Inverse Problemscited by 9
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  16. 2022Phi-DVAE: Physics-Informed Dynamical Variational Autoencoders for Unstructured Data Assimilationcited by 5
  17. 2024SplitVAEs: Decentralized scenario generation from siloed data for stochastic optimization problemscited by 2
  18. 2025Single-shot prediction of parametric partial differential equationscited by 2

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