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

arXiv: https://arxiv.org/abs/2609.33715 · HTML: https://mlsanity.com/paper/2609.33715/

## 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.

## Builds on (most cited first, 30 of 30)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Dynamic Convolution: Attention over Convolution Kernels](https://mlsanity.com/paper/1912.03458/) (2019)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Sobolev Training for Neural Networks](https://mlsanity.com/paper/1706.04859/) (2017)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Convolutional Kernel Networks](https://mlsanity.com/paper/1406.3332/) (2014)
- [Geom-DeepONet: A Point-cloud-based Deep Operator Network for Field Predictions on 3D Parameterized Geometries](https://mlsanity.com/paper/2403.14788/) (2024)
- [Domain Agnostic Fourier Neural Operators](https://mlsanity.com/paper/2305.00478/) (2023)
- [Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loads](https://mlsanity.com/paper/2306.03645/) (2023)
- [Approximation and Learning with Deep Convolutional Models: a Kernel Perspective](https://mlsanity.com/paper/2102.10032/) (2021)
- [Phase-Field DeepONet: Physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionals](https://mlsanity.com/paper/2302.13368/) (2023)
- [Extracting Interpretable Physical Parameters from Spatiotemporal Systems using Unsupervised Learning](https://mlsanity.com/paper/1907.06011/) (2019)
- [FLD: Fourier Latent Dynamics for Structured Motion Representation and Learning](https://mlsanity.com/paper/2402.13820/) (2024)
- [Solving PDEs on Unknown Manifolds with Machine Learning](https://mlsanity.com/paper/2106.06682/) (2021)
- [Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equations](https://mlsanity.com/paper/2303.02304/) (2023)
- [Variational Autoencoding of PDE Inverse Problems](https://mlsanity.com/paper/2006.15641/) (2020)
- [Quantitative approximation results for complex-valued neural networks](https://mlsanity.com/paper/2102.13092/) (2021)
- [Phi-DVAE: Physics-Informed Dynamical Variational Autoencoders for Unstructured Data Assimilation](https://mlsanity.com/paper/2209.15609/) (2022)
- [SplitVAEs: Decentralized scenario generation from siloed data for stochastic optimization problems](https://mlsanity.com/paper/2409.12328/) (2024)
- [Single-shot prediction of parametric partial differential equations](https://mlsanity.com/paper/2505.09063/) (2025)

## 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.