# Diffeomorphism Neural Operator for various domains and parameters of partial differential equations

arXiv: https://arxiv.org/abs/2402.12475 · HTML: https://mlsanity.com/paper/2402.12475/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 45 references that are in the map.
- First posted on arXiv 19 Feb 2024.

## Builds on (most cited first, 26 of 26)

- [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)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [Deep transfer operator learning for partial differential equations under conditional shift](https://mlsanity.com/paper/2204.09810/) (2022)
- [Physics-based Deep Learning](https://mlsanity.com/paper/2109.05237/) (2021)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
- [Group Equivariant Fourier Neural Operators for Partial Differential Equations](https://mlsanity.com/paper/2306.05697/) (2023)
- [Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains](https://mlsanity.com/paper/2104.10873/) (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)
- [Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoils](https://mlsanity.com/paper/2302.00807/) (2023)
- [Physics-enhanced deep surrogates for partial differential equations](https://mlsanity.com/paper/2111.05841/) (2021)

## Built on by (most cited first, 15 of 15)

- [Physics-Informed Geometry-Aware Neural Operator](https://mlsanity.com/paper/2408.01600/) (2024)
- [Geometric Operator Learning with Optimal Transport](https://mlsanity.com/paper/2507.20065/) (2025)
- [Fusion-DeepONet: A Data-Efficient Neural Operator for Geometry-Dependent Hypersonic and Supersonic Flows](https://mlsanity.com/paper/2501.01934/) (2025)
- [Discovering Physical Directions in Weight Space: Composing Neural PDE Experts](https://mlsanity.com/paper/2605.14546/) (2026)
- [Single vs. Multiple Branches in DeepONet and S-DeepONet: Network Architecture Follows Coupling in Multiphysics Systems](https://mlsanity.com/paper/2507.03660/) (2025)
- [Physics-informed Partitioned Coupled Neural Operator for Complex Networks](https://mlsanity.com/paper/2410.21025/) (2024)
- [Transferable Foundation Models for Geometric Tasks on Point Cloud Representations: Geometric Neural Operators](https://mlsanity.com/paper/2503.04649/) (2025)
- [Neural Shape Operator Surrogates -- Expression Rate Bounds](https://mlsanity.com/paper/2604.18012/) (2026)
- [A unified framework for geometry-independent operator learning in cardiac electrophysiology simulations](https://mlsanity.com/paper/2512.01702/) (2025)
- [A scalable neural bundle map for multiphysics prediction in lithium-ion battery across varying configurations](https://mlsanity.com/paper/2603.17209/) (2026)
- [Residual Attention Physics-Informed Neural Networks for Robust Multiphysics Simulation of Steady-State Electrothermal Energy Systems](https://mlsanity.com/paper/2603.23578/) (2026)
- [UFO: A Domain-Unification-Free Operator Framework for Generalized Operator Learning](https://mlsanity.com/paper/2605.12700/) (2026)
- [SpectONet: A Physics-Guided Spectral Deep Operator Network for Euler-Bernoulli Beam Dynamics](https://mlsanity.com/paper/2607.25790/) (2026)
- [Beyond Arbitrary Geometry: Topology Generalization In neural PDE Operators](https://mlsanity.com/paper/2609.05860/) (2026)
- [Learning PDE Dynamics between Submanifolds Using Green's Observation Operators](https://mlsanity.com/paper/2610.01697/) (2026)

## Abstract (arXiv)

In scientific and engineering applications, solving partial differential equations (PDEs) across various parameters and domains normally relies on resource-intensive numerical methods. Neural operators based on deep learning offered a promising alternative to PDEs solving by directly learning physical laws from data. However, the current neural operator methods were limited to solve PDEs on fixed domains. Expanding neural operators to solve PDEs on various domains hold significant promise in medical imaging, engineering design and manufacturing applications, where geometric and parameter changes are essential. This paper presents a novel neural operator learning framework for solving PDEs with various domains and parameters defined for physical systems, named diffeomorphism neural operator (DNO). The main idea is that a neural operator learns in a generic domain which is diffeomorphically mapped from various physics domains expressed by the same PDE. In this way, the challenge of operator learning on various domains is transformed into operator learning on the generic domain. The generalization performance of DNO on different domains can be assessed by a proposed method which evaluates the geometric similarity between a new domain and the domains of training dataset after diffeomorphism. Experiments on Darcy flow, pipe flow, airfoil flow and mechanics were carried out, where harmonic and volume parameterization were used as the diffeomorphism for 2D and 3D domains. The DNO framework demonstrated robust learning capabilities and strong generalization performance across various domains and parameters.