# DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains

arXiv: https://arxiv.org/abs/2402.07250 · HTML: https://mlsanity.com/paper/2402.07250/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 62 references that are in the map.
- First posted on arXiv 11 Feb 2024.

## Builds on (most cited first, 19 of 19)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [A Point Set Generation Network for 3D Object Reconstruction from a Single Image](https://mlsanity.com/paper/1612.00603/) (2016)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [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)
- [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)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
- [LNO: Laplace Neural Operator for Solving Differential Equations](https://mlsanity.com/paper/2303.10528/) (2023)
- [Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loads](https://mlsanity.com/paper/2306.03645/) (2023)
- [Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework](https://mlsanity.com/paper/2204.04238/) (2022)
- [Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoils](https://mlsanity.com/paper/2302.00807/) (2023)
- [Complex Functional Maps : a Conformal Link Between Tangent Bundles](https://mlsanity.com/paper/2112.09546/) (2021)
- [A rule-based method to model myocardial fiber orientation in cardiac biventricular geometries with outflow tracts](https://mlsanity.com/paper/1809.08297/) (2018)

## Built on by (most cited first, 16 of 16)

- [Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation](https://mlsanity.com/paper/2404.12355/) (2024)
- [Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networks](https://mlsanity.com/paper/2402.15406/) (2024)
- [VICON: Vision In-Context Operator Networks for Multi-Physics Fluid Dynamics Prediction](https://mlsanity.com/paper/2411.16063/) (2024)
- [Geometric Operator Learning with Optimal Transport](https://mlsanity.com/paper/2507.20065/) (2025)
- [LeMON: Learning to Learn Multi-Operator Networks](https://mlsanity.com/paper/2408.16168/) (2024)
- [DeepONet as a Multi-Operator Extrapolation Model: Distributed Pretraining with Physics-Informed Fine-Tuning](https://mlsanity.com/paper/2411.07239/) (2024)
- [Graph Fourier Neural Kernels (G-FuNK): Learning Solutions of Nonlinear Diffusive Parametric PDEs on Multiple Domains](https://mlsanity.com/paper/2410.04655/) (2024)
- [Diffeomorphic Latent Neural Operators for Data-Efficient Learning of Solutions to Partial Differential Equations](https://mlsanity.com/paper/2411.18014/) (2024)
- [A deformation-based framework for learning solution mappings of PDEs defined on varying domains](https://mlsanity.com/paper/2412.01379/) (2024)
- [Neural Shape Operator Surrogates -- Expression Rate Bounds](https://mlsanity.com/paper/2604.18012/) (2026)
- [A Neural Operator based on Dynamic Mode Decomposition](https://mlsanity.com/paper/2507.01117/) (2025)
- [Convex Neural Energy Elements: Monolithic Finite-Element Assembly of Geometry-Parameterized Neural Operators with Stability and Error Guarantees](https://mlsanity.com/paper/2608.02036/) (2026)
- [MENO: Memory-Efficient Neural Operator](https://mlsanity.com/paper/2609.27739/) (2026)
- [Structure-Adaptive Tree Field Integrators](https://mlsanity.com/paper/2609.34025/) (2026)
- [Neural Harmonic Measure Operator](https://mlsanity.com/paper/2609.35752/) (2026)
- [METRO: Metric-Enhanced Token Routing Operator](https://mlsanity.com/paper/2610.05100/) (2026)

## Abstract (arXiv)

The solution of a PDE over varying initial/boundary conditions on multiple domains is needed in a wide variety of applications, but it is computationally expensive if the solution is computed de novo whenever the initial/boundary conditions of the domain change. We introduce a general operator learning framework, called DIffeomorphic Mapping Operator learNing (DIMON) to learn approximate PDE solutions over a family of domains Omegaθθ, that learns the map from initial/boundary conditions and domain Omegaθ to the solution of the PDE, or to specified functionals thereof. DIMON is based on transporting a given problem (initial/boundary conditions and domain Omegaθ) to a problem on a reference domain Omega0, where training data from multiple problems is used to learn the map to the solution on Omega0, which is then re-mapped to the original domain Omegaθ. We consider several problems to demonstrate the performance of the framework in learning both static and time-dependent PDEs on non-rigid geometries; these include solving the Laplace equation, reaction-diffusion equations, and a multiscale PDE that characterizes the electrical propagation on the left ventricle. This work paves the way toward the fast prediction of PDE solutions on a family of domains and the application of neural operators in engineering and precision medicine.