# MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs

arXiv: https://arxiv.org/abs/2107.03673 · HTML: https://mlsanity.com/paper/2107.03673/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 26 references that are in the map.
- First posted on arXiv 8 Jul 2021.

## Builds on (most cited first, 13 of 13)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Training behavior of deep neural network in frequency domain](https://mlsanity.com/paper/1807.01251/) (2018)
- [Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains](https://mlsanity.com/paper/2007.11207/) (2020)
- [Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning](https://mlsanity.com/paper/2008.13333/) (2020)
- [A Phase Shift Deep Neural Network for High Frequency Approximation and Wave Problems](https://mlsanity.com/paper/1909.11759/) (2019)
- [Machine Learning and Computational Mathematics](https://mlsanity.com/paper/2009.14596/) (2020)

## Built on by (most cited first, 24 of 24)

- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Overview frequency principle/spectral bias in deep learning](https://mlsanity.com/paper/2201.07395/) (2022)
- [FMint: Bridging Human Designed and Data Pretrained Models for Differential Equation Foundation Model](https://mlsanity.com/paper/2404.14688/) (2024)
- [Learning Specialized Activation Functions for Physics-informed Neural Networks](https://mlsanity.com/paper/2308.04073/) (2023)
- [BI-GreenNet: Learning Green's functions by boundary integral network](https://mlsanity.com/paper/2204.13247/) (2022)
- [Generating synthetic data for neural operators](https://mlsanity.com/paper/2401.02398/) (2024)
- [Neural Q-learning for solving PDEs](https://mlsanity.com/paper/2203.17128/) (2022)
- [Approximation of Functionals by Neural Network without Curse of Dimensionality](https://mlsanity.com/paper/2205.14421/) (2022)
- [Capturing the Diffusive Behavior of the Multiscale Linear Transport Equations by Asymptotic-Preserving Convolutional DeepONets](https://mlsanity.com/paper/2306.15891/) (2023)
- [Green Multigrid Network](https://mlsanity.com/paper/2407.03593/) (2024)
- [Orthogonal greedy algorithm for linear operator learning with shallow neural network](https://mlsanity.com/paper/2501.02791/) (2025)
- [Learning Singularity-Encoded Green's Functions with Application to Iterative Methods](https://mlsanity.com/paper/2509.11580/) (2025)
- [A model-data asymptotic-preserving neural network method based on micro-macro decomposition for gray radiative transfer equations](https://mlsanity.com/paper/2212.05523/) (2022)
- [Bayesian Inversion with Neural Operator (BINO) for Modeling Subdiffusion: Forward and Inverse Problems](https://mlsanity.com/paper/2211.11981/) (2022)
- [Laplace-fPINNs: Laplace-based fractional physics-informed neural networks for solving forward and inverse problems of subdiffusion](https://mlsanity.com/paper/2304.00909/) (2023)
- [Learning Green's Function Efficiently Using Low-Rank Approximations](https://mlsanity.com/paper/2308.00350/) (2023)
- [Stabilizing and Solving Unique Continuation Problems by Parameterizing Data and Learning Finite Element Solution Operators](https://mlsanity.com/paper/2412.04409/) (2024)
- [Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimization](https://mlsanity.com/paper/2412.04596/) (2024)
- [Energy stable neural network for gradient flow equations](https://mlsanity.com/paper/2309.10002/) (2023)
- [Macroscopic auxiliary asymptotic preserving neural networks for the linear radiative transfer equations](https://mlsanity.com/paper/2403.01820/) (2024)
- [A Hybrid Kernel-Free Boundary Integral Method with Operator Learning for Solving Parametric Partial Differential Equations In Complex Domains](https://mlsanity.com/paper/2404.15242/) (2024)
- [STAResNet: a Network in Spacetime Algebra to solve Maxwell's PDEs](https://mlsanity.com/paper/2408.13619/) (2024)
- [Learning PDE Dynamics between Submanifolds Using Green's Observation Operators](https://mlsanity.com/paper/2610.01697/) (2026)

## Abstract (arXiv)

In this paper, we propose a a machine learning approach via model-operator-data network (MOD-Net) for solving PDEs. A MOD-Net is driven by a model to solve PDEs based on operator representation with regularization from data. For linear PDEs, we use a DNN to parameterize the Green's function and obtain the neural operator to approximate the solution according to the Green's method. To train the DNN, the empirical risk consists of the mean squared loss with the least square formulation or the variational formulation of the governing equation and boundary conditions. For complicated problems, the empirical risk also includes a few labels, which are computed on coarse grid points with cheap computation cost and significantly improves the model accuracy. Intuitively, the labeled dataset works as a regularization in addition to the model constraints. The MOD-Net solves a family of PDEs rather than a specific one and is much more efficient than original neural operator because few expensive labels are required. We numerically show MOD-Net is very efficient in solving Poisson equation and one-dimensional radiative transfer equation. For nonlinear PDEs, the nonlinear MOD-Net can be similarly used as an ansatz for solving nonlinear PDEs, exemplified by solving several nonlinear PDE problems, such as the Burgers equation.