# DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modeling

arXiv: https://arxiv.org/abs/2502.06250 · HTML: https://mlsanity.com/paper/2502.06250/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 66 references that are in the map.
- First posted on arXiv 10 Feb 2025.

## Builds on (most cited first, 30 of 39)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [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)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [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)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [On Robustness of Neural Ordinary Differential Equations](https://mlsanity.com/paper/1910.05513/) (2019)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [Physics-informed deep learning for incompressible laminar flows](https://mlsanity.com/paper/2002.10558/) (2020)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Inverse-design of nonlinear mechanical metamaterials via video denoising diffusion models](https://mlsanity.com/paper/2305.19836/) (2023)
- [Numerical Solution of Inverse Problems by Weak Adversarial Networks](https://mlsanity.com/paper/2002.11340/) (2020)
- [A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables](https://mlsanity.com/paper/2006.01789/) (2020)
- [Physics informed WNO](https://mlsanity.com/paper/2302.05925/) (2023)
- [Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems](https://mlsanity.com/paper/1912.12976/) (2019)
- [Semi-supervised Invertible Neural Operators for Bayesian Inverse Problems](https://mlsanity.com/paper/2209.02772/) (2022)
- [Fully probabilistic deep models for forward and inverse problems in parametric PDEs](https://mlsanity.com/paper/2208.04856/) (2022)
- [Physics-Informed Geometry-Aware Neural Operator](https://mlsanity.com/paper/2408.01600/) (2024)

## Built on by (most cited first, 5 of 5)

- [A unified physics-informed generative operator framework for general inverse problems](https://mlsanity.com/paper/2511.03241/) (2025)
- [The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems](https://mlsanity.com/paper/2606.04804/) (2026)
- [When Does AI for PDEs Yield Scientific Evidence?](https://mlsanity.com/paper/2608.22504/) (2026)
- [Reinforcement Learning Closures for Underresolved Partial Differential Equations using Synthetic Data](https://mlsanity.com/paper/2505.11308/) (2025)
- [GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media](https://mlsanity.com/paper/2602.14642/) (2026)

## Abstract (arXiv)

Solving parametric partial differential equations (PDEs) and associated PDE-based, inverse problems is a central task in engineering and physics, yet existing neural operator methods struggle with high-dimensional, discontinuous inputs and require large amounts of em labeled training data. We propose the Deep Generative Neural Operator (DGenNO), a physics-aware framework that addresses these challenges by leveraging a deep, generative, probabilistic model in combination with a set of lower-dimensional, latent variables that simultaneously encode PDE-inputs and PDE-outputs. This formulation can make use of unlabeled data and significantly improves inverse problem-solving, particularly for discontinuous or discrete-valued input functions. DGenNO enforces physics constraints without labeled data by incorporating as virtual observables, weak-form residuals based on compactly supported radial basis functions (CSRBFs). These relax regularity constraints and eliminate higher-order derivatives from the objective function. We also introduce MultiONet, a novel neural operator architecture, which is a more expressive generalization of the popular DeepONet that significantly enhances the approximating power of the proposed model. These innovations make DGenNO particularly effective for challenging forward and inverse, PDE-based problems, such as those involving multi-phase media. Numerical experiments demonstrate that DGenNO achieves higher accuracy across multiple benchmarks while exhibiting robustness to noise and strong generalization to out-of-distribution cases. Its adaptability, and the ability to handle sparse, noisy data while providing probabilistic estimates, make DGenNO a powerful tool for scientific and engineering applications.