Deep Generative Neural Operator

The DGenNO is a physics-informed, probabilistic framework for solving parametric PDEs and challenging inverse problems, especially with high-dimensional, discontinuous inputs, or sparse, noisy observations. Unlike traditional neural operators, DGenNO leverages a deep generative model with low-dimensional latent variables that jointly encode PDE inputs and outputs. DGenNO enforces physics constraints without labeled data through virtual observables and weak-form residuals based on compactly supported radial basis functions (CSRBFs), avoiding higher-order derivatives (saving computing efforts) and relaxing regularity requirements (efficient in dealing with irregular inputs). This enables:

Efficient use of unlabeled data, i.e., input-output pairs
Handling discontinuous or discrete-valued inputs
Efficient in solving inverse problems with sparse, noisy observation (observation positions are totally free).
Robust generalization to out-of-distribution cases
Provides probabilistic predictions with uncertainty quantification

We also introduce MultiONet, a powerful extension of DeepONet that significantly improves the expressiveness of operator learning without introducing new network architectures and many training parameters.

(1) The DGenNO framework (without any labeled input-output training pairs)

<img src="./Docs/DGM4DGNO.png" alt="DGenNO" width="800" height='300'/> Figure 1: The Deep Generative Neural Operator (DGenNO) framework.

(2) DGenNO for inverse problems with noisy, sparse observations

<img src="./Docs/DGNO4Inverse.png" alt="DGenNO4Inverse" width="800" height='300'/> Figure 2: Solving the inverse problem with DGenNO.

(3) The MultiONet architecture

<img src="./Docs/MultiONet.png" alt="MultiONet" width="800" height='600'/> Figure 3: a) DeepONet architecture vs. b) MultiONet architecture.

📌 Benchmark Problems

We evaluate the DNO frameworks on the following PDEs:

1. Burger’s Equation

Goal: Learn the operator mapping initial condition $a(x):=u(x,t=0)$ to the solution $u(x,t)$.

2. Darcy’s Flow

Goal: Learn the mapping from the permeability field $a(x)$ to the pressure field $u(x)$. We considered two cases: (1) Smooth $a(x)$ and (2) Piecewise-constant $a(x)$.

3. Stokes Flow with a Cylindrical Obstacle

Goal: Learn the mapping from in-flow velocity ${\bf u}_0 = (a(x), 0)$ to the pressure field $u(x)$.

4. Inverse Discontinuity Coefficient in Darcy’s Flow

We also consider the inverse problem of reconstructing the piecewise-constant permeability field $a(x)$ from sparse, noisy observations of $u$. This problem has important applications in subsurface modeling and medical imaging.

🔗 Data Availability

Training data and testing data can be downloaded from Google Drive (all trained forward and inverse models can also be downloaded here).

Related Resources

The implementation of Deep Generative Neural Operator (DGenNO) and other popular deep neural operator (DNO) methods (e.g., DeepONet, FNO, PI-DeepONet, and PINO) can also be found on the Github repository: https://github.com/yaohua32/Deep-Neural-Operators-for-PDEs.

📖 Citation

@article{zang2025dgenno,
  title={DGenNO: a novel physics-aware neural operator for solving forward and inverse PDE problems based on deep, generative probabilistic modeling},
  author={Zang, Yaohua and Koutsourelakis, Phaedon-Stelios},
  journal={Journal of Computational Physics},
  volume={538},
  pages={114137},
  year={2025},
  publisher={Elsevier}
}

DGenNO

Install / Use

README