Deep-Neural-Operators-for-PDEs

This repository implements several deep neural operator (DNO) frameworks for solving parametric partial differential equations (PDEs) and related inverse problems. These methods aim to learn an operator $\mathcal{G}$ that maps input parameters (e.g., coefficients, boundary/initial conditions) to the corresponding PDE solutions.

🎯 Problem Formulation

Given a parametric PDE $\mathcal{N}[u;a]=0$, where $\mathcal{N}$ represents the PDE, $a$ denotes the input function (e.g., coefficients or boundary/initial conditions), $u$ is the corresponding PDE solution. The goal of deep neural operators is to approximate the mapping: $\mathcal{G}: a \in \mathcal{A} \rightarrow u \in \mathcal{U}$.

🏆 Implemented Deep Neural Operator Frameworks

DNO methods can be categorized into data-driven and physics-aware approaches based on whether they incorporate physics constraints during training.

1. Data-Driven DNOs (Supervised Learning)

These methods learn the operator solely from labeled training pairs $(a,u)$.

• DeepONet: A pioneering architecture using branch and trunk networks.
• Fourier Neural Operator (FNO): Employs Fourier transformations for efficient global convolutions.
• MultiONet: Enhances DeepONet with shortcut connections, significantly improving approximation power with minimal parameter increase.

2. Physics-Aware DNOs (Self-Supervised Learning)

These methods incorporate physics constraints (i.e., PDE residuals) into the training process, reducing data requirements and improving accuracy and generalization.

• PI-DeepONet: Extends DeepONet by adding PDE residuals as training constraints. However, it requires higher regularity of inputs/outputs, making it unsuitable for singular or discontinuous inputs/outputs.
• PINO: A physics-informed extension of FNO that leverages PDE residuals but struggles with complex geometries and high-dimensional problems due to its reliance on regular, fine meshes for Fourier transformation and derivative approximation.
• PI-MultiONet: A physics-informed version of MultiONet, improving accuracy while reducing labeled data requirements.
• Deep Generative Neural Operator (DGenNO): A novel framework leveraging deep generative modeling and probabilistic latent variables to handle complex physics-based problems, including inverse problems. DGenNO offers several key advantages:
  • Enable to learn purely from physics constraints.
  • Effectively solves parametric PDEs and inverse problems with discontinuous inputs.
  • Provides probabilistic estimates and robust performance with sparse, noisy data in solving inverse problems.
  • Uses weak-form PDE residuals based on compactly supported radial basis functions (CSRBFs), reducing regularity constraints.

📌 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.

📌 Remark: Due to the challenging nature of this inverse problem, the above DNO frameworks are unable to solve it except for the Deep Generative Neural Operator (DGenNO) method. Therefore, we have only implemented DGenNO for this inverse problem.

🔗 Data Availability

• All Physics-aware DNOs in this repository are trained exclusively using physics information (i.e., without labeled (a, u) pairs).
• Training data (only for data-driven DNOs) and testing data can be downloaded from Google Drive.

🔬 Future Work and Related Resources

This repository is an ongoing project, and more DNO frameworks and PDE applications will be added. We welcome contributions and collaborations!

Related resources:

• Instructions and an implementation of the MultiONet method can be found at https://github.com/yaohua32/MultiONet
• Instructions and an implementation of the DeepONet method can be found at https://github.com/yaohua32/DeepONet
• Instructions and an implementation of the FNO (Fourier Neural Operator) method can be found at https://github.com/yaohua32/Fourier-Neural-Operator

📖 Citation

If you find this work useful or are interested in our DGenNO or PI-MultiONet methods, please cite our paper:

@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}
}