Federated scientific machine learning for approximating functions and solving differential equations with data heterogeneity

The data and code for the paper H. Zhang, L. Liu, K. Weng, & L. Lu. Federated scientific machine learning for approximating functions and solving differential equations with data heterogeneity. IEEE Transactions on Neural Networks and Learning Systems, 36 (10), 18104–18117, 2025.

Datasets

Data generation scripts are available in the data folder:

• Antiderivative
• Burgers equation
• Diffusion-reaction equation
• Lid-driven cavity flow

Also you may check the helper functions in data_assignment file for data generation for federated learning setting for 1D and 2D problem. Details are shown in Sec.II in the paper.

Code

All codes are provided in the folder src, including training centralized baseline models, federated models and extrapolation models. The code depends on the deep learning package DeepXDE.

FL for Function Approximation

• 1D Gramacy&Lee function
• 2D Schaffer function

FedPINN

• Poisson equation
• Helmholtz equation
• Allen Cahn equation
• Multiclient Allen Cahn
• Inverse Navier-Stokes equation
• Inverse Diffusion-reaction equation

FedDeepONet

• Antiderivative
• Burgers equation
• Diffusion-reaction equation
• Lid-driven cavity flow

Cite this work

If you use this data or code for academic research, you are encouraged to cite the following paper:

@ARTICLE{11053778,
  author  = {Zhang, Handi and Liu, Langchen and Weng, Kangyu and Lu, Lu},
  title   = {Federated Scientific Machine Learning for Approximating Functions and Solving Differential Equations With Data Heterogeneity}, 
  journal = {IEEE Transactions on Neural Networks and Learning Systems}, 
  year    = {2025},
  volume  = {36},
  number  = {10},
  pages   = {1-14},
  doi     = {10.1109/TNNLS.2025.3580409}
}

Question

To get help on how to use the data or code, simply open an issue in the GitHub "Issues" section.