SA-PINNs (Self-Adaptive Physics-Informed Neural Networks) assign learnable positive attention weights to observation points, boundary points, and PDE collocation points, enabling the model to focus automatically on regions that are difficult to fit during training.

This model package provides examples for the one-dimensional Laplace equation, the two-dimensional Helmholtz equation, and the one-dimensional time-dependent Burgers equation.

Paper: Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism
https://arxiv.org/abs/2009.04544

Model Description

SA-PINNs use min–max saddle-point optimization: network parameters reduce the weighted physics loss through gradient descent, while attention parameters increase the weights assigned to difficult locations through gradient ascent. The attention weights are constructed from normalized exp(alpha), ensuring that they remain positive with a mean close to 1. The Adam stage updates both the network and attention parameters, whereas the L-BFGS stage freezes the attention parameters and refines the network.

Use Cases

Use Case	Description
One-dimensional Laplace equation	Examine how attention focuses on collocation points with high residuals
Two-dimensional Helmholtz equation	Solve a two-dimensional analytical solution with high-frequency spatial variation
Burgers equation	Train jointly on initial conditions, boundary conditions, and nonlinear PDE residuals
Adaptive-weight research	Compare a standard PINN with pointwise soft-attention weighting

Usage

1. OneCode

Use the online OneCode environment for an intelligent, one-click AI for Science (AI4S) programming experience:

Launch OneCode for one-click AI4S programming

2. Manual Setup

Hardware Requirements

A GPU or DCU is recommended for training.
A CPU can be used for small-scale pipeline validation.
DCU users must install DTK and a PyTorch environment compatible with the target cluster.

Download the Model Package

bash

hf download OneScience-Group/SA-PINNs --local-dir ./SA-PINNs
cd SA-PINNs

Set Up the Runtime Environment

DCU Environment

bash

# Activate DTK and Conda first
conda create -n onescience311 python=3.11 -y
conda activate onescience311
pip install onescience[cfd-dcu] -i http://mirrors.onescience.ai:3141/pypi/simple/ --trusted-host mirrors.onescience.ai

GPU Environment

bash

# Activate Conda first
conda create -n onescience311 python=3.11 -y libstdcxx-ng=12 libgcc-ng=12 gcc_linux-64=12 gxx_linux-64=12
conda activate onescience311
pip install onescience[cfd-gpu] -i http://mirrors.onescience.ai:3141/pypi/simple/ --trusted-host mirrors.onescience.ai

Training Data

All three examples generate observation points, boundary points, and PDE collocation points from analytical equations and random sampling, with no dependency on external data files. The Burgers example does not include a reference solution and therefore outputs only the predicted field and attention distribution.

Training

Select an example with --case:

bash

python scripts/train.py --case laplace
python scripts/train.py --case helmholtz
python scripts/train.py --case burgers

Checkpoints and training histories are saved to weight/ and result/ by default. Command-line arguments such as --epochs, --lbfgs-iters, --n-pde, and --device can override values in conf/config.yaml.

Model Weights

This repository provides weights trained for all three examples in the weight/ directory.

Inference, Evaluation, and Visualization

After training the selected example, run:

bash

python scripts/inference.py --case laplace
python scripts/inference.py --case helmholtz
python scripts/inference.py --case burgers

Predictions, error plots, and PDE attention distributions are saved to the result/ directory. The Laplace and Helmholtz examples report relative L2 errors. Default model and training parameters can be modified in conf/config.yaml.

Official OneScience Resources

Platform	OneScience Repository	Skills Repository
Gitee	https://gitee.com/onescience-ai/onescience	https://gitee.com/onescience-ai/oneskills
GitHub	https://github.com/onescience-ai/OneScience	https://github.com/onescience-ai/oneskills

Citations and License

McClenny, L. and Braga-Neto, U. Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism. arXiv:2009.04544, 2020.
This model package is released under the Apache-2.0 license and retains attribution to the original paper.