README.md

---
frameworks:
  - pytorch
language:
  - en
license: apache-2.0
tags:
  - OneScience
  - BPINNs
  - Bayesian-physics-informed-neural-networks
  - partial-differential-equations
  - Laplace
tasks:
  - pde-solving
---
<p align="center">
  <strong>
    <span style="font-size: 30px;">BPINNs</span>
  </strong>
</p>

# Model Overview

BPINNs (Bayesian Physics-Informed Neural Networks) incorporate physical-equation constraints into Bayesian neural networks to jointly estimate equation solutions and predictive uncertainty from noisy data. This model package provides an example for solving a one-dimensional Laplace equation:

```text
u_xx + pi^2 sin(pi x) = 0,  x in [0, 1]
u(x) = sin(pi x)
```

Paper: B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data  
https://doi.org/10.1016/j.jcp.2020.109913

# Model Description

BPINNs are trained jointly on observational data, boundary conditions, and PDE residuals. By default, the model is optimized with Adam and then refined with L-BFGS. During inference, predictive means and standard deviations can be computed from multiple parameter states stored in a checkpoint. The default training workflow produces only one optimized state; full Bayesian uncertainty quantification requires posterior parameter states obtained through Hamiltonian Monte Carlo (HMC), variational inference, or ensemble sampling.

# Use Cases

| Use Case | Description |
| :---: | :--- |
| One-dimensional Laplace equation | Train the solution jointly from observation points, boundary points, and collocation points |
| PDE modeling with noisy data | Fit noisy observations subject to physical constraints |
| Optimizer comparison | Compare the effects of Adam and L-BFGS on PINN convergence |
| Posterior prediction | Compute predictive means and standard deviations from multiple parameter states |

# 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](https://web-2069360198568017922-iaaj.ksai.scnet.cn:58043/home)

## 2. Manual Setup

**Hardware Requirements**

- A GPU or DCU can accelerate full training; a CPU is sufficient for pipeline validation.
- DCU users must install DTK and a PyTorch environment compatible with the target cluster.

### Download the Model Package

```bash
modelscope download --model OneScience/BPINNs --local_dir ./BPINNs
cd BPINNs
```

### 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

This example automatically generates interior observation points, boundary points, PDE collocation points, and a test grid from the analytical solution; no external data files are required. The number of observation and collocation points, noise level, and computational domain can be configured in `conf/config.yaml`.

### Training

```bash
python scripts/train.py
```

By default, training produces the base weights at `weight/bpinn_laplace1d.pt` and saves the training history to `result/training_history.npz`.

### Model Weights

This repository provides weights trained on the one-dimensional Laplace equation in the `weight/` directory.

### L-BFGS Refinement

```bash
python scripts/refine.py
```

By default, refinement produces `weight/bpinn_laplace1d_refined.pt`.

### Inference, Evaluation, and Visualization

```bash
python scripts/inference.py
```

Inference loads the refined weights when available and otherwise falls back to the base weights. Predictions and visualizations are saved to `result/`. Model, training, loss, and inference parameters can all 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

- Yang, L., Meng, X., and Karniadakis, G. E. B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data. Journal of Computational Physics, 425, 109913, 2021.
- This model package is released under the Apache-2.0 license and retains attribution to the original paper.