Scalable $h$-adaptive probabilistic solver for time-independent and time-dependent systems

This repository contains the implementation for the paper:

Scalable $h$-adaptive probabilistic solver for time-independent and time-dependent systems (Link)

Overview

The code implements a scalable Gaussian process probabilistic solver (GPPS) for partial differential equations that combines a Gaussian process (GP) representation of the solution, a stochastic dual descent (SDD) algorithm for fast inference, and a clustering-based active learning strategy for $h$-adaptive refinement. The method applies to both time-independent (steady-state) and time-dependent (space–time) PDEs, and returns not only numerical solutions but also rigorous posterior uncertainty quantification (UQ). Unlike standard GP-based solvers, the approach in the paper is computationally scalable, and high-dimensional problems.

Example

Case Study 2: Poisson equation in 3D domain

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Usage

1. Clone the repository:

   git clone https://github.com/csccm-iitd/GPPS.git
   cd GPPS

2. Install the required dependencies:

   pip install -r requirements.txt
   

Requirements

To run the code, ensure that you have the following dependencies installed:

• Python 3.12
• PyTorch
• NumPy
• SciPy
• Matplotlib
• Other libraries specified in requirements.txt

Citation

If you use this code in your research, please cite the following paper:

@misc{thakur2025scalablehadaptiveprobabilisticsolver,
      title={Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems}, 
      author={Akshay Thakur and Sawan Kumar and Matthew Zahr and Souvik Chakraborty},
      year={2025},
      eprint={2508.09623},
      archivePrefix={arXiv},
      primaryClass={stat.ML},
      url={https://arxiv.org/abs/2508.09623}, 
}