𝕌pdes

𝕌pdes is a general-purpose library for mesh-free PDE simulation and control. There is no faster way to explore the realm of PDEs !

Features

𝕌pdes leverages Radial Basis Functions (RBFs) and JAX to provide the following features:

• User-centric design: no need to re-implement a solver for each new PDE
• Lightning fast mesh-free simulation via Radial Basis Functions
• Robust differentiable simulation via JAX, and portable across CPU, GPU, and TPU
• Support for Dirichlet, Neumann, Robin, and Periodic boundary conditions
• Automatic generation of normals from 2D GMSH meshes

𝕌pdes is easily extendable, with additional features added frequently.

Getting started

The package is available on PyPI. You can install it with

pip install updes

The example below illustrates how to solve the Laplace equation with Dirichlet and Neumann boundary conditions:

import updes
import jax.numpy as jnp

# Create a mesh-free cloud of points on a unit square
facet_types={"South":"n", "West":"d", "North":"d", "East":"d"}
cloud = updes.SquareCloud(Nx=30, Ny=20, facet_types=facet_types)

# Define the differential operator (left-hand side of the PDE)
def my_diff_operator(x, center, rbf, monomial, fields):
    return updes.nodal_laplacian(x, center, rbf, monomial)

# Define the right-hand side of the PDE
def my_rhs_operator(x, centers, rbf, fields):
    return 0.0

# Set a sin function as the Dirichlet BC on the North, and zero everywhere else
sine = lambda coord: jnp.sin(jnp.pi * coord[0])
zero = lambda coord: 0.0
boundary_conditions = {"South":zero, "West":zero, "North":sine, "East":zero}

# Solve the Laplace equation with a JIT-compiled solver
sol = updes.pde_solver_jit(diff_operator=my_diff_operator, 
                    rhs_operator=my_rhs_operator, 
                    cloud=cloud, 
                    boundary_conditions=boundary_conditions, 
                    rbf=updes.polyharmonic,
                    max_degree=1)

# Visualize the solution
cloud.visualize_field(sol.vals, cmap="jet", projection="3d", title="RBF solution")

𝕌pdes can handle much complicated cases with little modifications to the code above. Check out the documentation and Python interactive notebooks in the demos folder !

To-Dos

1. Logo, contributors guide, and developer documentation
2. Improved ill-conditioned linear systems for RBF-FD (i.e. support_size != "max")
3. More introductory examples in the documentation :
   • Integration with neural networks and Equinox
   • Non-linear and multi-dimensional PDEs
   • Adjoint schemes for fluid flows
4. Better point generation with accurate geometry and normals:
   • USD format
   • GMSH tutorial
5. Support for 3D radial basis functions

We welcome contributions from the community. Please feel free to open an issue or a pull request.

Dependencies

• Core: JAX - GMSH - Lineax - Matplotlib - Seaborn - Scikit-Learn
• Optional: PyVista - FFMPEG - QuartoDoc

See the pyproject.toml file the specific versions of the dependencies.

Cite us !

If you use this software, please cite us with the following BibTeX entry:

@inproceedings{nzoyem2023comparison,
  title={A comparison of mesh-free differentiable programming and data-driven strategies for optimal control under PDE constraints},
  author={Nzoyem Ngueguin, Roussel Desmond and Barton, David AW and Deakin, Tom},
  booktitle={Proceedings of the SC'23 Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis},
  pages={21--28},
  year={2023}}