🌐 e3j

Euclid-equivariant operations and harmonic polynomials for JAX.

This library is a fast and full-featured Euclidean equivariance backend which can be used in place of e3nn and e3x to replace slow operations in Machine Learned Interatomic Potentials (MLIPs) with carefully optimized and open-source CUDA and Pallas kernels for GPU and TPU.

The equivariance backend of our MLIP library is e3j as of mlip 0.2.0.

Note: e3j is currently in pre-release, with version 0.1.0 planned for early June 2026. Additional CUDA kernels and dedicated Pallas kernels for TPU will be rolled out progressively.

Installation

Pulling from PyPI

The e3j package is available on PyPI. It consists of a thin JAX-based Python API which can run on CPU, GPU and TPU, supporting Python versions from 3.11 to 3.14 included.

For efficiency on GPU, our CUDA binaries are bundled as the e3j_ops package on PyPI. The compatible version of the binaries should be pulled by requiring the "e3j[ops]" extra:

# requirements.txt
e3j[ops] >= 0.1.0b0
jax[cuda13_local] ~= 0.8.0

See JAX installation instructions for more information on JAX versions and their CUDA support. We recommend using a version of JAX above 0.7.0 and CUDA 13.

Building from source

Our dependencies are managed with uv. After cloning the repository, you can build from source by running run one of:

# Existing CUDA 13 install with `e3j_ops` kernels:
uv sync --group cuda13_local --extra ops
# Install CUDA 13 via pip and the `exp` group for benchmarks:
uv sync --group cuda13 --extra ops

The Python build internally relies on CMake, scikit-build and pybind11. You can also look at the Makefile for alternate recipes to build kernels, C++ tests and the Python bindings.

The e3j_ops Python package only contains our CUDA binaries and bindings to their associated XLA handlers. It is not meant to be used as standalone until its ABI is reported stable.

Project structure

The JAX primitives wrapping our custom XLA handlers are defined in the e3j.ops subpackage of e3j, provided the e3j_ops binaries can be found in the environment.

• src/e3j : Python source
  • core : Non-parameterized equivariant operations
  • linen : Parameterized equivariant operations as flax.linen.Module instances
  • ops : JAX primitives and AD rules around the XLA-FFI bindings of e3j_ops
• tests : Python test suite, based on pytest
  • test_ops : XLA-FFI bindings tests marked with "e3j_ops"
• Makefile : build recipes for CUDA/C++ objects and tests
• lib/e3j_ops : CUDA/C++ source for the e3j_ops subpackage
  • cuda : custom kernel implementations
  • ffi : XLA and Python binding boilerplate
  • tests : pure C++ tests for CUDA kernels
  • CMakeLists.txt : holistic build recipe for pyproject.toml integration with scikit-build

Contributing

Although it is too early for e3j to accept significant external contributions, bug reports or questions are very welcome via GitHub issues and discussions.

Citing

If you use e3j within your work, we kindly ask you to cite the following preprint:

@article{Peltre26-e3j,
    title   = {{E3J}: an Efficient and Open-Source Euclidean Equivariance Backend},
    author  = {Peltre, Olivier and Picard, Armand and Pichard, Adrien and Giacomoni, Luca and Braganca, Miguel and Heyraud, Valentin and Brunken, Christoph and Tilly, Jules},
    journal = {preprint},
    year    = {2026},
    url     = {(preprint)}
  }
}