OpenEquivariance

[PyTorch Examples] [JAX Examples] [Citation and Acknowledgements]

OpenEquivariance is a CUDA and HIP kernel generator for the Clebsch-Gordon tensor product, a key kernel in rotation-equivariant deep neural networks. It implements some of the tensor products that e3nn supports commonly found in graph neural networks (e.g. Nequip or MACE). To get started with PyTorch, ensure that you have PyTorch and GCC 9+ available before installing our package via

pip install openequivariance

We provide up to an order of magnitude acceleration over e3nn perform on par with the latest version of NVIDIA cuEquivariance, which has a closed-source kernel package. We also offer fused equivariant graph convolutions that can reduce computation and memory consumption significantly.

For detailed instructions on tests, benchmarks, MACE / Nequip, and our API, check out the documentation.

⭐️ JAX: Our latest update brings support for JAX. For NVIDIA GPUs, install it (after installing JAX) with the following two commands strictly in order:

pip install openequivariance[jax]
pip install openequivariance_extjax --no-build-isolation

For AMD GPUs:

pip install openequivariance[jax]
JAX_HIP=1 pip install openequivariance_extjax --no-build-isolation

See the section below for example usage and our API page for more details.

PyTorch Examples

Here's a CG tensor product implemented by e3nn:

import torch
import e3nn.o3 as o3

gen = torch.Generator(device='cuda')

batch_size = 1000
X_ir, Y_ir, Z_ir = o3.Irreps("1x2e"), o3.Irreps("1x3e"), o3.Irreps("1x2e") 
X = torch.rand(batch_size, X_ir.dim, device='cuda', generator=gen)
Y = torch.rand(batch_size, Y_ir.dim, device='cuda', generator=gen)

instructions=[(0, 0, 0, "uvu", True)]

tp_e3nn = o3.TensorProduct(X_ir, Y_ir, Z_ir, instructions,
        shared_weights=False, internal_weights=False).to('cuda')
W = torch.rand(batch_size, tp_e3nn.weight_numel, device='cuda', generator=gen)

Z = tp_e3nn(X, Y, W)
print(torch.norm(Z))

And here's the same tensor product using openequivariance. We require that your tensors are stored on a CUDA device for this to work:

import openequivariance as oeq

problem = oeq.TPProblem(X_ir, Y_ir, Z_ir, instructions, shared_weights=False, internal_weights=False)
tp_fast = oeq.TensorProduct(problem, torch_op=True)

Z = tp_fast(X, Y, W) # Reuse X, Y, W from earlier
print(torch.norm(Z))

Our interface for oeq.TPProblem is almost a strict superset of o3.TensorProduct (two key differences: we impose internal_weights=False and add support for multiple datatypes). You can pass e3nn Irreps instances directly or use oeq.Irreps, which is identical.

We recommend reading the e3nn documentation and API reference first, then using our kernels as drop-in replacements. We support most "uvu" and "uvw" tensor products; see this section for an up-to-date list of supported configurations.

Important: For many configurations, our kernels return results identical to e3nn up to floating point roundoff (this includes all "uvu" problems with multiplicity 1 for all irreps in the second input). For other configurations (e.g. any "uvw" connection modes), we return identical results up to a well-defined reordering of the weights relative to e3nn.

If you're executing tensor products as part of a message passing graph neural network, we offer fused kernels that save both memory and compute time:

from torch_geometric import EdgeIndex

node_ct, nonzero_ct = 3, 4

# Receiver, sender indices for message passing GNN
edge_index = EdgeIndex(
                [[0, 1, 1, 2],  # Receiver 
                 [1, 0, 2, 1]], # Sender 
                device='cuda',
                dtype=torch.long)

X = torch.rand(node_ct, X_ir.dim, device='cuda', generator=gen)
Y = torch.rand(nonzero_ct, Y_ir.dim, device='cuda', generator=gen)
W = torch.rand(nonzero_ct, problem.weight_numel, device='cuda', generator=gen)

tp_conv = oeq.TensorProductConv(problem, torch_op=True, deterministic=False) # Reuse problem from earlier
Z = tp_conv.forward(X, Y, W, edge_index[0], edge_index[1]) # Z has shape [node_ct, z_ir.dim]
print(torch.norm(Z))

If you can guarantee EdgeIndex is sorted by receiver index and supply the transpose permutation, we can provide even greater speedup (and deterministic results) by avoiding atomics:

_, sender_perm = edge_index.sort_by("col")            # Sort by sender index 
edge_index, receiver_perm = edge_index.sort_by("row") # Sort by receiver index

# Now we can use the faster deterministic algorithm
tp_conv = oeq.TensorProductConv(problem, torch_op=True, deterministic=True) 
Z = tp_conv.forward(X, Y[receiver_perm], W[receiver_perm], edge_index[0], edge_index[1], sender_perm) 
print(torch.norm(Z))

Note: you don't need Pytorch geometric to use our kernels. When deterministic=False, the sender and receiver indices can have arbitrary order.

JAX Examples

After installation, use the library as follows. Set OEQ_NOTORCH=1 in your environment to avoid the PyTorch import in the regular openequivariance package.

import jax
import os

os.environ["OEQ_NOTORCH"] = "1"
import openequivariance as oeq

seed = 42
key = jax.random.PRNGKey(seed)

batch_size = 1000
X_ir, Y_ir, Z_ir = oeq.Irreps("1x2e"), oeq.Irreps("1x3e"), oeq.Irreps("1x2e")
problem = oeq.TPProblem(X_ir, Y_ir, Z_ir, [(0, 0, 0, "uvu", True)], shared_weights=False, internal_weights=False)


node_ct, nonzero_ct = 3, 4
edge_index = jax.numpy.array(
    [
        [0, 1, 1, 2],
        [1, 0, 2, 1],
    ],
    dtype=jax.numpy.int32, # NOTE: This int32, not int64
)

X = jax.random.uniform(key, shape=(node_ct, X_ir.dim), minval=0.0, maxval=1.0, dtype=jax.numpy.float32)
Y = jax.random.uniform(key, shape=(nonzero_ct, Y_ir.dim),
                        minval=0.0, maxval=1.0, dtype=jax.numpy.float32)
W = jax.random.uniform(key, shape=(nonzero_ct, problem.weight_numel),
                        minval=0.0, maxval=1.0, dtype=jax.numpy.float32)

tp_conv = oeq.jax.TensorProductConv(problem, deterministic=False)
Z = tp_conv.forward(
    X, Y, W, edge_index[0], edge_index[1]
)
print(jax.numpy.linalg.norm(Z))

Citation and Acknowledgements

If you find this code useful, please cite our paper:

@inbook{openequivariance,
author={Vivek Bharadwaj and Austin Glover and Aydin Buluc and James Demmel},
title={An Efficient Sparse Kernel Generator for O(3)-Equivariant Deep Networks}, 
booktitle = {SIAM Conference on Applied and Computational Discrete Algorithms (ACDA25)},
chapter = {},
url={https://arxiv.org/abs/2501.13986},
publisher={Society for Industrial and Applied Mathematics},
year={2025}
}

Our codebase includes a lightweight clone of e3nn's frontend interface (in particular, the TensorProduct and Irreps classes). We removed references to Pytorch and separated the implementation from the problem description (for future frontend support outside of torch). We also extracted the Wigner 3j tensor generating code from QuTiP. Thank you to the current developers and maintainers!

Copyright

Copyright (c) 2025, The Regents of the University of California, through Lawrence Berkeley National Laboratory (subject to receipt of any required approvals from the U.S. Dept. of Energy). All rights reserved.

If you have questions about your rights to use or distribute this software, please contact Berkeley Lab's Intellectual Property Office at IPO@lbl.gov.

NOTICE. This Software was developed under funding from the U.S. Department of Energy and the U.S. Government consequently retains certain rights. As such, the U.S. Government has been granted for itself and others acting on its behalf a paid-up, nonexclusive, irrevocable, worldwide license in the Software to reproduce, distribute copies to the public, prepare derivative works, and perform publicly and display publicly, and to permit others to do so.