Clifford-Steerable Convolutional Neural Networks (2402.14730v3)

Abstract: We present Clifford-Steerable Convolutional Neural Networks (CS-CNNs), a novel class of $\mathrm{E}(p, q)$-equivariant CNNs. CS-CNNs process multivector fields on pseudo-Euclidean spaces $\mathbb{R}^{p,q}$. They cover, for instance, $\mathrm{E}(3)$-equivariance on $\mathbb{R}^3$ and Poincar\'e-equivariance on Minkowski spacetime $\mathbb{R}^{1,3}$. Our approach is based on an implicit parametrization of $\mathrm{O}(p,q)$-steerable kernels via Clifford group equivariant neural networks. We significantly and consistently outperform baseline methods on fluid dynamics as well as relativistic electrodynamics forecasting tasks.

• The paper introduces CS-CNNs that extend steerable CNN frameworks to pseudo-Euclidean spaces, ensuring equivariance to complex symmetry groups.
• The paper utilizes Clifford algebra and implicit parameterization via equivariant MLPs to overcome the limitations of conventional analytical methods.
• The paper demonstrates robust empirical gains over baselines in tasks such as fluid dynamics and electrodynamics, highlighting significant sample efficiency.

Clifford-Steerable Convolutional Neural Networks

The paper presents a novel class of Convolutional Neural Networks (CNNs) termed Clifford-Steerable Convolutional Neural Networks (CS-CNNs). These networks are designed to process multivector fields on pseudo-Euclidean spaces denoted by $R^{p,q}$. They offer a notable advancement by being equivariant to the symmetries in these spaces, such as $(n)$-equivariance on $R^n$ and Poincaré-equivariance on Minkowski spacetime $R^{1,3}$—addressing the long-standing challenge of designing CNNs that respect these symmetries.

Equivariance and Symmetry

The central premise of the paper hinges on the concept of equivariance, which is critical for ensuring that a neural network's predictions respect the inherent symmetries present in the data. In physical systems, this often translates to the requirement that the underlying differential equations or the predictor respect transformation properties like rotations (isometries) in Euclidean space $R^3$ or Lorentz transformations in Minkowski spacetime $R^{1,3}$.

CS-CNNs extend the framework of steerable CNNs, previously limited to Euclidean spaces and compact groups, to pseudo-Euclidean spaces with mixed signatures. This generalization is non-trivial due to the non-compact nature of groups like $(p,q)$ which necessitate novel approaches both in theory and implementation.

Clifford Algebra and Kernel Parametrization

The core of these networks is their reliance on Clifford algebra to achieve equivariance. The Clifford algebra offers a rich mathematical structure that can naturally incorporate the geometric properties of the base space. For an input vector space $V = R^{p,q}$, the Clifford algebra $(V, \eta)$ encompasses vectors, scalars, and higher-grade multivectors, providing a versatile feature space for deep learning models.

The steerable kernels in CS-CNNs are implicitly parameterized via neural networks that are equivariant to the Clifford group. This implicit parameterization approach overcomes the limitations of analytical solutions which would be cumbersome and highly specialized for each group structure. Instead, the implicit method leverages equivariant multi-layer perceptrons (MLPs) to satisfy the equivariance constraints inherently.

Achievements and Results

CS-CNNs are evaluated on tasks such as fluid dynamics forecasting on $R^2$, and electrodynamics simulations on $R^3$ and $R^{1,2}$. They consistently outperform baseline methods, including conventional steerable CNNs and non-equivariant Clifford CNNs. This performance is robust across various dataset sizes, demonstrating significant sample efficiency. In particular, CS-CNNs retain full symmetry properties while providing substantial empirical gains.

Practical and Theoretical Implications

The implications of CS-CNNs extend both practically and theoretically. Practically, they offer improved prediction capabilities and sample efficiency for complex physical systems like fluid dynamics and electrodynamics, which are described by partial differential equations invariant under certain symmetries. Theoretically, CS-CNNs broaden the scope of equivariant deep learning models to incorporate pseudo-Euclidean symmetries, a leap from the previously explored Euclidean and compact Lie group settings.

Future Directions

Future developments could focus on the generalizations to other types of geometric features and more intricate symmetries, potentially even beyond pseudo-Euclidean spaces. Additionally, the models could be further optimized for specific applications in physics and engineering, such as better integration with numerical solvers for PDEs. Another promising direction is the exploration of CS-CNNs in more generalized settings, such as Riemannian and pseudo-Riemannian manifolds, enhancing their capability to handle a wider range of physical systems.

Conclusion

CS-CNNs represent a significant advancement in the domain of geometric deep learning by integrating Clifford algebra with the framework of steerable CNNs. Their ability to handle the complex symmetries of pseudo-Euclidean spaces while ensuring computational efficiency and robust performance underlines their potential both as a research tool and in practical applications. This work opens new avenues for the exploration and application of equivariant neural networks to model intricate physical systems accurately and efficiently.