Incorporating equivariance as an inductive bias into deep learning architectures to take advantage of the data symmetry has been successful in multiple applications, such as chemistry and dynamical systems. In particular, roto-translations are crucial for effectively modeling geometric graphs and molecules, where understanding the 3D structures enhances generalization. However, equivariant models often pose challenges due to their high computational complexity. In this paper, we introduce REMUL, a training procedure for approximating equivariance with multitask learning. We show that unconstrained models (which do not build equivariance into the architecture) can learn approximate symmetries by minimizing an additional simple equivariance loss. By formulating equivariance as a new learning objective, we can control the level of approximate equivariance in the model. Our method achieves competitive performance compared to equivariant baselines while being 10 faster at inference and 2.5 at training. Equivariant machine learning models have achieved notable success across various domains, such as computer vision (Weiler et al., 2018; Yu et al., 2022), dynamical systems (Han et al., 2022; Xu et al., 2024), chemistry (Satorras et al., 2021; Brandstetter et al., 2022), and structural biology (Jumper et al., 2021). Equivariant machine learning models benefit from this inductive bias by explicitly leveraging symmetries of the data during the architecture design. Typically, such architectures have highly constrained layers with restrictions on the form and action of weight matrices and nonlinear activations (Batzner et al., 2022; Batatia et al., 2022). This may come at the expense of higher computational cost, making it sometimes challenging to scale equivariant architectures, particularly those relying on spherical harmonics and irreducible representations (Thomas et al., 2018; Fuchs et al., 2020; Liao & Smidt, 2023; Luo et al., 2024).

This complicates brute force approaches, making them virtually unfeasible for even moderately small molecules. In this paper we tackle the problem of generating conformers of a molecule in 3D space given Systematic methods, like OMEGA (Hawkins et al., 2010), its molecular graph. We parameterize these conformers offer rapid processing through rule-based generators and as continuous functions that map elements curated torsion templates. Despite their efficiency, these from the molecular graph to points in 3D models typically fail on complex molecules, as they often space. We then formulate the problem of learning overlook global interactions and are tricky to extend to to generate conformers as learning a distribution inputs like transition states or open-shell molecules. Classic over these functions using a diffusion generative stochastic methods, like molecular dynamics (MD) and model, called Molecular Conformer Fields Markov chain Monte Carlo (MCMC), rely on extensively exploring (MCF). Our approach is simple and scalable, and the energy landscape to find low-energy conformers.

We present Manifold Diffusion Fields (MDF), an approach to learn generative models of continuous functions defined over Riemannian manifolds. Leveraging insights from spectral geometry analysis, we define an intrinsic coordinate system on the manifold via the eigen-functions of the Laplace-Beltrami Operator. MDF represents functions using an explicit parametrization formed by a set of multiple input-output pairs. Our approach allows to sample continuous functions on manifolds and is invariant with respect to rigid and isometric transformations of the manifold. Empirical results on several datasets and manifolds show that MDF can capture distributions of such functions with better diversity and fidelity than previous approaches.