Daily Papers

We introduce Adjoint Sampling, a highly scalable and efficient algorithm for learning diffusion processes that sample from unnormalized densities, or energy functions. It is the first on-policy approach that allows significantly more gradient updates than the number of energy evaluations and model samples, allowing us to scale to much larger problem settings than previously explored by similar methods. Our framework is theoretically grounded in stochastic optimal control and shares the same theoretical guarantees as Adjoint Matching, being able to train without the need for corrective measures that push samples towards the target distribution. We show how to incorporate key symmetries, as well as periodic boundary conditions, for modeling molecules in both cartesian and torsional coordinates. We demonstrate the effectiveness of our approach through extensive experiments on classical energy functions, and further scale up to neural network-based energy models where we perform amortized conformer generation across many molecular systems. To encourage further research in developing highly scalable sampling methods, we plan to open source these challenging benchmarks, where successful methods can directly impact progress in computational chemistry.

We revisit the geometric theory of defects. In the differential-geometric models of defects that have been adopted since the 1950s, dislocations have been associated with torsion, disclinations with the full curvature, and point defects with the first kind trace of non-metricity. The mainstream formulation exhibits several conceptual and technical shortcomings, most notably a hierarchy inconsistency, the non-exictence of a genuine metric formulation, and the potential emergence of Ostrogradsky-type instabilities. These issues have motivated us to develop a new framework, namely a generalized teleparallel geometric theory of defects. In our model, dislocations are identified with the trace of torsion, disclinations with the second kind trace of the non-metricity, and point defects with the first kind trace of the non-metricity. In addition, we retain the scalar part torsion as a free parameter for describing some possible unknown degrees of freedom in the theory of defects. The proposed geometric theory of defects is free from all of the aforementioned drawbacks and is therefore worthy of further investigation. To ensure the coherence and completeness of the discussion, we begin our analysis with elastic deformations, then summarize the existing metric-affine geometric theory of defects, and finally proceed to our original contribution, namely the new theory introduced here. We formulate the entire theory in Eulerian coordinates. Naturally, all results can be reformulated in Lagrangian coordinates as well. All analyses and formulae are expressed in the language of exterior algebra and are carried out in coordinate-independent orthonormal frames.

A complete theory of null Cartan normal helices in Minkowski space-time E^4_1 is developed. Two algebraic conditions, obtained by successive differentiation of the helix invariant along a unit C-constant normal field, fully characterize null Cartan helices; the quadratic condition yields two mutually orthogonal helix axes in the Lorentzian metric. Special field types are analyzed and null Cartan cubics are shown to be normal helices. On a timelike hypersurface, a Darboux frame with six curvature functions is constructed from first principles, the normal isophotic condition is shown to reduce to a linear first-order ODE, and the existence of normal silhouettes in E^4_1 is established.

In this paper we tackle the problem of generating conformers of a molecule in 3D space given its molecular graph. We parameterize these conformers as continuous functions that map elements from the molecular graph to points in 3D space. We then formulate the problem of learning to generate conformers as learning a distribution over these functions using a diffusion generative model, called Molecular Conformer Fields (MCF). Our approach is simple and scalable, and achieves state-of-the-art performance on challenging molecular conformer generation benchmarks while making no assumptions about the explicit structure of molecules (e.g. modeling torsional angles). MCF represents an advance in extending diffusion models to handle complex scientific problems in a conceptually simple, scalable and effective manner.

We present THEMol (Torsion, Hessian, Energy of Molecules), a massive open-source collection of quantum mechanical properties tailored for closed-shell organic molecules, with up to 50 heavy atoms. THEMol includes a Hessian subset with more than 3 million relaxed geometries with Hessian matrices, a TorsionScan subset with nearly 100 million constrained relaxed geometries with energies and forces, and relaxation-trajectory subsets (HessianRelax and TorsionScanRelax) that together comprise about 3 billion DFT calculations. The chemical space sampling is comprehensive, spanning twelve essential elements and diverse molecular architectures relevant to drug discovery, electrolytes, ionic liquids, and beyond. The dataset also features exhaustive conformational sampling through the TorsionScan and TorsionScanRelax subsets, including comprehensive in-ring and non-ring torsional scans. Furthermore, it contains an extensive library of Hessian matrices, computed at relaxed geometries, to capture critical second-derivative information of the potential energy landscape. Additionally, we supply electron density-derived atomic multipoles computed via the Minimal Basis Iterative Stockholder partition scheme. Organized into five distinct subsets (Hessian, TorsionScan, HessianRelax, TorsionScanRelax, and MBIS), the data encompasses optimized geometries, relaxation trajectories, and derived molecular properties. We anticipate that this massive and diverse dataset will significantly empower the development of highly accurate and transferable molecular potentials.

We present an algorithm to compute the torsion component Pic^τX of the Picard scheme of a smooth projective variety X over a field k. Specifically, we describe Pic^τX as a closed subscheme of a projective space defined by explicit homogeneous polynomials. Furthermore, we compute the group scheme structure on Pic^τX. As applications, we provide algorithms to compute various homological invariants. Among these, we compute the abelianization of the geometric étale fundamental group π^{{et}}_1(X_{k}, x)^{ab}. Moreover, we determine the Galois module structure of the first étale cohomology groups H^1_{{et}}(X_{k}, Z/nZ) without requiring n to be prime to the characteristic of k.