SO(3)-equivariant networks are the dominant models for machine learning interatomic potentials (MLIPs). The key operation of such networks is the Clebsch-Gordan (CG) tensor product, which is computationally expensive. To accelerate the computation, we develop tensor decomposition networks (TDNs) as a class of approximately equivariant networks whose CG tensor products are replaced by low-rank tensor decompositions, such as the CANDECOMP/PARAFAC (CP) decomposition. With the CP decomposition, we prove (i) a uniform bound on the induced error of SO(3)-equivariance, and (ii) the universality of approximating any equivariant bilinear map. To further reduce the number of parameters, we propose path-weight sharing that ties all multiplicity-space weights across the O(L^3) CG paths into a single shared parameter set without compromising equivariance, where L is the maximum angular degree. The resulting layer acts as a plug-and-play replacement for tensor products in existing networks, and the computational complexity of tensor products is reduced from O(L^6) to O(L^4). We evaluate TDNs on PubChemQCR, a newly curated molecular relaxation dataset containing 105 million DFT-calculated snapshots. We also use existing datasets, including OC20, and OC22. Results show that TDNs achieve competitive performance with dramatic speedup in computations. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

We present an evaluation of 32-bit POSIT arithmetic through its implementation as accelerators on FPGAs and GPUs. POSIT, a floating-point number format, adaptively changes the size of its fractional part. We developed hardware designs for FPGAs and software for GPUs to accelerate linear algebra operations using Posit(32,2) arithmetic. Our FPGA- and GPU-based accelerators in Posit(32,2) arithmetic significantly accelerated the Cholesky and LU decomposition algorithms for dense matrices. In terms of numerical accuracy, Posit(32,2) arithmetic is approximately 0.5 - 1.0 digits more accurate than the standard 32-bit format, especially when the norm of the elements of the input matrix is close to 1. Evaluating power consumption, we observed that the power efficiency of the accelerators ranged between 0.043 - 0.076 Gflops/watts for the LU decomposition in Posit(32,2) arithmetic. The power efficiency of the latest GPUs as accelerators of Posit(32,2) arithmetic is better than that of the evaluated FPGA chip.

Ground-state 3D geometries of molecules are essential for many molecular analysis tasks. Modern quantum mechanical methods can compute accurate 3D geometries but are computationally prohibitive. Currently, an efficient alternative to computing ground-state 3D molecular geometries from 2D graphs is lacking. Here, we propose a novel deep learning framework to predict 3D geometries from molecular graphs. To this end, we develop an equilibrium message passing neural network (EMPNN) to better capture ground-state geometries from molecular graphs. To provide a testbed for 3D molecular geometry analysis, we develop a benchmark that includes a dataset with precise ground-state geometries of approximately 4 million molecules. Experimental results show that EMPNN can efficiently predict more accurate ground-state 3D geometries than RDKit and other deep learning methods. Results also show that the proposed framework outperforms self-supervised learning methods on property prediction tasks.

General Matrix Multiplication (GEMM),$C=\alpha AB+\beta C$with input matrices$A, B$and scalar parameters$\alpha,\beta$, is widely used in scientific and engineering computations. Its precision is critical in determining the accuracy of the target applications. The requirements for the number of bits to represent floating-point (FP) numbers depend on individual applications. As the IEEE 754 standard [1] defines, FP formats and arithmetic are available in various precision, such as binary32 and binary64. Operations with higher precision like binary128 are desired for specific applications represented by Semidefinite Programming (SDP), a natural extension of linear programming that aims to minimize linear functions subject to certain constraints. Since binary128 is hardly supported as hardware, the performance of applications relying on its arithmetic is typically 100 to 1000x slower than that only relying on binary64. Therefore, acceleration of binary128 arithmetic is crucial to solve SDP and applications demanding binary128 fast.

Semidefinite programming (SDP) is one of the most important problems among optimization problems at present. It is relevant to a wide range of fields such as combinatorial optimization, structural optimization, control theory, economics, quantum chemistry, sensor network location and data mining. The capability to solve extremely large-scale SDP problems will have a significant effect on the current and future applications of SDP. In 1995, Fujisawa et al. started the SDPA(Semidefinite programming algorithm) Project aimed at solving large-scale SDP problems with high numerical stability and accuracy. SDPA is one of the main codes to solve general SDPs. SDPARA is a parallel version of SDPA on multiple processors with distributed memory, and it replaces two major bottleneck parts (the generation of the Schur complement matrix and its Cholesky factorization) of SDPA by their parallel implementation. In particular, it has been successfully applied to combinatorial optimization and truss topology optimization. The new version of SDPARA (7.5.0-G) on a large-scale supercomputer called TSUBAME 2.0 at the Tokyo Institute of Technology has successfully been used to solve the largest SDP problem (which has over 1.48 million constraints), and created a new world record. Our implementation has also achieved 533 TFlops in double precision for large-scale Cholesky factorization using 2,720 CPUs and 4,080 GPUs.

A SemiDefinite Programming (SDP) problem is one of the most central problems in mathematical optimization. SDP provides an effective computation framework for many research fields. Some applications, however, require solving a large-scale SDP whose size exceeds the capacity of a single processor both in terms of computation time and available memory. SDPARA (SemiDefinite Programming Algorithm paRAllel package) [Yamashita et al. 2003b] was designed to solve such large-scale SDPs. Its parallel performance is outstanding for general SDPs in most cases. However, the parallel implementation is less successful for some sparse SDPs obtained from applications such as Polynomial Optimization Problems (POPs) or Sensor Network Localization (SNL) problems, since this version of SDPARA cannot directly handle sparse Schur Complement Matrices (SCMs). In this article we improve SDPARA by focusing on the sparsity of the SCM and we propose a new parallel implementation using the formula-cost-based distribution along with a replacement of the dense Cholesky factorization. We verify numerically that these features are key to solving SDPs with sparse SCMs more quickly on parallel computing systems. The performance is further enhanced by multithreading and the new SDPARA attains considerable scalability in general. It also finds solutions for extremely large-scale SDPs arising from POPs which cannot be obtained by other solvers.