Tensor Train Decomposition Achieves Global Optimisation, Overcoming Dimensionality Challenges for Clusters

Researchers are tackling the longstanding problem of optimising the structure of atomic clusters, a challenge complicated by the exponential increase in potential energy minima as cluster size grows. Konstantin Sozykin, Nikita Rybin, and Andrei Chertkov, all from the Skolkovo Institute of Science and Technology, alongside Anh-Huy Phan et al., present a new framework utilising Tensor Train decomposition to overcome this ‘curse of dimensionality’. Their innovative approach combines algebraic and probabilistic sampling strategies with physically-constrained encoding, allowing them to identify global minima in Lennard-Jones clusters of up to 45 atoms and accurately optimise 20-atom carbon clusters using machine-learned potentials. This work establishes Tensor Train decomposition as a powerful technique for molecular structure prediction and offers a versatile solution for high-dimensional optimisation across computational material science.

The research team combined two distinct TT-based strategies, TTOpt and PROTES, employing both maximum volume sampling and generative sampling techniques to efficiently navigate the complex potential energy landscape of these clusters. Researchers achieved this by representing high-dimensional potential energy surfaces with low-rank TT decompositions, significantly reducing computational demands. This allowed for a more thorough exploration of the potential energy landscape, leading to the discovery of stable cluster configurations.

The combination of algebraic and probabilistic TT-based methods, coupled with physically-constrained encoding, provides a robust and efficient approach to tackling complex energy minimization challenges. Experiments using Lennard-Jones clusters, ranging from 5 to 45 atoms, confirmed the method’s ability to accurately identify global minima. The successful optimization of 20-atom carbon clusters, validated against quantum-accurate simulations, highlights the potential of this framework for real-world materials design and discovery. The researchers utilized the generalized maximum volume principle and single-shot relaxation via the L-BFGS-B algorithm to refine candidate structures identified through TT decomposition.
The framework’s versatility is further enhanced by its potential compatibility with other tensor decomposition schemes, including Canonical Polyadic Decomposition, Tucker decomposition, and Hierarchical Tucker decomposition. Moreover, the team suggests that tensor networks traditionally used in quantum physics, such as Projected Entangled Pair States and the Multiscale Entanglement Renormalization Ansatz, could also be integrated into the framework. The study pioneered the use of TT-decomposition for high-dimensional optimization, representing multidimensional arrays as interconnected, smaller tensors to reduce computational demands and memory requirements. Experiments employed generalized maximum volume principles and probabilistic sampling techniques, followed by single-shot relaxation using the L-BFGS-B algorithm to refine candidate minima. Researchers primarily adopted the TT-format due to its computational convenience, but the framework is designed to be adaptable to other tensor decomposition schemes, including Canonical Polyadic Decomposition and Hierarchical Tucker decomposition. The system delivers high-precision structural predictions, with optimized structures determined to within the fitting error of the MTP model. Analysis revealed a trade-off between computational efficiency and reliability, dependent on cluster size, encoding scheme, and initialisation strategy. Clusters ranging from 5 to 45 atoms were analysed, categorised as small (5, 15 atoms), medium (16, 30 atoms), and large (31, 45 atoms) to explore different optimisation regimes. The particularly challenging LJ38 cluster, known for its double-funnel energy landscape, was included as a benchmark. Performance was quantified using PROTES Calls (PC), representing search efficiency, and the success rate (SRt), indicating the percentage of runs locating the global minimum.

Results demonstrate that for clusters up to 26 atoms, agnostic initialisation paired with Simple Relative encoding achieved 100% success rates with reduced computational budgets, such as 2,147 evaluations for LJ26. This configuration rapidly identifies promising regions without physical bias, representing an “efficiency-optimised” mode. However, for larger clusters (Na ≥30), physically-constrained initialisation became crucial, transitioning the method into a “reliability-optimised” mode. Although requiring more evaluations, this approach maintained non-zero success rates where agnostic initialisation failed.

Notably, for LJ45, only physically-constrained initialisation with Constrained Relative encoding yielded a successful outcome, achieving 33% SR with 52,300 evaluations. The selection of encoding schemes further modulates the efficiency-reliability trade-off. Simple Relative encoding generally enhanced efficiency for smaller clusters, while Constrained Relative encoding provided superior reliability for challenging systems, achieving 33% success on LJ38 compared to 16% with the simple variant, and 33% on LJ45 versus 0%. The TTOpt algorithm with direct encoding consistently achieved 100% success rates but at higher computational costs, demonstrating the robustness of the underlying TT-approach. The method achieved modest success rates (16-33%) on the notoriously difficult LJ38 system, confirming its ability to navigate complex energy landscapes, and accurately reproduced the global minimum structure, as substantiated by its pairwise distance distribution.

Physically-Constrained Tensor Decomposition for Atomic Clusters enables efficient

The key contribution of this research is a methodological framework that establishes a new approach for high-dimensional optimization, incorporating physical constraints directly into the tensor decomposition process. This physically-constrained formulation reduces the search space dimensionality while maintaining mathematical rigor and aligning with chemical intuition. The authors acknowledge limitations in automatically tuning discretization parameters and initialization strategies. Future research will focus on developing adaptive schemes and hybrid approaches combining tensor methods with existing optimization frameworks like USPEX and Basin Hopping, potentially advancing automated materials discovery pipelines.