[Paper Review] Global Optimization of Atomic Clusters via Physically-Constrained Tensor Train Decomposition

Konstantin Sozykin, Nikita Rybin+6 more|arXiv (Cornell University)|Jan 26, 2026

Machine Learning in Materials Science0 citations

TL;DR

The paper proposes a TT-based framework combining algebraic (TTOpt) and probabilistic (PROTES) strategies to globally optimize atomic clusters by exploiting low-rank tensor representations of potential energy surfaces, with physically-constrained encodings.

ABSTRACT

The global optimization of atomic clusters represents a fundamental challenge in computational chemistry and materials science due to the exponential growth of local minima with system size (i.e., the curse of dimensionality). We introduce a novel framework that overcomes this limitation by exploiting the low-rank structure of potential energy surfaces through Tensor Train (TT) decomposition. Our approach combines two complementary TT-based strategies: the algebraic TTOpt method, which utilizes maximum volume sampling, and the probabilistic PROTES method, which employs generative sampling. A key innovation is the development of physically-constrained encoding schemes that incorporate molecular constraints directly into the discretization process. We demonstrate the efficacy of our method by identifying global minima of Lennard-Jones clusters containing up to 45 atoms. Furthermore, we establish its practical applicability to real-world systems by optimizing 20-atom carbon clusters using a machine-learned Moment Tensor Potential, achieving geometries consistent with quantum-accurate simulations. This work establishes TT-decomposition as a powerful tool for molecular structure prediction and provides a general framework adaptable to a wide range of high-dimensional optimization problems in computational material science.

Motivation & Objective

• Motivate and tackle the global optimization of atomic clusters whose energy landscapes have exponentially many local minima.
• Introduce a TT-based framework to represent the energy surface as a low-rank tensor to mitigate the curse of dimensionality.
• Develop physically-constrained encoding schemes to incorporate molecular constraints into discretization.
• Combine algebraic and probabilistic TT-based optimization methods to identify near-global minima.
• Demonstrate effectiveness on Lennard-Jones clusters up to 45 atoms and carbon clusters with a machine-learning potential.

Proposed method

• Represent the cluster energy E(x) on a discretized grid as a d-dimensional tensor E in TT-format to enable low-rank storage and operations.
• Use two TT-based optimization strategies: TTOpt (maxvol-based algebraic optimization) and PROTES (probabilistic sampling with TT-based density estimation).
• TTOpt processes tensor unfoldings and iteratively selects submatrices of maximum volume to locate near-global minima, with potentially adaptive ranks.
• PROTES builds a TT-decomposed probability distribution p_theta(n) proportional to the squared TT-tensor to sample candidate configurations, then updates TT cores via SGD using top-k energy candidates.
• Encode particle coordinates via multiple discretization schemes (Direct, Relative, and Bit encodings) to incorporate translational/rotational invariances and physical spacing constraints.
• Physically-constrained initializations with PROTES and various encoding schemes are employed to ensure realistic, non-overlapping configurations.

Experimental results

Research questions

• RQ1 Can a tensor-train representation capture the low-rank structure of high-dimensional potential energy surfaces for atomic clusters?
• RQ2 Do TTOpt and PROTES provide complementary strengths for global optimization on rugged energy landscapes?
• RQ3 How do physically-constrained encoding schemes influence search efficiency and solution quality in TT-based optimization?
• RQ4 Is the framework effective for Lennard-Jones clusters up to 45 atoms and carbon clusters modeled with a machine-learned interatomic potential?
• RQ5 What is the accuracy of TT-based minima relative to quantum-accurate simulations or high-fidelity potentials?

Key findings

• Global minima for Lennard-Jones clusters with up to 45 atoms were identified using the proposed TT-based framework.
• The method achieved geometries for 20-atom carbon clusters that are consistent with quantum-accurate simulations when using a machine-learned Moment Tensor Potential.
• The combination of algebraic TTOpt and probabilistic PROTES leverages low-rank structures and information geometry to tackle high-dimensional optimization.
• Physically-constrained encodings effectively reduce search space and enforce reasonable interatomic distances and angular configurations.
• The TT-decomposition enables efficient storage and computation for high-dimensional optimization problems in computational material science.