SpinPSO Framework in Magnetic Optimization

• SpinPSO is a computational optimization method that extends classical PSO to the continuous spin space (S²) by incorporating physical spin dynamics.
• It employs a discrete Landau–Lifshitz–Gilbert update rule blended with stochastic cognitive and social forces to efficiently converge on noncollinear magnetic ground states.
• The framework is integrated with atomate, pymatgen, and VASP for high-throughput DFT simulations on HPC clusters, reducing iterations and enhancing robustness.

The SpinPSO framework designates a class of computational and optimization methodologies that integrate physical spin dynamics into stochastic and agent-based numerical schemes. Most prominently, SpinPSO refers to a hybrid meta-heuristic optimization approach for identifying noncollinear magnetic ground states from first-principles, extending classical Particle Swarm Optimization (PSO) onto the space of continuous spin vectors $\mathcal{S}^2$ and incorporating atomistic spin dynamics equations. Distinct SpinPSO formulations also appear in the context of uncertainty quantification for beam and spin dynamics via the stochastic Galerkin method, as well as in integrators for multicomponent Schrödinger–Poisson systems with internal spin degrees of freedom. This article focuses on the agent-based optimization framework introduced by (Moore et al., 2023), but highlights conceptual links to related SpinPSO implementations where spin effects and stochasticity are central.

1. Mathematical Structure of SpinPSO Optimization

SpinPSO extends PSO to cases where optimization variables are $N_{\rm at}$-site configurations $\{\mathbf{s}_j\}$, with each spin $\mathbf{s}_j\in\mathbb{R}^3$ constrained to the unit sphere ($\mathcal{S}^2$). The target is to solve

$\min_{\{\mathbf{s}_j\}} E(\{\mathbf{s}_j\}),$

where $E$ is typically the total electronic energy from noncollinear Density Functional Theory (DFT) under fixed spin directions, evaluated using VASP within the atomate and pymatgen ecosystem. For benchmarking and theoretical analysis, one may use a classical Heisenberg Hamiltonian $$\mathcal{H}=-\sum_{j\neq k}J_{jk}\,\mathbf{s}_j\cdot\mathbf{s}_k$$ as the objective.

Each agent in the swarm maintains its own spin configuration. The PSO update, replacing Cartesian velocity, is governed by a discrete-time Landau–Lifshitz–Gilbert (LLG) equation: $$\mathbf{s}_{i,j}^{n+1} = \mathbf{s}_{i,j}^n - \gamma \Delta\tau\,\mathbf{s}_{i,j}^n \times \mathbf{h}^n_{i,j} - \alpha \Delta\tau\,\mathbf{s}_{i,j}^n \times ( \mathbf{s}_{i,j}^n \times \mathbf{h}^n_{i,j} ),$$ with explicit renormalization $$\mathbf{s}^{n+1}_{i,j} \leftarrow \mathbf{s}^{n+1}_{i,j} / \|\mathbf{s}^{n+1}_{i,j}\|$$. The effective field $\mathbf{h}_{i,j}^n$ is a stochastic linear combination of the agent’s personal best $\tilde{\mathbf{s}}_{i,j}^n$ and the global swarm best $\hat{\mathbf{s}}_{j}^n$: $$\mathbf h_{i,j}^n = \frac{a_c \sigma_c \tilde{\mathbf s}_{i,j}^n + a_s \sigma_s \hat{\mathbf s}_j^n}{\| a_c \sigma_c \tilde{\mathbf s}_{i,j}^n + a_s \sigma_s \hat{\mathbf s}_j^n \| }, \qquad \sigma_c, \sigma_s \sim \mathcal{U}(0,1).$$ For the unique global-best agent, the random perturbation is replaced by actual DFT-computed energy gradients, i.e., the local effective field $-\partial E/\partial\mathbf{s}_{i,j}^n$ is blended with the conventional PSO field for faster local convergence.

2. Algorithmic Workflow and HPC Realization

The SpinPSO workflow is fully integrated into atomate and pymatgen for high-throughput, parallel simulation. The main stages are:

• Input Preparation: User supplies structure files and (optionally) precomputed exchange parameters. DFT parameters for noncollinear calculations are specified via the environment.
• Swarm Initialization: Each agent’s spin configuration is randomly assigned on $\mathcal{S}^2$ per atom. Energies and personal bests are initialized in parallel across all agents using DFT.
• PSO Iterations: At each meta-time step,
  • All agents’ spins are updated via the LLG-based PSO rule.
  • The global-best agent receives a DFT-based effective field as a gradient step; all others use stochastic blends of cognition and social information.
  • New DFT calculations run in parallel to evaluate all agents’ energies.
  • Bests are updated and convergence criteria (energy improvement, stagnation, or iteration count) are assessed.
• Final Local Relaxation: The identified global best configuration may be polished by direct DFT relaxation using only the true LLG gradient.

Each PSO layer iteration requires $N_p$ parallel DFT calculations (one per agent). Within each VASP run, standard MPI/OpenMP parallelism is employed. The framework is suitable for allocation on tier-1 HPC nodes—per-batch wall-time is governed by DFT, with typical single-iteration durations of 1–3 hours per 32-core node.

3. Physical Principles and Motivation

SpinPSO’s extension of swarm algorithms to the differentiable manifold $\mathcal{S}^2$ leverages both stochastic global search and physically-motivated local dynamics. The adoption of the Landau–Lifshitz–Gilbert evolution ensures that search steps are compatible with the geometry of spin space, providing precession for exploration and damping for rapid (local) convergence. The stochastic combination of cognition and social terms preserves the global exploration strength of GCPSO-type algorithms while the DFT true-gradient replacement for the best agent’s update ensures superior final convergence.

Key parameters controlling evolution are:

• Damping $\alpha$ (typically $\alpha\gtrsim3\gamma$ for efficient alignment),
• Critically, the gradient feedback on the global best agent, which is not present in earlier GCPSO or Firefly-type metaheuristics.

4. Performance Benchmarks and Materials Applications

SpinPSO demonstrates robust convergence to experimentally determined magnetic ground states in diverse materials with complex, low-symmetry, noncollinear spin orderings. On the Heisenberg model, the gradient-informed step allows convergence to minima in fewer than 10 iterations (mean $\sim7$). For materials such as MnPtGa, YMnO$_3$, FeF$_3$, and various kagome systems, convergence in $\mathcal O(10)$–$\mathcal O(20)$ iterations is typical, with results matching experimental and previous theoretical characterizations within few-degree accuracy for spin angles.

Compared to previous agent-based optimization schemes such as Firefly, SpinPSO achieves substantially faster and more robust convergence, requiring approximately half the number of iterations and showing stronger consistency across random seeds. For cells with 100–200 atoms, a full run with 16 agents over 15 iterations completes in $\sim$45 wall-clock hours on as many nodes.

5. Software Implementation and Workflow Integration

SpinPSO is implemented entirely atop the pymatgen (magnetism module) and atomate workflow infrastructure, with VASP as the DFT backend. The framework automates parallel job submission, error handling, and data provenance via FireWorks (atomate). Integration allows for large-scale hyperparameter sweeps (agent count, step sizes, damping constants, blend weights) and batch runs across multiple materials. DFT energies and gradients are computed within constraints to prescribed spin directions, using VASP’s effective field implementation.

6. Related Approaches and Theoretical Extensions

While the term SpinPSO in (Moore et al., 2023) denotes an agent-based optimizer for noncollinear magnetic structure identification, similar terminology appears for stochastic Galerkin-based uncertainty quantification in beam and spin dynamics (Slim et al., 2017) and for integrators of multicomponent Schrödinger–Poisson systems with spin-spin couplings (Jain et al., 2022). In the optimization context, the key innovation is embedding isotropic spin geometry and Landau–Lifshitz–Gilbert physics within the particle swarm paradigm; in the stochastic Galerkin and Schrödinger–Poisson settings, the framework revolves around efficient and conservative evolution of coupled spinor or spin-dependent ODE/PDE systems, leveraging polynomial chaos expansions, Strang splitting, or analytic kick-step solvers.

A plausible implication is that the success of the agent-based SpinPSO approach for magnetic structure identification may motivate further generalizations to other optimization problems on curved manifolds—where constraint geometry and local dynamics are nontrivial—both within and beyond the domain of magnetism. The mathematical integration of analytic or physics-based motion into metaheuristic frameworks is a recurrent theme across these various instantiations.