GO-Diff: Diffusion-Based Global Optimization

• GO-Diff is a diffusion-based method designed to directly generate thermodynamically favorable atomic configurations using a Boltzmann-weighted score matching loss.
• It employs reverse-time sampling with a force predictor to steer structures toward deep energy minima while reducing energy evaluation costs.
• The method features amortized optimization, enabling pretrained models to transfer knowledge across related systems for faster convergence.

GO-Diff is a diffusion-based global optimization method designed to learn and directly sample low-energy atomic configurations without requiring prior data or explicit relaxation steps. The algorithm is trained from scratch using a Boltzmann-weighted score-matching loss, which leverages only the known energy function to guide generation toward thermodynamically favorable regions in the configuration space. GO-Diff operates via a two-stage loop of self-sampling and model refinement, progressively improving its ability to target low-energy structures. It supports "amortized optimization," meaning pretrained models on one system can be reused for related systems, enabling faster convergence on new optimization tasks without retraining from scratch (Rønne et al., 15 Oct 2025).

1. Diffusion-Based Generation for Structure Optimization

GO-Diff utilizes score-based diffusion models, which are generative models that gradually add and then remove noise to sample from complex distributions. In this context, the diffusion process is used to sample atomic configurations from the region of the potential energy surface (PES) corresponding to low energy. The forward SDE process perturbs candidate atomic structures, while the reverse process leverages a trainable score network to drive samples toward configurations that maximize the Boltzmann distribution:

$$\pi_T(x) = \frac{\exp(-E(x)/T)}{Z_T}, \quad Z_T = \int \exp(-E(x)/T) dx$$

where $E(x)$ is the energy of configuration $x$, $T$ is temperature, and $Z_T$ is the partition function.

2. Reverse-Time Sampling and Force-Field Guidance

During sampling, the reverse SDE is solved using the score network $s_t(x, t) \approx \nabla_x \log p_t(x)$, producing atomic configurations progressively closer to the target distribution. The predictor-corrector algorithm is augmented with a force predictor head $F_t(x)$, which injects direct force information into the sampling trajectory:

$\Delta x = \alpha (1 - t)^\zeta F_t(x)$

Here, $\alpha$ is an adaptive step size and $\zeta$ controls scaling near the end of the diffusion process ($t \to 0$). This guidance steers final structures toward deep energy minima, improving sampling quality without requiring explicit structure relaxation.

3. Boltzmann-Weighted Score Matching Loss and Replay Buffer

Because low-energy structures are rare and no labeled dataset is available, GO-Diff frames training as score matching on the Boltzmann distribution, using importance sampling over a dynamically updated buffer of previously generated structures. The loss is given by:

$$L_\theta^\text{Boltzmann} = \mathbb{E}_{t \sim U(0,1)} \left[ \lambda(t) \, \mathbb{E}_{x_0 \sim q, x_t \sim p_{t|0}(\cdot|x_0)} \left\{ w(E) \left\| s_\theta(x_t, t) - \nabla_{x_t} \log p_{t|0}(x_t|x_0) \right\|_2^2 \right\} \right]$$

with normalized Boltzmann weight

$$w(E) = \frac{\exp(-E/T)}{\sum_{i \in \mathcal{B}} \exp(-E^{(i)}/T)}$$

where $\mathcal{B}$ is the replay buffer of configurations.

4. Self-Sampling and Iterative Model Refinement

GO-Diff operates iteratively:

• Self-Sampling: The current diffusion model samples new atomic configurations via reverse SDE. Their energies are evaluated and added to the buffer.
• Model Refinement: The score network is updated using the buffer, weighted via Boltzmann factors, with the temperature $T$ annealed over time. Early iterations with higher $T$ facilitate exploration, while later, lower $T$ focus sampling on deep energy wells.

This loop progressively improves the model’s ability to concentrate sampling in low-energy regions with competitive efficiency compared to standard relaxation-based structural search algorithms.

5. Amortized Optimization and Model Transfer

A key feature of GO-Diff is amortized optimization. Once pretrained on one system, the model retains chemical and energetic priors (e.g., surface geometries, bonding motifs) and can be transferred to related systems. This dramatically accelerates convergence in new tasks: for example, a pretrained model on a Pt addatom problem quickly finds the stable Pt heptamer on Pt(111) with fewer than 2,560 energy evaluations, compared to 10,000 for random structure search (RSS).

6. Comparative Evaluation and Efficiency

Compared to traditional optimization pipelines (RSS, basin-hopping, simulated annealing), GO-Diff provides competitive structure search capability with significantly fewer energy evaluations. The absence of explicit relaxation during sampling and direct guidance by the energy function streamlines the search for thermodynamically favorable atomic configurations.

7. Formulation Summary

The central mathematical elements are:

• Boltzmann distribution for target sampling:

$\pi_T(x) = \frac{\exp(-E(x)/T)}{Z_T}$

• Boltzmann-weighted score-matching objective:

$$L_\theta^\text{Boltzmann} = \mathbb{E}_{\text{sampling}} [ w(E) \| s_\theta(x_t, t) - \nabla_{x_t} \log p_{t|0}(x_t|x_0) \|_2^2 ]$$

• Force-guided predictor step:

$\Delta x = \alpha (1 - t)^\zeta F_\theta(x)$

This principled combination of score-based generative modeling, Boltzmann weighting, and self-sampling/refinement enables effective global optimization in atomic configuration space, circumventing the limitations of needing labeled datasets or explicit relaxation algorithms.

Applications and Limitations

GO-Diff is applicable to structural optimization problems in computational chemistry, materials science, and condensed matter, particularly when only an energy function is available and prior data is scarce. While the method dramatically reduces required energy evaluations and supports fast model transfer, its precision and generalization rely on the fidelity of the score-based modeling and the adequacy of the replay buffer sampling. As with any generative optimization framework, quality hinges on the energy landscape complexity and the degree of exploration allowed by the annealing schedule and temperature sequence.

GO-Diff exemplifies a modern approach to data-free, amortized global structure optimization driven by diffusion modeling directly steered by the physical energetics of the system, with performance validated against classical optimization strategies (Rønne et al., 15 Oct 2025).