Classical techniques suffer from limited applicability or huge computational costs.

Both provide lower bounds for the partition function by utilizing the so-called gauge transformation which modifies factors of GM while keeping the partition function invariant. Moreover, we prove that both G-MF and G-BP are exact for GMs with a single loop of a special structure, even though the bare MF and BP perform badly in this case.

Despite the popularity of these algorithms, it is well understood that the Sum-of-Squares (SOS) hierarchy based on semidefinite programming (SDP) can provide superior guarantees.

The scaled-dot-product attention (SDPA) mechanism is a core component of modern deep learning, but its mathematical form is often motivated by heuristics. This work provides a first-principles justification for SDPA. We first show that the attention forward pass is the exact solution to a degenerate, one-sided Entropic Optimal Transport (EOT) problem, which seeks a distribution that maximizes similarity while being maximally entropic. This optimization perspective has a direct consequence for the backward pass. We prove that the standard gradient computed via backpropagation is mathematically identical to an advantage-based policy gradient, a variance-reduced update rule from reinforcement learning. Crucially, we demonstrate that the EOT formulation of the forward pass induces a specific information geometry on the space of attention distributions. It is this geometry, characterized by the Fisher Information Matrix, that dictates the precise form of the learning gradient, revealing the advantage-based update as a natural consequence of the optimization problem being solved. This unified view reveals SDPA as a principled mechanism where the forward pass performs optimal inference and the backward pass implements a rational, manifold-aware learning update.

Designing mechanical mechanisms to trace specific paths is a classic yet notoriously difficult engineering problem, characterized by a vast and complex search space of discrete topologies and continuous parameters. We introduce MechaFormer, a Transformer-based model that tackles this challenge by treating mechanism design as a conditional sequence generation task. Our model learns to translate a target curve into a domain-specific language (DSL) string, simultaneously determining the mechanism's topology and geometric parameters in a single, unified process. MechaFormer significantly outperforms existing baselines, achieving state-of-the-art path-matching accuracy and generating a wide diversity of novel and valid designs. We demonstrate a suite of sampling strategies that can dramatically improve solution quality and offer designers valuable flexibility. Furthermore, we show that the high-quality outputs from MechaFormer serve as excellent starting points for traditional optimizers, creating a hybrid approach that finds superior solutions with remarkable efficiency.

Generative machine learning has emerged as a powerful tool for design representation and exploration. However, its application is often constrained by the need for large datasets of existing designs and the lack of interpretability about what features drive optimality. To address these challenges, we introduce a systematic framework for constructing training datasets tailored to generative models and demonstrate how these models can be leveraged for interpretable design. The novelty of this work is twofold: (i) we present a systematic framework for generating archetypes with internally homogeneous but mutually heterogeneous inputs that can be used to generate a training dataset, and (ii) we show how integrating generative models with Bayesian optimization can enhance the interpretability of the latent space of admissible designs. These findings are validated by using the framework to design a flow battery manifold, demonstrating that it effectively captures the space of feasible designs, including novel configurations while enabling efficient exploration. This work broadens the applicability of generative machine-learning models in system designs by enhancing quality and reliability.

Large language models (LLMs) are increasingly being applied to black-box optimization tasks, from program synthesis to molecule design. Prior work typically leverages in-context learning to iteratively guide the model towards better solutions. Such methods, however, often struggle to balance exploration of new solution spaces with exploitation of high-reward ones. Recently, test-time training (TTT) with synthetic data has shown promise in improving solution quality. However, the need for hand-crafted training data tailored to each task limits feasibility and scalability across domains. To address this problem, we introduce MiGrATe-a method for online TTT that uses GRPO as a search algorithm to adapt LLMs at inference without requiring external training data. MiGrATe operates via a mixed-policy group construction procedure that combines on-policy sampling with two off-policy data selection techniques: greedy sampling, which selects top-performing past completions, and neighborhood sampling (NS), which generates completions structurally similar to high-reward ones. Together, these components bias the policy gradient towards exploitation of promising regions in solution space, while preserving exploration through on-policy sampling. We evaluate MiGrATe on three challenging domains-word search, molecule optimization, and hypothesis+program induction on the Abstraction and Reasoning Corpus (ARC)-and find that it consistently outperforms both inference-only and TTT baselines, demonstrating the potential of online TTT as a solution for complex search tasks without external supervision.