相关论文: Gold-Medal-Level Olympiad Geometry Solving with Ef…

Bayesian optimization is a principled optimization strategy for a black-box objective function. It shows its effectiveness in a wide variety of real-world applications such as scientific discovery and experimental design. In general, the…

Bayesian optimization is a data-efficient technique that has been shown to be extremely powerful to optimize expensive, black-box, and possibly noisy objective functions. Many applications involve optimizing probabilities and mixtures which…

In this chapter, we identify fundamental geometric structures that underlie the problems of sampling, optimisation, inference and adaptive decision-making. Based on this identification, we derive algorithms that exploit these geometric…

Using AI to write formal proofs for mathematical problems is a challenging task that has seen some advancements in recent years. Automated systems such as Lean can verify the correctness of proofs written in formal language, yet writing the…

Partial Differential Equations (PDEs) underpin many scientific phenomena, yet traditional computational approaches often struggle with complex, nonlinear systems and irregular geometries. This paper introduces the AMG method, a Multi-Graph…

Combinatorial Optimization (CO) has been a long-standing challenging research topic featured by its NP-hard nature. Traditionally such problems are approximately solved with heuristic algorithms which are usually fast but may sacrifice the…

A computation method of algebraic local cohomology with parameters, associated with zero-dimensional ideal with parameter, is introduced. This computation method gives us in particular a decomposition of the parameter space depending on the…

Large Language Models (LLMs) have driven substantial progress in artificial intelligence in recent years, exhibiting impressive capabilities across a wide range of tasks, including mathematical problem-solving. Inspired by the success of…

The art of quantum algorithm design is highly nontrivial. Grover's search algorithm constitutes a masterpiece of quantum computational software. In this article, we use methods of geometric algebra (GA) and information geometry (IG) to…

Although Hamiltonian Monte Carlo has proven an empirical success, the lack of a rigorous theoretical understanding of the algorithm has in many ways impeded both principled developments of the method and use of the algorithm in practice. In…

Motivated by the sign problem in several systems, we have developed a geometric simulation algorithm based on the strong coupling expansion which can be applied to abelian pure gauge models. We have studied the algorithm in the U(1) model…

Proprietary AI systems have recently demonstrated impressive capabilities on complex proof-based problems, with gold-level performance reported at the 2025 International Mathematical Olympiad (IMO). However, the training pipelines behind…

Algorithms for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. The algorithms are based on global fixed point iterations which apply to families of functions satisfying…

During the last ten years, increasing efforts were made to improve and simplify the process from Computer Aided Design (CAD) modeling to a numerical simulation. It has been shown that the transition from one model to another, i.e. the…

Algebraic multigrid (AMG) is conventionally applied in a black-box fashion, agnostic to the underlying geometry. In this work, we propose that using geometric information -- when available -- to assist with setting up the AMG hierarchy is…

We propose a new hybrid topology optimization algorithm based on multigrid approach that combines the parallelization strategy of CPU using OpenMP and heavily multithreading capabilities of modern Graphics Processing Units (GPU). In…

The continuous computational power growth in the last decades has made solving several optimization problems significant to humankind a tractable task; however, tackling some of them remains a challenge due to the overwhelming amount of…

This paper describes a massively parallel algebraic multigrid method based on non-smoothed aggregation. It is especially suited for solving heterogeneous elliptic problems as it uses a greedy heuristic algorithm for the aggregation that…

We present both $hp$-a priori and $hp$-a posteriori error analysis of a mixed-order hybrid high-order (HHO) method to approximate second-order elliptic problems on simplicial meshes. Our main result on the $hp$-a priori error analysis is a…

We present miniF2F, a dataset of formal Olympiad-level mathematics problems statements intended to provide a unified cross-system benchmark for neural theorem proving. The miniF2F benchmark currently targets Metamath, Lean, Isabelle…