The quadratic unconstrained binary optimization (QUBO) is a well-known NP-hard problem that takes an $n\times n$ matrix $Q$ as input and decides an $n$-dimensional 0-1 vector $x$, to optimize a quadratic function. Existing learning-based models that always formulate the solution process as sequential decisions suffer from high computational overload. To overcome this issue, we propose a neural solver called the Value Classification Model (VCM) that formulates the solution process from a classification perspective. It applies a Depth Value Network (DVN) based on graph convolution that exploits the symmetry property in $Q$ to auto-grasp value features. These features are then fed into a Value Classification Network (VCN) which directly generates classification solutions. Trained by a highly efficient model-tailored Greedy-guided Self Trainer (GST) which does not require any priori optimal labels, VCM significantly outperforms competitors in both computational efficiency and solution quality with a remarkable generalization ability. It can achieve near-optimal solutions in milliseconds with an average optimality gap of just 0.362\% on benchmarks with up to 2500 variables. Notably, a VCM trained at a specific DVN depth can steadily find better solutions by simply extending the testing depth, which narrows the gap to 0.034\% on benchmarks. To our knowledge, this is the first learning-based model to reach such a performance.

Preprint August 2025 - This version has not been peer - reviewed . Abstract The progress of Large Language Models (LLMs) like ChatGPT raises the question of how they can be integrated into education. One hope is that they can support mathematics learning, including word - problem solving. Since LLMs can handle textual input with ease, they appear well - suited for solving mathematical word problems. Yet their real competence, whether they can make sense of the real - world context, and the implications for classrooms remain unclear. We conducted a scoping review from a mathematics - education perspective, including three parts: a technical overview, a systematic review of word problems used in research, and a state - of - the - art empirical evaluation of LLMs on mathematical word problems. First, in the technical overview, we contrast the conceptualization of word problems and their solution processes between LLMs and students. In computer - science research this is typically labeled mathematical reasoning, a term that does not align with usage in mathematics education. Second, our literature review of 213 studies shows that the most popular word - problem corpora are dominated by s - problems, which do not require a consideration of realities of their real - world context. Finally, our evaluation of GPT - 3.5 - turbo, GPT - 4o - mini, GPT - 4.1, o3, and GPT - 5 on 287 word problems shows that most recent LLMs solve these s - problems with near - perfect accuracy, including a perfect score on 2 0 problems from PISA. LLMs still showed weaknesses in tackling problems where the real - world context is problematic or non - sensical. In sum, we argue based on all three aspects that LLMs have mastered a superficial solution process but do not make sense of word problems, which potentially limits their value as instructional tools in mathematics classroom s. Keywords LLM; word - problem solving; AI; mathematical reasoning; modelling 1 Introduction In the last couple of years, the rapid improvement of Large Language Models (LLMs) has led to an unprecedented interest in educational research in artificial intelligence in general, and of LLMs in particular (Kasneci et al., 2023) . However, while LLMs excel at producing, translating and reviewing text, they are not natively designed for processing numerical information, calculating, or proving (Chang et al., 2024) . C ompared to other tasks, solving mathematical problems is relatively difficult for LLMs (Testolin, 2024) . This is also true for mathematical word - problems solving.

Geometry problem solving (GPS) requires models to master diagram comprehension, logical reasoning, knowledge application, numerical computation, and auxiliary line construction. This presents a significant challenge for Multimodal Large Language Models (MLLMs). However, existing benchmarks for evaluating MLLM geometry skills overlook auxiliary line construction and lack fine-grained process evaluation, making them insufficient for assessing MLLMs' long-step reasoning abilities. To bridge these gaps, we present the GeoLaux benchmark, comprising 2,186 geometry problems, incorporating both calculation and proving questions. Notably, the problems require an average of 6.51 reasoning steps, with a maximum of 24 steps, and 41.8% of them need auxiliary line construction. Building on the dataset, we design a novel five-dimensional evaluation strategy assessing answer correctness, process correctness, process quality, auxiliary line impact, and error causes. Extensive experiments on 13 leading MLLMs (including thinking models and non-thinking models) yield three pivotal findings: First, models exhibit substantial performance degradation in extended reasoning steps (nine models demonstrate over 50% performance drop). Second, compared to calculation problems, MLLMs tend to take shortcuts when solving proving problems. Third, models lack auxiliary line awareness, and enhancing this capability proves particularly beneficial for overall geometry reasoning improvement. These findings establish GeoLaux as both a benchmark for evaluating MLLMs' long-step geometric reasoning with auxiliary lines and a guide for capability advancement. Our dataset and code are included in supplementary materials and will be released.

The quadratic unconstrained binary optimization (QUBO) is a well-known NP-hard problem that takes an n\times n matrix Q as input and decides an n -dimensional 0-1 vector x, to optimize a quadratic function. Existing learning-based models that always formulate the solution process as sequential decisions suffer from high computational overload. To overcome this issue, we propose a neural solver called the Value Classification Model (VCM) that formulates the solution process from a classification perspective. It applies a Depth Value Network (DVN) based on graph convolution that exploits the symmetry property in Q to auto-grasp value features. These features are then fed into a Value Classification Network (VCN) which directly generates classification solutions.

Modern Mixed Integer Linear Programming (MILP) solvers use the Branch-and-Bound algorithm together with a plethora of auxiliary components that speed up the search. In recent years, there has been an explosive development in the use of machine learning for enhancing and supporting these algorithmic components. Within this line, we propose a methodology for predicting the optimal objective value, or, equivalently, predicting if the current incumbent is optimal. For this task, we introduce a predictor based on a graph neural network (GNN) architecture, together with a set of dynamic features. Experimental results on diverse benchmarks demonstrate the efficacy of our approach, achieving high accuracy in the prediction task and outperforming existing methods. These findings suggest new opportunities for integrating ML-driven predictions into MILP solvers, enabling smarter decision-making and improved performance.

The recurrent neural network has been greatly developed for effectively solving time-varying problems corresponding to complex environments. However, limited by the way of centralized processing, the model performance is greatly affected by factors like the silos problems of the models and data in reality. Therefore, the emergence of distributed artificial intelligence such as federated learning (FL) makes it possible for the dynamic aggregation among models. However, the integration process of FL is still server-dependent, which may cause a great risk to the overall model. Also, it only allows collaboration between homogeneous models, and does not have a good solution for the interaction between heterogeneous models. Therefore, we propose a Distributed Computation Model (DCM) based on the consortium blockchain network to improve the credibility of the overall model and effective coordination among heterogeneous models. In addition, a Distributed Hierarchical Integration (DHI) algorithm is also designed for the global solution process. Within a group, permissioned nodes collect the local models' results from different permissionless nodes and then sends the aggregated results back to all the permissionless nodes to regularize the processing of the local models. After the iteration is completed, the secondary integration of the local results will be performed between permission nodes to obtain the global results. In the experiments, we verify the efficiency of DCM, where the results show that the proposed model outperforms many state-of-the-art models based on a federated learning framework.