1. Introduction

With the rapid development of information technology, technological breakthroughs in the fields of big data, cloud computing, artificial intelligence and other fields continue to emerge, which has brought about profound changes in all walks of life. In the field of operations research, mathematical modelling and optimization analysis methods play a crucial role in solving complex problems. Therefore, for mathematical formulas in operations research, we need to find it and distinguish it, and identify the named entities in the formulas accurately, which helps to improve the efficiency and accuracy of operations research.

Latex is a coding language for the representation of formulas in operations research, and it is also involved in other industries or professions related to mathematical formulas. Nonetheless, the topic of named entity recognition in operations research has seen comparatively little investigation and effort, and no sizeable and accessible datasets have been compiled. As a result, dealing with current problems in the field of formulas in operations research with existing techniques achieves mediocre results due to the lack of sufficient reference and experimental data.

Named Entity Recognition is a fundamental task for several natural language processing applications such as knowledge base question and answer systems, machine translation, information retrieval, sentiment analysis, and knowledge mapping. Finding specific entities, such as names of individuals, locations, organizations, etc. from a piece of text, and distinguish them from non-entities is the main objective of NER. Named entity recognition of latex texts makes further research possible. Accordingly, this paper constructs a dataset from the text of operations research courses in schools and related digital resources, and conducts research on entity recognition in operations research on the basis of the dataset.

This paper proposes a mathematical formula named entity recognition model consisting of BERT, BiLSTM and Transformer, for addressing the above issues. The BERT model is used to obtain the vector representation of each character, which is spliced with the vector representation of a randomly initialized instance query, and then jointly fed into BERT for encoding, and then the query semantics are enhanced by using One-Way Self-Attention so that the query can be modelled in terms of its connections with each other. This is followed by feature extraction through the BiLSTM layer and transformer layer, and finally the final predicted labelling results are obtained by finding the optimal allocation by finding the minimum cost matrix for the allocation problem.

2. Related Work

The research progress in named entity recognition techniques can be divided into the following stages:

The first stage is the use of dictionary and rule based approach. Firstly, a candidate dictionary is obtained based on statistical analysis, and then manual screening is used while extracting the important terms in the domain. Using the a priori knowledge of the lexicon, potential entities in the sentence are matched, after which they are filtered according to rules. Rule-based approaches tend to rely too much on manually defined rules and templates, and thus may have limitations in their coverage for complex linguistic expressions and diverse inputs. Moreover, manually constructed rules are subjective and prone to bias and errors.

The second stage is the statistical machine learning based approach, in which researchers begin to try to use statistical models for named entity recognition. For instance, in a research finished by Yu et al. [1], they utilized Hidden Markov Model (HMM) for the recognition of Chinese named entities; Huang et al. [2] suggested an approach combining Support Vector Machine (SVM) with transformation-based error-driven learning for biological entity recognition; A named entity recognition technique based on conditional random fields was proposed by Feng et al. [3]. Statistical machine learning based methods no longer rely on hand-constructed tedious rules, however, they need a large number of training sets with clear labels, which still takes a lot of effort and resources.

The third stage is deep learning based named entity recognition. With the theory and application of deep learning gradually coming into people's attention, in addition to the improvement of algorithms and computer performance, the depth and width of neural networks [4] are also increasing. As a result, many neural network structures have emerged that are particularly well known today, the Convolutional Neural Network (CNN) [5], the Recurrent Neural Network (RNN) [6], the Long Short Term Memory Network (LSTM), and even more intricate deep learning models like BERT (Bidirectional Encoder Representation from Transformer) [7]. Deep learning models can learn and extract features from data on their own, in contrast to standard machine learning techniques that require feature extraction by hand. Deep learning greatly saves the human resources required for feature fusion, and deep learning models can automatically learn complex language patterns with strong generalization ability, which enables them to be better applied to practical tasks. For example, unidirectional Long Short Term Memory (LSTM) networks [8] are widely used in NER tasks due to their strong sequential feature extraction ability and are often combined with CRF (LSTM-CRF [9]) to achieve better recognition results. However, since unidirectional LSTM networks is limited to extracting unidirectional text features, Lample et al. [10] then proposed a Bidirectional LSTM (BiLSTM) network on this basis to obtain global contextual deep features, and then combined with CRF to form a BiLSTM-CRF neural network, it enhances the model's effect even further, and since then the model has gradually become the mainstream model for deep learning to solve NER problems in various fields. For example, Zhou et al. [11] promoted knowledge mining of ancient Chinese medicine books by extract text features using the BiLSTM-CRF method; Cheng et al. [12] applied the BiLSTM-CRF model to the field of ancient Chinese literature, and realized the processing tasks of automatic sentence breakage, automatic word division, lexical annotation and other processing tasks of the ancient Chinese information, and achieved very good results.

Meanwhile there are many scholars who have improved and innovated the BiLSTM-CRF model. For example, Huang et al. [13] introduced an external cybersecurity lexicon to enhance the features of cybersecurity texts based on the BiLSTM-CRF model, and obtained a favorable outcome on the cybersecurity dataset. On the field of agriculture, Zhou et al. [14] firstly processed the long text of the dataset into short text, and then input it into the ERNIE model for encoding to get this representation that preserves semantic associations, and subsequently enter it into BiLSTM-CRF to address the issues of low efficiency of the soil fertility named entity recognition method and poor text processing effect. Regarding the study of Chinese, Li et al. [15] introduced Hybrid Attention mechanism (Hybrid Attention) into BiLSTM-CRF model to achieve good semantic analysis ability and accurately represent the negation semantics in a sentence.

The Iterative Expanded Convolutional Neural Network (IDCNN), which can effectively extract local features across a wide acceptance domain, was also applied for Named Entity Recognition for the first time by Strubell et al. [16]. In order to solve the named entity identification issue in electronic medical records, Chen et al. [17] presented an attention mechanism based on the IDCNN-CRF model, and the special step-size of the inflated convolution can extract the text features more accurately with excellent recognition results.

To improve the word vectors' semantic representation, scholars have proposed pre-trained language models. Peters et al. [18] proposed ELMo model based on BiLSTM structure, which is able to extract bi-directional textual features. Radford et al. [19] proposed GPT (Generative Pre-trained Transformer) model, which is able to capture more distant semantic features.is able to capture more distant semantic information, but because it is a unidirectional model, it is unable to obtain global contextual information. Therefore, Devlin et al. [8] from Google team proposed the BERT model with bidirectional Transformer encoder structure. This boosts its performance on named entity recognition tests and further refines the semantic representation of word vectors. For example, Li et al. [20] utilized the BERT-CRF model to the joint extraction of maize breeding entity relationships, which provides an effective data base for the construction of maize breeding knowledge graph and other downstream tasks; Zheng et al. [21] utilized the BiLSTM-CRF model based on BERT for web content monitoring to identify sensitive words and variants, and the recognition effect is improved compared with other models; Yu et al. [22] proposed an ancient poetic place name recognition model, referred to as DABERT-CRF, using a data enhancement method while integrating BERT-CRF, as a way to promote further research on Chinese classical literature; Li et al. [23] used BERT and BiLSTM, combined with a bilinear attention mechanism, to successfully improve the semantic information and attain favorable outcomes in the Chinese recognition of medically named entities. Good results have been achieved on it.

The NER task was redefined as a machine-reading task in recent years by Luo et al. [24], Mengge et al. [25] and Zheng et al. [26], who demonstrated strong performance on both flat and nested datasets. With the goal to extract entities, they create type-specific queries based on external knowledge and treat phrases as contexts. They create PER-specific queries in natural language form, for instance, for the statement "U.S. President Donald Trump is enjoying his vacation in Miami" in order to extract PER entities like "U.S. President" and "Donald Trump". However, only one type of entity can be extracted for each inference because searches are type-specific. This approach overlooks the inherent relationships between different entity types in addition to producing ineffective predictions. Furthermore, Type-specific searches are manually constructed using external knowledge, which makes realistic scenarios using hundreds of entity types difficult.

3. Methodology: A mathematical formula named entity recognition method combining enhanced dynamic allocation of labels

Figure 1 Structure of the model.

Based on the similarities between sequences of mathematical formulas and textual information, in everyday use mathematical formulas are often in the form of latex encoding, thus allowing useful features to be learned from the data in a textual form. BERT-formula is a hybrid model architecture based on the BERT model encoding. A vector representation of each character is first computed, spliced with a randomly initialized vector representation of the instance query, and then jointly fed into BERT for encoding, followed by the use of unidirectional Self-Attention, which allows the queries to model connections with each other, enhancing the query semantics [27]. This is followed by feature extraction through BiLSTM layer and transformer layer to improve the model's generalization capacity even more. Finally, a dynamic label assignment mechanism is designed to ascertain the best possible result for the assignment, and label assignment is regarded as a one-to-many Linear Assignment Problem (LAP) (Burkard and Cela, 1999) [28]. The final predicted labelling outcome is obtained by finding the minimum cost matrix of the allocation problem and finding the optimal allocation.

Figure 1 displays the model's framework as suggested in this paper. There are three components to it: encoder, entity prediction and dynamic label assignment, where entity classification and entity localization are the two subtasks of entity prediction. The input of the encoder part comes from textual information as well as instance queries that can learn global semantic information, and the vector representation is obtained by using embedding to extract rich syntactic and semantic elements; the entity prediction part mainly accomplishes the boundary prediction of the entity as well as the category prediction of the entity, and if more than one prediction for the same entity occurs, The one chosen to be kept is the one with the highest probability value; the dynamic label entity assignment mainly According to the allocation cost generated in the previous part to form the Cost matrix, and then further use the algorithm of the linear allocation problem to calculate the label allocation matrix with the minimum cost, so as to achieve the instance query and the allocation of entity labels, and get the prediction results of each entity to complete the recognition task.

To indicate an example of training, we use $(X,Y)$, where $X$ is a sentence with $N$ words, and the three represent the index of the left boundary of the $k$-th entity, the index of the right boundary, and the index of the entity type, respectively. In our study, $M\left(>G\right)$ globally learnable instance queries are set up, it extracts an entity from the phrase in each case. During training, they can independently learn query semantics and are initialized at random. Therefore, the challenge involves using the learnable instance inquiry $I$ to extract entity $Y$ from an input sentence $X$.

3.4 Training objectives

After the above steps, we as well compute the entity prediction results for M instance queries and get their labels when the total allocation cost is minimum. We specify the classification loss and boundary loss for training model. We utilize the binary cross entropy function to get the loss values for left and right border prediction:

$L_b=-{\sum }_{\delta \in \{l,r\}}{\sum }_{i=0}^{M-1}{\sum }_{j=0}^{N-1}\left[{\hat{Y}}_i^c=c\right]\log P_{ic}^t+\left[{\hat{Y}}_i^{\delta }\ne c\right]\log \left(1-P_{ic}^t\right)$

We employ the cross entropy function for entity categorization to calculate the loss value:

$$L_b=-\sum _{i=0}^{M-1}\sum _{c\in \epsilon }\left[{\hat{Y}}_i^t=c\right]\log P_{ic}^t$$

where $1\left[\omega \right]$ represents an indicator function that $\omega$ takes 1 when true and 0 otherwise.

Following Al-Rfou et al. [29] and Carion et al. [30], after every character level transformer layer, we add an entity pointer and an entity classifier, allowing us to obtain two loss values at each layer. As a result, the overall loss on training set $D$ can be described as:

$$L=\sum _D\sum _{\tau =1}^L{L}_t^{\tau }+L_b^{\tau }$$

where $L_t^{\tau }$, $L_b^{\tau }$ are the classification loss and boundary loss of the $\tau$ layer, respectively. In order to predict, we solely use entity prediction at the last layer for prediction.

Figure 2 Knowledge graph based learning path recommendation scheme map.

4. Research on Learning Path Recommendation Based on Knowledge Graph

Knowledge graphs (KGs) serve as a pedagogical and computational scaffold for structuring fragmented knowledge domains, enabling the generation of logically coherent learning paths that respect intrinsic educational dependencies such as prerequisite sequences, hierarchical inclusions, and conceptual analogies. Unlike traditional methods, which often prioritize isolated content delivery, KGs formalize knowledge topology through nodes (e.g., "Gradient Descent") and edges (e.g., "is_prerequisite_for"), allowing systematic traversal from a target node (e.g., "Convolutional Neural Networks") to prerequisite foundations (e.g., "Linear Algebra" or "Image Processing"). This structural rigor ensures learning continuity by mandating that no critical intermediate concepts are omitted, thereby preserving the integrity of pedagogical logic. For instance, omitting "Backpropagation" in a path toward "Deep Reinforcement Learning" would violate prerequisite dependencies, undermining learning efficacy. To operationalize this, a hybrid recommendation strategy is proposed, integrating inclusion-relation prioritization and multi-criteria node ranking. First, inclusion relations (e.g., "Machine Learning Fundamentals" $\supset$ "Supervised Learning") are extracted using subgraph isolation:

where $G_{\text{include}}$ denotes the subgraph of nodes directly containing or contained by the target node $v_{\text{target}}$. Subsequently, candidate nodes are ranked by a composite metric balancing association strength and learning cost. Association strength is quantified via a weighted combination of inverse shortest-path distance $\text{Distance}(v,v_{\text{target}})$ and node centrality $\text{Degree}\left(v\right)$;

where $\alpha$ and $\beta$ calibrated through learner feedback or A/B testing. Nodes with higher scores are prioritized, while ties are resolved using learning cost $C\left(v\right)$, modeled as a function of historical learner engagement ($t_{\text{avg}}$) and resource complexity $RC\left(v\right)$:

where $\gamma$ adjusts the weight of temporal versus cognitive load. This framework aligns with graph-based recommendation paradigms such as Zhang et al. [31] graph convolutional network (GCN) approach, which leverages node embeddings to capture latent educational dependencies, and Zheng et al. [32] ant colony optimization variant, dynamically balancing exploration of novel concepts and exploitation of known pathways. Furthermore, Shi et al. [33] validate the necessity of multidimensional KGs, demonstrating that integrating node centrality (e.g., PageRank scores) and learner-specific factors (e.g., proficiency levels) enhances path personalization for learners, as measured by retention metrics. The methodology's robustness is further reinforced by its compatibility with ontology-enriched KGs, where named entity recognition (NER) refines node categorization—for example, distinguishing "Bayesian Inference" as a theoretical versus applied concept—ensuring granular alignment with curricular taxonomies [34]. Figure 2 shows the general process of learning path recommendation based on knowledge graph.

5. Experiments

In this paper, we use manual annotation to extract more than four thousand latex mathematical formulas from the text of about 100,000 words of operations research textbook, which contains more than 10,000 entities and annotate the latex formulas, and there are five types of entity categories defined, namely: Function operator, Unary logical operator, Binary relation operator, Delimiter and Special operator. Table 1 shows certain entities' pertinent descriptions and examples.

6. Conclusion

This research investigates named entity recognition of mathematical formulas based on text in the field of operations research, and proposes a model for named entity recognition of mathematical formulas, BERT-formula. an efficient feature representation of mathematical formulas using embedding is spliced with a vector representation of randomly initialized instance queries, which are then jointly input to BERT for encoding, and then encoded using one-way Self-Attention, which allows the queries to model connections with each other and enhances the query semantics. After that, feature extraction is performed through BiLSTM layer and transformer layer, and finally the final predicted labeling results are obtained by finding the optimal allocation by finding the minimum cost matrix for the allocation problem. The experimental findings demonstrate that, in comparison to the conventional NER technique, the inference speed of the approach presented in this study is significantly enhanced, and it is also superior in terms of recognition results. The identified formula entities can further support adaptive learning path recommendations by mapping domain-specific knowledge components to targeted educational resources. The named entity recognition of operations research formulas achieved in this research establishes a strong basis for downstream NLP tasks in the field of operations research, and the example queries in this paper do not rely on external knowledge, but throughout the training phase, acquire the query semantics associated with the entity type and location, which saves a lot of manual consumption; the feature of not relying on the knowledge related to a specific domain also makes the method can be easily applied to other domains. At the same time, for the problem of fewer datasets of text labeling of mathematical formulas in the field of operations research, a dataset of operations research latex formulas is constructed, which contains nearly five thousand data, and it is anticipated that it will contribute in some way to the advancement of operations research. Subsequently, the size of the dataset will be further expanded to further verify the transferability of the model and apply the model to other fields.