The relationship between Ordinary Differential Equations (ODEs) and machine learning is natural: while ODEs act as mathematical models encoding fundamental laws, machine learning emerges as a system capable of inferring patterns from complex data. In essence, ODEs allow data to be generated from the model, whereas machine learning strategies enable the derivation of models from data, with strategies ranging from kernel methods (García, 2022) and neural networks (Chen et al., 2018) to transformers (Becker et al., 2023; d’Ascoli et al., 2024), which are the basis of large-scale language models (LLMs). This empirical landscape suggests that the use of LLMs for ODE exam evaluation is not simply a technological convenience, but a logical extension of machine learning’s ability to interpret and validate symbolic reasoning, particularly in this case.

The integration of AI into ODE education offers a powerful tool for bridging procedural skills with conceptual mastery. While recent observational studies highlight that LLMs still face significant hurdles in complex mathematical reasoning (Collins et al., 2024), emerging frameworks in educational data mining demonstrate their potential for providing automated and formative feedback in problem-solving tasks (Worden et al., 2024). This synergy supports the development of neuro-symbolic systems where LLMs manage natural language while symbolic engines ensure algebraic precision.

The implementation of automated assessment systems (Gnanaprakasam & Lourdusamy, 2024; Korthals et al., 2025; Mendonca et al., 2025) responds to the need to scale educational feedback without compromising consistency, objectivity or comprehensiveness. By integrating a symbolic engine with an LLM, this approach aligns with the principles of adaptive assessment, in which the system’s ability to diagnose underlying reasoning allows for feedback personalization beyond simple binary correction (Shuste & Zapata-Rivera, 2012). Thus, the identification of logical milestones facilitates a transition toward personalized education by determining whether a discrepancy stems from a specific operational error or a theoretical deficiency.

In large class sizes, manual grading is prone to fatigue and subjective variability. Automation ensures uniform rubrics and facilitates immediate formative feedback, which is essential to prevent conceptual errors.

In this context, LLMs based on the transformer architecture (Vaswani et al., 2017) have redefined the processing of hybrid content where natural language and formal technical notation converge. When applied to automated exam assessment, this technology facilitates scalability and can support a transition toward personalized education. By employing these tools, it is possible to deconstruct the development of a solution into logical milestones, enabling the identification of whether a discrepancy stems from a specific operational error or an underlying theoretical deficiency (Wei et al., 2022; Zhou et al., 2023).

However, the application of LLMs in the exact sciences presents accuracy challenges arising from their stochastic nature (Lee et al., 2025). This can lead to divergences in algebraic reasoning (Huang et al., 2025), where the model generates sequences that, although linguistically plausible, lack formal validity. To mitigate this risk, this article proposes a hybrid architecture where the LLM acts as a semantic orchestrator and the symbolic processor functions as a deterministic verification anchor. This collaboration ensures that the interpretive flexibility of the language model is backed by the absolute rigor of computational calculation.

In this framework, we hypothesize that a neuro-symbolic architecture, integrating the semantic orchestration of LLMs with the deterministic verification of a Computer Algebra System (CAS), allows for the automated assessment of ODEs while maintaining mathematical rigor and pedagogical fairness. This synergy is expected to bridge the gap between probabilistic reasoning and symbolic accuracy, enabling a human-like ‘error-drift’ analysis.

Thus, in this work we present a novel neuro-symbolic framework for the automated assessment of ODE exams by integrating LLMs, Gemini (Google DeepMind, 2024) in this case, with CAS, SymPy (Meurer et al., 2017) in this case.

To show one way of addressing this problem, we have organized the article into five sections, organized as follows: Section 2 analyzes the specific challenges of qualifying ODEs, such as sequential dependency and the non-uniqueness in the representation of solutions; Section 3 details the proposed methodology to implement the neuro-symbolic strategy, describing the semantic extraction flow and the evaluator configuration using a structured system prompt; Section 4 shows a real-world case study; Section 5 presents the final remarks.

The assessment of ODEs remains a significant challenge for classical AI systems, which often struggle with the multi-step reasoning and precise symbolic manipulation required in STEM subjects (Tan et al., 2025). As noted by Tan, the assessment of STEM subjects presents unique structural challenges, particularly in maintaining mathematical consistency and in addressing the black-box nature of deep learning models. Classical systems frequently fail to provide the explainable, step-by-step validation necessary for complex mathematical derivations, a gap that persists in current automated grading technologies.

In the particular case of ODEs, one of the main obstacles is sequential dependency or the cascade effect: solving an ODE is a graph of dependencies in which a minor error in an intermediate step, such as calculating an integrating factor, invalidates the final numerical result. However, this does not necessarily imply that the logic of the subsequent procedure is incorrect; an expert human evaluator is capable of performing an error propagation analysis to assign partial scores, a capability that traditional automated systems lack.

Another critical challenge lies in the non-uniqueness of the solution form, since, due to trigonometric identities or properties of logarithms or other functions, a correct answer can be expressed in multiple visually distinct ways. Conventional systems often fail to recognize these identities, requiring an exact character match rather than validating the mathematical identity of the function. In addition, the technical validation of an ODE requires checking whether the student’s proposal satisfies the fundamental differential operator (L[y] = g(x)), a symbolic verification that classic correctors do not integrate, limiting their ability to offer deep and fair pedagogical feedback.

To overcome these limitations, the integration of LLMs and CAS emerges as a robust solution. Although the LLM offers the semantic flexibility needed to interpret the nuances of student language, the symbolic engine ensures mathematical precision by providing a formal verification layer that mitigates the risk of model hallucinations.

This neuro-symbolic integration seeks to enhance evaluative accuracy while minimizing generative biases. From this perspective, the neuro-symbolic integration is proposed as a robust solution to potentially mitigate generative biases and enhance evaluative accuracy. This synergy is projected to facilitate the automation of complex pedagogical tasks, such as error-drift analysis and the validation of non-unique solutions.

The proposed methodology is based on a deterministic symbolic evaluation approach that goes beyond simple text comparison to focus on the logical validity of the mathematical procedure. The process begins with the segmentation of the student’s response into critical milestones. Subsequently, symbolic extraction is performed using the SymPy library to translate natural language into exact computational variables. The main innovation lies in error drift detection: if the system detects a fault in step n, it generates code to verify whether the subsequent steps are consistent with that initial error instead of automatically invalidating the entire exam. Finally, a litmus test is applied using the differential operator L[y] = g(x) and an algorithmic identity check (simplify(Student – Ref) == 0), ensuring that any solution mathematically equivalent to the reference is accepted, regardless of its visual form.

The study was conducted using a dataset consisting of n = 18 complete exams. Since each exam consists of three multi-step ODE problems, the analysis covered a total of 54 detailed solution units. The participants were university students from the Faculty of Engineering at the Andres Bello Catholic University (Caracas, Venezuela), enrolled in Computer Engineering, Civil Engineering, and Telecommunications Engineering.

In the following, one student’s exam will be used as a representative case study of the proposed assessment methodology. This exam was selected because one problem is solved well and partial errors are made in the others. We believe this offers a useful perspective on our strategy.

It should be noted that the performance of the proposed strategy observed in this particular case study exam is similar to that of the rest of the group evaluated, which allows us to generalize the conclusions obtained. In this way, and in order to optimize the length of the article and avoid an excessive load of images from the exam, only a detailed analysis of one of its answers will be presented, which will serve as an illustrative model of the interaction between the linguistic model and the symbolic calculation engine.

To make the presentation lighter, we will divide it into three parts: i) the presentation of the problem to the student, the reference solution and rubric for human assessment, ii) the student’s response, and iii) the response of the automatic evaluation system.

The integration of an LLM and a CAS, interconnected through a chain-of-thought framework, represents a robust solution to the dichotomy between contextual interpretation and algorithmic rigor in the sciences.

Although LLMs enable fluent reasoning, the interpretation of ambiguous statements, and the diagnosis of conceptual errors in natural language, the symbolic engine serves as a deterministic anchor that executes mathematical operations without the risk of hallucinations. This architecture allows scientific problems to be approached with the cognitive flexibility required to understand human-led processes, combined with the computational precision indispensable for validating results, effectively bridging the gap between theoretical intuition and technical accuracy.

The integration of a symbolic engine to verify the student’s chain of thought suggests the potential to refine the quality of feedback by helping to distinguish between minor algebraic slips and fundamental conceptual gaps. Regarding student learning, the capacity to receive partial credit through ‘Error-Drift’ analysis appears to offer a more supportive environment that acknowledges logical consistency even when initial errors occur. From a teaching perspective, this framework might serve as a complementary tool for instructional practice, possibly mitigating some of the subjective variability and fatigue typically associated with manual grading in large class sizes.

Table 5 summarizes our beliefs about the advantages of the hybrid approach at a macroscopic level, compared with traditional methods and the use of pure LLMs.

In conclusion, this hybrid approach represents a robust solution for scaling personalized education in exact sciences, ensuring that feedback is both semantically consistent and mathematically accurate.