# GM-PRM: A Generative Multimodal Process Reward Model for Multimodal Mathematical Reasoning
> Corresponding author.
Abstract
Multimodal Large Language Models (MLLMs) demonstrate remarkable capabilities but often struggle with complex, multi-step mathematical reasoning, where minor errors in visual perception or logical deduction can lead to complete failure. While Process Reward Models (PRMs) offer step-by-step supervision, existing multimodal PRMs are limited to being binary verifiers that can identify but not correct errors, offering little explanatory power. To address these deficiencies, we introduce the Generative Multimodal Process Reward Model (GM-PRM), a novel paradigm that transforms the PRM from a passive judge into an active reasoning collaborator. Instead of a simple scalar score, GM-PRM provides a fine-grained, interpretable analysis of each reasoning step, evaluating its step intent, visual alignment, and logical soundness. More critically, GM-PRM is trained to generate a corrected version of the first erroneous step it identifies. This unique corrective capability enables our new test-time inference strategy, Refined Best-of-N (Refined-BoN). This framework actively enhances solution quality by using the PRMâs generated correction to guide the policy model toward a more promising reasoning trajectory, thereby improving the diversity and correctness of the solution pool. We demonstrate that GM-PRM achieves state-of-the-art results on multiple multimodal math benchmarks, significantly boosting policy model performance with remarkable data efficiency, requiring only a 20K-sample training dataset. Our code will be released upon acceptance.
1 Introduction
The advent of Multimodal Large Language Models (MLLMs) has marked a significant milestone in artificial intelligence, demonstrating remarkable capabilities in integrating and understanding visual and textual information (Caffagni et al. 2024; Yan et al. 2024c; Yan and Lee 2024; Huo et al. 2024; Zheng et al. 2024b). While these models excel at general-purpose tasks such as image captioning and visual question answering, they often falter when confronted with complex, multi-step reasoning challenges, particularly within specialized domains like mathematics (Wang et al. 2024; Yan et al. 2024a, 2025a; Ahn et al. 2024). Solving multimodal mathematical problems requires not only accurate perception of visual elements (e.g., geometric figures, function graphs) but also a rigorous, step-by-step logical deduction process (Shi et al. 2024; Zhuang et al. 2025; Yan et al. 2025b). Minor errors in either image interpretation or logical inference can cascade, leading to entirely incorrect final answers.
<details>
<summary>figures/comparison.png Details</summary>
### Visual Description
## Diagram: Comparison of Reward Models
### Overview
The image presents a comparative diagram illustrating three different reward models: Outcome Reward Model (ORM), Multimodal Process Reward Model (PRM), and GM-PRM (a proposed model). The diagram highlights the flow of information and the reward mechanisms associated with each model.
### Components/Axes
* **Title:** The image is divided into three sections, each representing a different reward model.
* (a) Outcome Reward Model
* (b) Multimodal Process Reward Model
* (c) GM-PRM (Ours)
* **Input:** Each model starts with an input.
* ORM: "Question" (represented by a blue box containing the equation E=mc²)
* PRM and GM-PRM: "Multimodal Math Qns" (represented by a blue box containing math equations)
* **Process:** The models process the input through a series of steps.
* ORM: A single "Output" arrow leading to "Answer" (green box).
* PRM and GM-PRM: A series of steps labeled "Step 1", "Step 2", ..., "Step T" (green boxes) leading to "Answer" (green box).
* **Reward:** Each model includes a reward mechanism.
* ORM, PRM, and GM-PRM: A robot icon labeled "ORM", "PRM", and "GM-PRM" respectively, receiving a "Reward" (orange dashed arrow).
* **Annotations:** Additional text annotations provide context and highlight key differences.
* ORM: "ONLY Reward Final Ans" (red text)
* PRM: "Limited Explainability", "No Correction Mechanism" (red text)
* GM-PRM: "1st incorrect step" (purple text), "After Correction" (purple arrow), "Refined BoN" (green text), "Refined & Corrected Version" (purple text)
* **Analysis & Judgement (GM-PRM):** A purple box at the bottom of the GM-PRM section contains the labels "Step Intent", "Image Alignment", and "Reasoning Logic".
### Detailed Analysis
**Outcome Reward Model (ORM):**
* The model takes a "Question" as input.
* The question is processed to produce an "Answer".
* The "ORM" agent receives a "Reward" only based on the final answer.
* The annotation "ONLY Reward Final Ans" emphasizes that the reward is solely based on the outcome.
**Multimodal Process Reward Model (PRM):**
* The model takes "Multimodal Math Qns" as input.
* The input is processed through a series of steps: "Step 1", "Step 2", ..., "Step T".
* The final step leads to an "Answer".
* The "PRM" agent receives a "Reward" based on the process.
* The annotations "Limited Explainability" and "No Correction Mechanism" highlight the limitations of this model.
**GM-PRM (Ours):**
* The model takes "Multimodal Math Qns" as input.
* The input is processed through a series of steps: "Step 1", "Step 2" (purple), ..., "Step T".
* The annotation "1st incorrect step" indicates a point where a correction mechanism is applied.
* The "After Correction" arrow shows the flow after a correction.
* The final step leads to an "Answer".
* The "GM-PRM" agent receives a "Reward" based on the refined process.
* The "Refined BoN" annotation suggests a refined version of something (likely a "Bag of Neurons" or similar concept).
* The "Refined & Corrected Version" annotation emphasizes the improvements made in this model.
* The "Analysis & Judgement" box indicates the model's ability to analyze and judge the steps involved.
### Key Observations
* The ORM is the simplest model, focusing only on the final outcome.
* The PRM considers the process but lacks explainability and a correction mechanism.
* The GM-PRM builds upon the PRM by incorporating a correction mechanism and analysis/judgment capabilities.
### Interpretation
The diagram illustrates the evolution of reward models for problem-solving, particularly in the context of multimodal math questions. The GM-PRM model is presented as an improvement over existing models by incorporating a correction mechanism and analysis/judgment capabilities. This suggests that the GM-PRM is more robust and capable of handling complex problems compared to the ORM and PRM models. The "Refined BoN" annotation implies that the GM-PRM utilizes a more sophisticated approach to processing information and generating solutions. The diagram highlights the importance of considering the process and incorporating feedback mechanisms in reward models for complex tasks.
</details>
Figure 1: Comparison among ORM (a), PRM (b), and our proposed GM-PRM (c) for multimodal math reasoning.
To mitigate these reasoning deficiencies, Process Reward Models (PRMs) have emerged as a promising paradigm (Gao et al. 2024; Zhong et al. 2025). Unlike outcome-based models that only reward correct final answers (shown in Figure 1 (a)), PRMs provide fine-grained supervision by evaluating the correctness of each intermediate step in a reasoning chain (Zheng et al. 2024a; Lambert et al. 2024; Yan et al. 2024b), as shown in Figure 1 (b). This approach has proven effective in the language domain (Zeng et al. 2025; Yuan et al. 2024; Zhang et al. 2025a). However, extending PRMs to the multimodal context presents unique challenges (Miao et al. 2025; Du et al. 2025; Li et al. 2025b; Cao et al. 2025). Existing multimodal PRMs often function as binary classifiers, assigning a simple correct/incorrect label to each step, which offers limited explanatory power. Furthermore, they typically possess the ability to identify errors but lack the mechanism to correct them, leaving the reasoning process fundamentally broken. This limitation constrains their utility, especially within mechanisms like Best-of-N (BoN) sampling, which remain passive selection processes over a static set of potentially flawed solutions.
In this work, we introduce a novel G enerative M ultimodal P rocess R eward M odel (GM-PRM) to address these limitations, as illustrated in Figure 1 (c). Our model transcends the role of a simple verifier and acts as an active reasoning collaborator. Instead of merely outputting a scalar score, our GM-PRM leverages its generative capabilities to produce a detailed, interpretable analysis for each reasoning step. This analysis deconstructs the step into three critical aspects: its fundamental step intent, the correctness of its image alignment, and the soundness of its reasoning logic. More importantly, our model is trained not only to identify errors but also to generate a refined, corrected version of the first incorrect step it encounters.
This unique corrective capability enables us to propose a new test-time inference strategy: the Refined Best-of-N (Refined-BoN) process. This dynamic framework moves beyond passive selection by actively enhancing the quality of candidate solutions. When our GM-PRM identifies a flawed step within a generated solution, it intervenes by providing a corrected step, which is then used to guide the policy model in generating a new, more promising reasoning trajectory. This iterative refinement process significantly improves the diversity and correctness of the solution pool, leading to a substantial boost in the policy modelâs problem-solving performance. Furthermore, we demonstrate that this powerful capability can be achieved with remarkable data efficiency, requiring a significantly smaller training dataset than previous approaches. Our primary contributions are as follows:
- We develop a generative multimodal PRM called GM-PRM that provides fine-grained, interpretable feedback for mathematical reasoning. It analyzes each stepâs purpose, image alignment, and logical validity, moving beyond simple binary classification to offer deeper insight into the modelâs thought process.
- We introduce a novel Refined-BoN framework that leverages the PRMâs generative power to actively correct errors at test time. It enhances the policy modelâs ability to find correct solutions by iteratively improving flawed reasoning paths.
- We demonstrate the effectiveness and data efficiency of GM-PRM, achieving state-of-the-art results on multiple multimodal math benchmarks. Our approach requires only a 20K sample dataset, highlighting the quality of data curation and the power of generative supervision strategy.
2 Related Work
Process Reward Models (PRMs)
PRMs have been proposed to evaluate the fine-grained step level for model reasoning. During the implementation process, annotating and obtaining a high-quality training dataset incurs a high cost. PRM800K (Lightman et al. 2023) is the first process supervision dataset completely annotated by humans. To mitigate annotation costs, Math-Shepherd (Wang et al. 2023) proposes Monte Carlo (MC) estimation, while OmegaPRM (Luo et al. 2024) leverages Monte Carlo Tree Search (MCTS) to automatically evaluate each reasoning step, both utilizing the generation capabilities of Large Language Models (LLMs). Subsequent research has enhanced the effectiveness of PRMs through various methods, including VersaPRM (Zeng et al. 2025), Implicit PRM (Yuan et al. 2024), OpenPRM (Zhang et al. 2025a), PQM (Li and Li 2024), PAV (Setlur et al. 2024), and others. In addition, GenRM (Zhao et al. 2025) utilizes the generation ability of reward models to analyze each reasoning step and obtain the score of each step by taking the probability of the special evaluation token. Furthermore, GenPRM (Zhao et al. 2025), ThinkPRM (Khalifa et al. 2025), R-PRM (She et al. 2025) extend the method of using model generation analysis to evaluate steps to PRMs. There are also many studies on benchmarks of PRMs such as ProcessBench (Zheng et al. 2024a), PRMBench (Song et al. 2025), and Socratic-PRMBench (Li et al. 2025a).
Multimodal PRMs
After achieving certain results and progress in the research of language modality in PRMs, research on PRMs has also begun to shift towards multimodal tasks. M-STAR (Liu et al. 2024a) proposes and implements multimodal PRM on multimodal problems. URSA (Luo et al. 2025) constructs a dataset by inserting errors and utilizes it to train a multimodal PRM. VisualPRM (Wang et al. 2025b) not only uses MC estimation to construct a multimodal VisualPRM400K training dataset, but also proposes a benchmark for multimodal PRMs called VisualProcessBench, which is entirely annotated by humans. Athena-PRM (Wang et al. 2025a) proposes using prediction consistency between strong and weak completers to enhance the quality of automatically annotated data based on MC estimation and improving multimodal PRM by ORM initialization and negative data up-sampling. Moreover, PRM-BAS (Hu et al. 2025), MM-PRM (Du et al. 2025) and DreamPRM (Cao et al. 2025) also improve the capability of multimodal PRMs. Although several studies have explored multimodal PRMs, applying them to multimodal tasks effectively remains certain challenges, such as insufficient interpretability of the labels assigned to each reasoning step and the inability to correct identified erroneous steps. In our work, we introduce a generative multimodal PRM, GM-PRM to solve the above problems.
<details>
<summary>figures/Case.png Details</summary>
### Visual Description
## Question Analysis: Geometry Translation
### Overview
The image presents a geometry problem involving the translation of a triangle on a coordinate plane. It includes a visual representation of the triangle, the problem statement, and two proposed solutions: an "Original Solution" and a "Refined Solution." The image also contains an analysis and judgement section (GM-PRM) that evaluates the original solution.
### Components/Axes
* **Question (Visual Part):** A coordinate plane with x and y axes ranging from 0 to 10 and 0 to 8 respectively. Triangle ABC is plotted on the plane.
* Vertex A is located at (3, 4).
* Vertex B is located at (1, 1).
* Vertex C is located at (4, 2).
* **Question (Textual Part):** The problem statement describes a translation of triangle ABC: "As shown in the diagram, triangle ABC is first translated 4 units to the right and then 3 units upward. The position of vertex A after the translation, represented as a coordinate pair, is ( )."
* Multiple-choice options are provided:
* A. (5, 4)
* B. (8, 5)
* C. (7, 7)
* D. No correct answer
* **Original Solution:** A step-by-step solution that incorrectly identifies the initial coordinates of vertex A.
* Step 1: General statement about translation.
* Step 2: Incorrectly claims vertex A is at the origin (0, 0).
* **Refined Solution:** A step-by-step solution that correctly identifies the initial coordinates of vertex A and applies the translation.
* Step 2: Correctly identifies vertex A's initial coordinates as (3, 4).
* Step 3: Applies the translation of 4 units to the right, resulting in (7, 4).
* Step 6: States the final answer as C.
* **GM-PRM: Analysis & Judgement:** An analysis of the original solution.
* Step intent analysis: Identifies the initial coordinates of vertex A.
* Image alignment analysis: Correctly states that the image shows vertex A at (3, 4), not (0, 0).
* Reasoning logic analysis: Incomplete.
### Detailed Analysis or Content Details
* **Visual Part:** The coordinate plane provides a visual representation of the triangle's initial position. The grid lines are spaced at integer intervals.
* **Textual Part:** The problem statement clearly defines the translation to be performed.
* **Original Solution:** The error lies in Step 2, where it incorrectly assumes vertex A is at the origin.
* **Refined Solution:** The refined solution correctly identifies the initial coordinates and applies the translation. Adding 4 to the x-coordinate of (3, 4) results in (7, 4). Then adding 3 to the y-coordinate of (7, 4) results in (7, 7).
* **GM-PRM:** The analysis correctly identifies the error in the original solution's Step 2.
### Key Observations
* The "Original Solution" makes a fundamental error in identifying the initial coordinates of vertex A.
* The "Refined Solution" provides a correct step-by-step solution to the problem.
* The GM-PRM analysis accurately pinpoints the error in the "Original Solution."
### Interpretation
The image highlights the importance of accurately interpreting visual information in geometry problems. The "Original Solution" fails because it misidentifies the coordinates of vertex A on the coordinate plane. The "Refined Solution" demonstrates the correct application of translation principles. The GM-PRM analysis serves as a validation tool, confirming the error in the initial approach. The problem demonstrates a simple geometric translation, but emphasizes the need for careful observation and accurate application of geometric principles.
</details>
Figure 2: The illustration of a representative example before and after applying GM-PRM. In particular, GM-PRM first judges the steps of the original solution generated by the policy model. Subsequently, GM-PRM finds that the second step is incorrect and refines the second step to generate the correct version. The correct steps are input to the policy model to generate the refined solution, and finally the correct answer is obtained.
3 Methodology
In this section, we first describe how to utilize PRMs and generative PRMs combined with the BoN method to improve the performance of policy models for mathematical problems in Section 3.1. Then, we introduce our process to implement multimodal generative PRM, including data construction and model training in Section 3.2. Finally, we propose a novel Refined-BoN framework for PRMs to enhance its performance beyond traditional BoN method in Section 3.3.
3.1 PRMs for Mathematical Problem
In this section, we present the implementation methods of PRM and GM-PRM, and provide formal explanations of their usage through mathematical notation.
Problem and Reasoning Steps Generation
Let $Q$ denote a mathematical problem. Firstly, an LLM $\pi$ is involved in solving the mathematical problem $Q$ . To facilitate reasoning, the problem is combined with a prompt $P$ , which includes specific instructions guiding the generation of a step-by-step reasoning process and a final answer. This composite input is then fed into the LLM. When generating a response, $\pi$ generates a sequence of reasoning steps, denoted as $R=\{r_{1},r_{2},...,r_{T}\}$ , where $T$ represents the total number of reasoning steps to the given mathematical problem. The above process can be explained as follows:
$$
R=\pi(Q\parallel P), \tag{1}
$$
where $\parallel$ denotes the concatenation of the problem $Q$ and the prompt $P$ , and $\pi(¡)$ represents the inference of LLM.
PRM
A single instance in a training dataset $\mathcal{D}$ to train a PRM comprises three components: (1) a problem statement, (2) a generated response consisting of multiple inference steps, and (3) a corresponding set of binary labels, each taking a value of either 0 or 1, indicating whether the associated reasoning step is incorrect or correct, respectively.
During training, the PRM is optimized using cross-entropy loss and supervised to align its predictions with the ground-truth labels. After being trained, the PRM model is capable of processing new reasoning steps generated by the LLM in response to a given mathematical problem, which means that the PRM is able to assign a scalar score to each individual reasoning step, reflecting the modelâs confidence in the correctness of each step:
$$
f_{\text{PRM}}:(Q,R)\mapsto(s_{1},s_{2},\dots,s_{T}), \tag{2}
$$
where $f_{\text{PRM}}:(¡)$ represents the inference of PRM, $s_{i}â[0,1]$ denotes the confidence score assigned to the $i$ -th reasoning step $r_{i}$ , and $T$ denotes the number of reasoning steps.
For generative PRM, the binary labels in the training dataset are replaced with textual analyses and judgments, each formulated as a textual choice such as âincorrectâ or âcorrectâ. During inference, generative PRM also generates textual critiques and judgments for each step.
GM-PRM
By extending generative PRMs from the textual modality to a multimodal setting, we introduce GM-PRM. In this setting, mathematical problems are represented using both textual and visual information. The input to the policy model comprises the image of the problem, its textual description and task-specific instructions, which are processed jointly to generate reasoning steps. Similarly, during both training and inference, it is essential to provide GM-PRM with inputs from both visual and textual modalities, enabling it to perform cross-modal analysis when assigning correctness labels to each reasoning step:
$$
f_{\text{{GM-PRM}}}:(Q,I,R)\mapsto(c_{1},j_{1},\dots,c_{T},j_{T}), \tag{3}
$$
where $f_{\text{{GM-PRM}}}:(¡)$ represents the inference of GM-PRM, $I$ denotes the image of the mathematical problem, $c_{i}$ denotes the critique of the $i$ -th reasoning step $r_{i}$ , and $j_{i}$ denotes the textual judgment assigned to the $i$ -th reasoning step $r_{i}$ .
3.2 Data Construction
In this section, we present our methodology employed to construct the training data for GM-PRM. The process consists of three key stages: (1) the selection of appropriate types and quantities of question data from the VisualPRM400K dataset (Wang et al. 2025b); (2) the generation of textual analysis and judgment data using GPT-4o; and (3) the filtering of the generated data through MC estimation and LLM-as-a-judge techniques to ensure quality and reliability.
Data Selection
VisualPRM400K is a large-scale dataset containing approximately 400,000 multimodal process supervision samples. In our work, we select plane geometry- and function-related problems from VisualPRM400K to construct a specialized subset and supplement it with corresponding textual analysis for training GM-PRM. This targeted subset with textual critiques supports the effective training of GM-PRM, yielding strong performance on geometric and function-based mathematical reasoning tasks.
Generation of Analysis and Judgment
To obtain textual analyses and judgments, we employ GPT-4o to critique each reasoning step from 4 key aspects: step intent, image alignment, reasoning logic, and step refinement.
The aspect of step intent indicates identifying the purpose of each reasoning step. This initial analysis establishes a foundation that allows GM-PRM to interpret and evaluate each reasoning step in context more effectively. Furthermore, this level of understanding facilitates subsequent error detection and correction tasks, thereby enhancing the overall effectiveness of GM-PRM.
The second aspect is image alignment. When MLLMs are used for inference in solving multimodal problems, MLLMs often make errors in image alignment, such as misidentifying parallel relationships or incorrectly annotating angles, which leads to flawed solutions. To address this, we employ GPT-4o to produce textual analysis and judgments in image alignment for inference steps, to form the dataset for training GM-PRM.
Reasoning logic is an indispensable presence in the step-by-step problem-solving process of MLLMs. However, the occurrences of logical inconsistencies and errors, such as miscalculations and incorrect inferences significantly impact the correctness of the reasoning steps and the final answers. Therefore, it is crucial for GM-PRM to be capable of identifying such logical flaws and making accurate judgments regarding the validity of the reasoning logic for each step. In our work, we employ GPT-4o to generate textual analysis and judgments of each step in reasoning logic to form the training dataset. The above process can be formulated as follows:
$$
\mathcal{F}:(Q,I,R\parallel P)\mapsto\{SI_{i},IA_{i},RL_{i},FJ_{i}\}_{i=1}^{t}, \tag{4}
$$
where $\mathcal{F}:(¡)$ represents the inference of GPT-4o, $SI_{i}$ denotes the textual analysis of step intent for the $i$ -th reasoning step, $t$ denotes the number of the first incorrect step or the last step, $1â¤slant tâ¤slant T$ , $IA_{i}=\{IAC_{i},IAJ_{i}\}$ denotes the analysis which contains critique $IAC_{i}$ and judgment $IAJ_{i}$ in image alignment of the $i$ -th reasoning step, $RL_{i}=\{RLC_{i},RLJ_{i}\}$ denotes the analysis which contains critique $RLC_{i}$ and judgment $RLJ_{i}$ in image alignment of the $i$ -th reasoning step, $FJ_{i}$ denotes the final judgment of the $i$ -th reasoning step.
Building on aforementioned three aspects, we further aim for GM-PRM to correct the first identified erroneous step. The above information enables GM-PRM to generate corrected reasoning steps that are logically coherent, visually accurate, and semantically aligned with the original step intent. The resulting corrected steps can then be used to construct more diverse and accurate inference solutions and ultimately produce more reliable final answers. In our work, we employ GPT-4o to generate a corrected version of the first identified error step in a reasoning process if the first error step is detected to exist by GPT-4o:
$$
\mathcal{F}:(Q,I,R\parallel P)\mapsto\begin{cases}RS,&\text{if incorrect step exists},\\
\emptyset,&\text{otherwise}.\end{cases} \tag{5}
$$
where $RS$ denotes refined step of the first error step in a reasoning process.
In summary, we design a structured prompt for GPT-4o to generate comprehensive analysis data across four dimensions based on the provided problems, associated images, and step-by-step solutions:
$$
\mathcal{F}:(Q,I,R\parallel P)\mapsto\mathcal{D}, \tag{6}
$$
where $\mathcal{D}$ denotes the generated training dataset:
$$
\mathcal{D}=\{(\{SI_{i}^{k},IA_{i}^{k},RL_{i}^{k},FJ_{i}^{k}\}_{i=1}^{t},RS^{k})\}_{k=1}^{K}, \tag{7}
$$
where $kâ\{1,2,...,K\}$ represents the $k$ -th sample in the dataset, and $K$ denotes the number of the training instances.
Data Filtering
The process of constructing training data using GPT-4o can be regarded as an implementation of LLM-as-a-judge methodology. Inspired by the combination of LLM-as-a-judge and MC estimation techniques (Zhang et al. 2025b), we employ the MC estimation technique proposed by Math-Shepherd (Wang et al. 2023) to effectively filter and curate the generated data.
Monte Carlo estimation is a strategy for automated annotation that leverages LLMs or MLLMs to generate multiple subsequent solutions for each step. When applying MC estimation to evaluate a step $r_{i}$ , we use an LLM or an MLLM as a âcompleterâ to finalize multiple subsequent reasoning processes from this step:
$$
f_{comleter}\mapsto\{(r_{i+1}^{j},\dots,r_{L_{j}}^{j},a^{j})\}_{j=1}^{m}, \tag{8}
$$
where $a^{j}$ is the final answer of the $j$ -th finalized solution and $L_{j}$ is the total number of steps.
Within MC estimation, one type of evaluation method is commonly applied: hard estimation. In hard estimation, a step $r_{i}$ is deemed correct if at least one subsequent solution reaches the correct final answer $a^{*}$ ; otherwise, it is considered incorrect:
$$
l_{i}^{HE}=\begin{cases}1,&\exists a_{j},a_{j}=a^{*},\\
0,&\text{otherwise}.\end{cases} \tag{9}
$$
In our data construction process, we employ hard estimation to label the correctness of individual reasoning steps. By integrating LLM-as-a-judge technique and MC estimation, we compare the labels acquired by MC estimation and judgments generated by GPT-4o. Data samples that receive consistent evaluations from both methods are selected as our final training dataset. By integrating these two methods, we aim to further enhance the reliability and quality of the training data, ensuring better performance of GM-PRM.
3.3 Refined-BoN Process
When applying Test-time Scaling (TTS) for LLMs and MLLMs, a widely adopted method is Best-of-N (BoN) approach. In the BoN process, a policy model is employed to generate N candidate solutions, which are then evaluated by reward models or self-consistency to select the optimal solution. However, during the BoN process, policy models are under identical prompting conditions when generating multiple solutions, which leads to the problem that the solutions often lack diversity and may exhibit limited correctness. In our work, we propose a novel Refined-BoN framework utilizing TTS techniques to enhance the diversity and accuracy of generated solutions, thereby improving the reasoning capabilities of policy models.
Refined-BoN Method
As shown in Figure 2, in Refined-BoN process, we first employ an MLLM as the policy model to generate $N/2$ initial solutions to a multimodal problem, and then these solutions are evaluated step-by-step by GM-PRM. For the subsequent $N/2$ solutions, the policy model generates them under varying conditions, informed by the evaluation of the preceding $N/2$ solutions: If GM-PRM identifies an incorrect reasoning step within a solution, it stops evaluating and refines the first erroneous step by generating a corrected version. This corrected step, along with all previously validated correct steps, is then input back into the policy model to continue the solution generation process. Conversely, if GM-PRM determines that all steps in a particular solution are correct, we employ the policy model to generate a new solution using the same prompt. Through this regeneration mechanism, we obtain the additional $N/2$ solutions. Subsequently, we employ GM-PRM to evaluate the subsequent $N/2$ solutions.
Solution Selection
After applying the Refined-BoN process, we obtain $N$ solutions for each problem, each accompanied by step-level correctness judgments. Moreover, we divide all the solutions into two categories: one where GM-PRM judges that it contains incorrect steps, and the other where GM-PRM judges that all its steps are correct. Furthermore, we take the probability of GM-PRM generating the associated âCorrectâ and âIncorrectâ tokens as the score of each step.
Among the $N$ generated solutions, if there exist solutions in which all reasoning steps are judged correct, we calculate the average of the scores of all steps in these solutions as the overall score of the solution, and select the solution with the highest average score as the optimal solution.
For N solutions to a problem, if GM-PRM determines that all $N$ solutions contain incorrect steps, we calculate the average score of all steps in a solution as the overall score of the solution, and select the solution with the highest overall score as the final answer.
4 Experiments
| MiniCPM-V2.6-8B + GM-PRM (Ours) Improvements | 44.3 51.0 +6.7 | 16.0 18.1 +2.1 | 18.9 24.4 +5.5 | 22.6 25.7 +3.1 | 38.6 51.0 +12.4 | 28.1 34.0 +5.9 |
| --- | --- | --- | --- | --- | --- | --- |
| Llama-3.2-11B-Vision | 44.5 | 14.3 | 16.5 | 28.4 | 46.1 | 30.0 |
| + GM-PRM (Ours) | 49.5 | 18.2 | 18.8 | 32.7 | 53.4 | 34.5 |
| Improvements | +5.0 | +3.9 | +2.3 | +4.3 | +7.3 | +4.5 |
| Qwen2.5-VL-7B | 63.2 | 25.1 | 32.8 | 35.0 | 60.6 | 43.3 |
| + GM-PRM (Ours) | 65.0 | 28.2 | 37.4 | 39.2 | 69.0 | 47.8 |
| Improvements | +1.8 | +3.1 | +4.6 | +4.2 | +8.4 | +4.5 |
| InternVL3-8B | 50.6 | 20.3 | 25.0 | 27.0 | 50.9 | 34.8 |
| + GM-PRM (Ours) | 55.7 | 22.2 | 31.7 | 33.4 | 59.2 | 40.4 |
| Improvements | +5.1 | +1.9 | +6.7 | +6.4 | +8.3 | +5.6 |
| InternVL3-38B | 68.3 | 34.9 | 37.8 | 40.1 | 66.4 | 49.5 |
| + GM-PRM (Ours) | 69.9 | 37.0 | 39.1 | 43.1 | 72.9 | 52.4 |
| Improvements | +1.6 | +2.1 | +1.3 | +3.0 | +6.5 | +2.9 |
| InternVL3-78B | 68.0 | 34.6 | 36.0 | 38.1 | 65.7 | 48.5 |
| + GM-PRM (Ours) | 70.7 | 37.1 | 40.6 | 39.9 | 72.2 | 52.1 |
| Improvements | +2.7 | +2.5 | +4.6 | +1.8 | +6.5 | +3.6 |
Table 1: Percentage accuracy scores (%) of multiple MLLMs across five datasets. For each MLLM, the first row shows the baseline, the second shows the final result with GM-PRM, and the third shows the improvement. Only positive improvements are underlined. The best results are highlighted in bold. All values are reported after rounding to three decimal places.
In this section, we introduce our experimental setup to assess GM-PRM under the Refined-BoN process on five multimodal mathematical benchmarks in Section 4.1. In addition, we present the results of our experiments and three conclusions analyzed from the results in Section 4.2. Finally, we show the ablation studies in Section 4.3.
4.1 Experimental Setup
Benchmarks.
We evaluate GM-PRM across five datasets, including MathVista (Lu et al. 2023), MathVision (Wang et al. 2024), MathVerse (Zhang et al. 2024), DynaMath (Zou et al. 2024) and WeMath (Qiao et al. 2024). The datasets contain diverse problem types, such as plane geometry, functions, puzzle tests, etc. We use Vision-Only subset of MathVerse dataset and Plane-Geometry subset of DynaMath.
Settings.
We employ GM-PRM as the critic model for Refined-BoN evaluation and set N to 8 by default. For MLLMs, we select six models as the policy models to generate step-by-step reasoning processes. When reasoning, we set the temperature of the policy models to 0.7 and top-p to 0.9. For comparison, we use the average accuracy of N sets of answers generated by policy models as baselines.
Training Details.
To train GM-PRM, we use Qwen2.5-VL-7B-Instruct as our base model and perform supervised fine-tuning (SFT) with all parameters trainable except for the frozen Vision Transformer (ViT) encoder. During the training process, we utilize bfloat16 mixed-precision and DeepSpeed with zero3 technology and set the training consists of 2 epochs. For batch size, the batch size on each training device is set to 2, and through gradient accumulation, the effective batch size is extended to 16. Moreover, we use two A800 GPUs to train GM-PRM, and the AdamW optimizer is used with an initial learning rate of $1Ă 10^{-5}$ . The learning rate schedule involves a linear warm-up with the warm-up ratio equal to 0.05 followed by linear decay.
4.2 Main Results
As shown in Table 1, integrating GM-PRM with the Refined-BoN process consistently improves performance across five benchmark datasets for six different MLLMs. On average, our method yields notable accuracy gains, with improvements of +5.9 for MiniCPM-V2.6-8B, +4.5 for Llama-3.2-11B-Vision, +4.5 for Qwen2.5-VL-7B, and +5.6 for InternVL3-8B.
A closer look at dataset-level results reveals that the improvements are not uniform. The WeMath benchmark shows the most significant enhancement, with MiniCPM-V2.6-8B improving by +12.4 points, highlighting the ability of our method to strengthen mathematical reasoning on challenging problems. Similarly, MathVerse and DynaMath exhibit consistent gains of +4.5â6.7 points across multiple models, suggesting that our approach particularly benefits datasets requiring complex symbolic manipulation and multi-step reasoning. In contrast, MathVision improvements are more modest (+1.9â3.9), indicating that the visual reasoning component may already be relatively strong in baseline models.
GM-PRM combined with the Refined-BoN process demonstrates strong generalization across diverse multimodal mathematical problems, with particularly remarkable gains in plane geometry tasks. As illustrated in Figure 3, even after excluding plane geometry and function problems, policy models still achieve notable improvements across the datasets. This indicates that although GM-PRM is primarily trained on a dataset composed of plane geometry and function problems, it generalizes effectively to other types of multimodal mathematical problems. Moreover, as shown by the averaged results in Figure 3, the improvements achieved by GM-PRM with Refined-BoN on plane geometry problems consistently exceed those on the overall dataset, function problems, and other categories, underscoring the exceptional effectiveness of our method in tackling plane geometry tasks.
<details>
<summary>figures/Question_types.png Details</summary>
### Visual Description
## Bar Chart: Improvements of Accuracy
### Overview
The image is a bar chart comparing the improvements in accuracy (%) of three models: MiniCPM-V2.6-8B, InternVL3-8B, and InternVL3-78B. The chart displays the improvements in accuracy for "Overall", "Plane Geometry", "Functions", and "Others" categories.
### Components/Axes
* **X-axis:** Model names: MiniCPM-V2.6-8B, InternVL3-8B, InternVL3-78B.
* **Y-axis:** Improvements of accuracy (%), ranging from 0 to 10.
* **Legend:** Located at the top of the chart.
* Orange: Overall
* Blue: Plane Geometry
* Green: Functions
* Yellow: Others
### Detailed Analysis
**MiniCPM-V2.6-8B:**
* Overall (Orange): Approximately 5.9%
* Plane Geometry (Blue): Approximately 9.2%
* Functions (Green): Approximately 5.4%
* Others (Yellow): Approximately 1.3%
**InternVL3-8B:**
* Overall (Orange): Approximately 5.6%
* Plane Geometry (Blue): Approximately 6.4%
* Functions (Green): Approximately 0.5%
* Others (Yellow): Approximately 3.3%
**InternVL3-78B:**
* Overall (Orange): Approximately 3.6%
* Plane Geometry (Blue): Approximately 5.9%
* Functions (Green): Approximately 5.6%
* Others (Yellow): Approximately 1.0%
### Key Observations
* MiniCPM-V2.6-8B shows the highest improvement in "Plane Geometry" accuracy compared to the other models.
* InternVL3-8B has a very low improvement in "Functions" accuracy.
* InternVL3-78B shows a more balanced improvement across "Overall", "Plane Geometry", and "Functions" compared to InternVL3-8B.
* The "Others" category consistently shows the lowest improvement across all models.
### Interpretation
The bar chart provides a comparative analysis of the accuracy improvements achieved by three different models across four categories. MiniCPM-V2.6-8B excels in "Plane Geometry", while InternVL3-8B struggles with "Functions". InternVL3-78B offers a more balanced performance. The consistently low improvement in the "Others" category suggests that this area may require further optimization across all models. The data suggests that the models have different strengths and weaknesses depending on the specific task or category.
</details>
Figure 3: Improvements of the average percentage accuracy (%) of multiple MLLMs across different question types in MathVista, MathVision and MathVerse datasets.
The Refined-BoN process yields disproportionately larger gains for models with lower baseline performance. As shown in Table 1, InternVL3-38B starts with the highest initial average accuracy among all policy models (49.5%) and achieves a modest improvement of +2.9 points (+5.9%). In contrast, Qwen2.5-VL-7B, which has the highest baseline accuracy (43.3%) among models with fewer than 12 billion parameters, improves by +4.5 points (+10.4%), surpassing the relative gains of InternVL3-38B. Notably, MiniCPM-V-2.6-8B demonstrates the most significant relative improvement, achieving +5.9 points (+21.0%), despite its lower initial score. These results suggest that models with weaker baseline performance benefit more from the refinement mechanism of GM-PRM with Refined-BoN, likely because the process effectively corrects errors in reasoning steps, leaving greater room for improvement.
<details>
<summary>figures/MiniCPM-BoN.png Details</summary>
### Visual Description
## Chart: Accuracy vs. Number of Solutions per Problem
### Overview
The image is a line chart comparing the accuracy of two methods, "GM-PRM" and "Self-Consistency," as the number of solutions per problem increases. The x-axis represents the number of solutions per problem, and the y-axis represents the accuracy in percentage.
### Components/Axes
* **X-axis:** "# Solutions per Problem" with markers at 1, 4, 6, and 8.
* **Y-axis:** "Accuracy (%)" with markers at 38, 43, 48, and 53.
* **Legend:** Located on the right side of the chart.
* Blue line with circular markers: "GM-PRM"
* Orange line with circular markers: "Self-Consistency"
### Detailed Analysis
* **GM-PRM (Blue):** The accuracy of GM-PRM generally increases as the number of solutions per problem increases.
* At 1 solution: approximately 39%
* At 4 solutions: approximately 47%
* At 6 solutions: approximately 48%
* At 8 solutions: approximately 52%
* **Self-Consistency (Orange):** The accuracy of Self-Consistency also increases as the number of solutions per problem increases, but at a slower rate than GM-PRM.
* At 1 solution: approximately 39%
* At 4 solutions: approximately 44%
* At 6 solutions: approximately 45%
* At 8 solutions: approximately 46%
### Key Observations
* GM-PRM consistently outperforms Self-Consistency across all tested numbers of solutions per problem.
* Both methods show an increase in accuracy as the number of solutions per problem increases, but the rate of increase diminishes as the number of solutions grows.
* The largest increase in accuracy for both methods occurs between 1 and 4 solutions per problem.
### Interpretation
The chart suggests that increasing the number of solutions per problem improves the accuracy of both GM-PRM and Self-Consistency methods. However, GM-PRM appears to be more effective in leveraging additional solutions to achieve higher accuracy. The diminishing returns observed as the number of solutions increases indicate that there may be a point beyond which adding more solutions provides only marginal improvements in accuracy. The data implies that GM-PRM is a superior method for this particular task, given its consistently higher accuracy across different numbers of solutions.
</details>
(a) MiniCPM-V-2.6-8B
<details>
<summary>figures/QwenVL-BoN.png Details</summary>
### Visual Description
## Line Chart: Accuracy vs. Number of Solutions per Problem
### Overview
The image is a line chart comparing the accuracy of two methods, "GM-PRM" and "Self-Consistency," as the number of solutions per problem increases. The x-axis represents the number of solutions per problem, ranging from 1 to 8. The y-axis represents accuracy, ranging from 60% to 70%.
### Components/Axes
* **X-axis:** "# Solutions per Problem" with markers at 1, 4, 6, and 8.
* **Y-axis:** "Accuracy (%)" with markers at 60, 62, 64, 66, 68, and 70.
* **Legend:** Located on the right side of the chart.
* Blue line with circular markers: "GM-PRM"
* Orange line with circular markers: "Self-Consistency"
### Detailed Analysis
* **GM-PRM (Blue):** The accuracy of GM-PRM increases as the number of solutions per problem increases.
* At 1 solution: approximately 61%
* At 4 solutions: approximately 66%
* At 6 solutions: approximately 68%
* At 8 solutions: approximately 69%
* **Self-Consistency (Orange):** The accuracy of Self-Consistency also increases as the number of solutions per problem increases, but at a slower rate than GM-PRM.
* At 1 solution: approximately 60.5%
* At 4 solutions: approximately 64.5%
* At 6 solutions: approximately 65.8%
* At 8 solutions: approximately 66.5%
### Key Observations
* GM-PRM consistently outperforms Self-Consistency across all tested numbers of solutions per problem.
* The increase in accuracy for GM-PRM appears to slow down as the number of solutions increases from 6 to 8.
* The increase in accuracy for Self-Consistency appears to be relatively linear across the tested range.
### Interpretation
The chart suggests that increasing the number of solutions per problem improves the accuracy of both GM-PRM and Self-Consistency methods. However, GM-PRM demonstrates a higher accuracy overall and a more significant initial improvement with increasing solutions. The diminishing returns observed for GM-PRM between 6 and 8 solutions might indicate a point of saturation or a need for further optimization beyond this range. The consistent, albeit slower, improvement of Self-Consistency suggests it may benefit from further exploration with even higher numbers of solutions per problem.
</details>
(b) Qwen2.5-VL-7B
<details>
<summary>figures/InternVL3-78B-BoN.png Details</summary>
### Visual Description
## Line Chart: Accuracy vs. Number of Solutions per Problem
### Overview
The image is a line chart comparing the accuracy of two methods, GM-PRM and Self-Consistency, as the number of solutions per problem increases. The x-axis represents the number of solutions per problem, ranging from 1 to 8. The y-axis represents the accuracy in percentage, ranging from 65% to 73%.
### Components/Axes
* **X-axis:** "# Solutions per Problem" with markers at 1, 4, 6, and 8.
* **Y-axis:** "Accuracy (%)" with markers at 65, 67, 69, 71, and 73.
* **Legend:** Located on the right side of the chart.
* Blue line: GM-PRM
* Orange line: Self-Consistency
### Detailed Analysis
* **GM-PRM (Blue Line):**
* The line starts at approximately 66% accuracy with 1 solution per problem.
* It increases sharply to approximately 71% accuracy at 4 solutions per problem.
* It increases slightly to approximately 71.5% accuracy at 6 solutions per problem.
* It increases slightly to approximately 72.5% accuracy at 8 solutions per problem.
* Trend: Generally increasing, with a significant jump between 1 and 4 solutions.
* **Self-Consistency (Orange Line):**
* The line starts at approximately 66% accuracy with 1 solution per problem.
* It increases to approximately 68% accuracy at 4 solutions per problem.
* It remains at approximately 68% accuracy at 6 and 8 solutions per problem.
* Trend: Increases initially, then plateaus.
### Key Observations
* GM-PRM consistently outperforms Self-Consistency across all numbers of solutions per problem.
* The accuracy of GM-PRM increases more significantly than Self-Consistency as the number of solutions increases from 1 to 4.
* Self-Consistency plateaus after 4 solutions per problem.
### Interpretation
The chart suggests that the GM-PRM method is more effective than Self-Consistency in improving accuracy as the number of solutions per problem increases. The significant increase in accuracy for GM-PRM between 1 and 4 solutions indicates that this method benefits more from having multiple solutions to consider. The plateauing of Self-Consistency suggests that it reaches a point of diminishing returns after a certain number of solutions. The data demonstrates that GM-PRM is a superior method for this particular problem, especially when multiple solutions are available.
</details>
(c) InternVL3-78B
Figure 4: The results of changing the value of N in the Refined-BoN process on the WeMath across different policies. As N increases, the effectiveness of GM-PRM in enhancing accuracy improves and surpasses that of Self-Consistency.
4.3 Hyperparameter & Ablation Study
Number of solution samples N in Refined BoN.
Following Test-time Scaling technique, we vary the number of N in the Refined-BoN process to evaluate the performance of GM-PRM in comparison to the Self-Consistency baseline.
Figure 4 depicts WeMath accuracy as the number of sampled solutions per problem ( $N$ ) increases from 1 to 8. Across all three backbonesâMiniCPMâVâ2.6â8B, Qwen2.5âVLâ7B, and InternVL3â78Bâboth GM-PRM and the SelfâConsistency (SC) baseline benefit from a larger sampling budget, yet GM-PRM exhibits a noticeably steeper growth curve.
Under the widely adopted Bestâofâ8 setting, GM-PRM delivers gains of 4.9 and 3.5 over SC on MiniCPMâVâ2.6â8B and Qwen2.5âVLâ7B, respectively. Even for the 78Bâparameter InternVL3, GM-PRM maintains a substantial 4.1 margin. These results indicate that the proposed refinement strategy not only scales to larger models but also converts additional candidate solutions into accuracy more effectively than SelfâConsistency, thereby underscoring the robustness and versatility of GM-PRM.
Furthermore, for MiniCPM-V-2.6-8B, GM-PRM surpasses the self-consistency baseline by 2.1, 2.2, and 4.9 points under the Best-of-4, Best-of-6, and Best-of-8 settings, respectively, indicating a steadily increasing performance gap between GM-PRM and self-consistency as N increases.
Methods for aggregating step scores.
For PRMs, the method used to aggregate step scores into an overall solution score plays a critical role. In this part, we compare several different aggregation strategies, including averaging step scores, selecting the maximum step score, and selecting the minimum step score. Since step-by-step solutions that contain steps judged incorrect are often not evaluated or scored for all steps, this experiment focuses exclusively on solutions where all steps are judged correct.
<details>
<summary>figures/Aggregation.png Details</summary>
### Visual Description
## Bar Chart: Model Accuracy Comparison
### Overview
The image is a bar chart comparing the accuracy of three different models: MiniCPM-V-2.6-8B, Qwen2.5-VL-7B, and InternVL3-8B. For each model, the chart displays four different accuracy metrics: Original, Average, Min, and Max. The y-axis represents accuracy in percentage, ranging from 0% to 60%.
### Components/Axes
* **X-axis:** Model names: MiniCPM-V-2.6-8B, Qwen2.5-VL-7B, InternVL3-8B.
* **Y-axis:** Accuracy (%), with scale markers at 0, 10, 20, 30, 40, 50, and 60.
* **Legend:** Located at the top of the chart, indicating the color-coding for each accuracy metric:
* Original: Grey
* Average: Blue
* Min: Teal
* Max: Light Green
### Detailed Analysis
**MiniCPM-V-2.6-8B:**
* Original (Grey): Approximately 28%
* Average (Blue): Approximately 34%
* Min (Teal): Approximately 34%
* Max (Light Green): Approximately 30%
**Qwen2.5-VL-7B:**
* Original (Grey): Approximately 43%
* Average (Blue): Approximately 48%
* Min (Teal): Approximately 47%
* Max (Light Green): Approximately 43%
**InternVL3-8B:**
* Original (Grey): Approximately 35%
* Average (Blue): Approximately 40%
* Min (Teal): Approximately 40%
* Max (Light Green): Approximately 37%
### Key Observations
* Qwen2.5-VL-7B generally has the highest accuracy across all metrics compared to the other two models.
* For all models, the "Average" and "Min" accuracy values are very close to each other.
* The "Original" accuracy is consistently lower than the "Average" and "Min" accuracy for all models.
* The "Max" accuracy is lower than "Average" and "Min" for MiniCPM-V-2.6-8B and InternVL3-8B, but similar to "Original" for Qwen2.5-VL-7B.
### Interpretation
The bar chart provides a comparative analysis of the accuracy of three different models under different conditions (Original, Average, Min, Max). The data suggests that Qwen2.5-VL-7B performs better overall in terms of accuracy compared to MiniCPM-V-2.6-8B and InternVL3-8B. The differences between "Average," "Min," and "Max" accuracy might indicate the variability in the model's performance under different test conditions or datasets. The "Original" accuracy likely represents the baseline performance without any specific optimizations or averaging. The proximity of "Average" and "Min" suggests a consistent lower bound on performance, while "Max" indicates the potential peak performance.
</details>
Figure 5: Average percentage accuracy (%) of MLLMs via different aggregation methods across five datasets.
The results are illustrated in the Figure 5. Across all policy models and datasets, we find that both averaging the step scores and selecting the minimum score significantly outperform the strategy of selecting the maximum score. This suggests that either the average or the minimum score provides a more accurate reflection of the overall quality of a solution than the maximum score. Between the minimum and average aggregation methods, we observe that averaging performs slightly better. This improvement may stem from the fact that the average score takes into account all problem-solving steps, providing a more comprehensive evaluation, whereas the minimum score reflects only the step with the lowest score and thus offers a less holistic assessment.
Refined-BoN vs. BoN.
The Refined-BoN process aims to enhance the diversity of N candidate solutions by refining the steps judged incorrect and integrating the refined steps with the steps judged correct into the prompt for the policy models. In this part, we use the Pass@k metric to evaluate the diversity and accuracy of policy models in generating multiple solutions to the given problems.
The results are summarized in the Table 2. Overall, the Refined-BoN process improves Pass@8 scores compared to the standard BoN process across multiple policy models and five benchmark datasets. Specifically, it increases the average Pass@8 values of MiniCPM-V-2.6-8B, Llama-3.2-11B-Vision, and InternVL3-8B by 0.9, 1.3, and 0.9 points, respectively, across the five datasets, demonstrating the effectiveness of the Refined-BoN approach in enhancing the diversity and correctness of the generated solutions.
| MiniCPM-V-2.6-8B Llama-3.2-11B-Vision InternVL3-8B | 62.5 62.7 65.3 | 63.4 64.0 66.2 | +0.9 +1.3 +0.9 |
| --- | --- | --- | --- |
Table 2: Average percentage Pass@8 scores of BoN and Refined-BoN across five datasets for different models.
5 Conclusion
In this work, we introduced GM-PRM, a novel paradigm that transforms the reward model from a passive judge into an active reasoning collaborator for multimodal mathematics. By providing fine-grained, interpretable analysis and, more critically, generating corrections for erroneous steps, GM-PRM moves beyond simple binary verification. This unique corrective capability powers our Refined Best-of-N (Refined-BoN) framework, which actively improves flawed reasoning trajectories at test time. Our experiments demonstrate that this approach achieves state-of-the-art results on multiple benchmarks, significantly boosting policy model performance with remarkable data efficiency. The consistent gains across diverse models and problem types underscore the robustness and generalizability of our method. This shift from passive error detection to generative, collaborative correction represents a crucial advance in multimodal reasoning.
\nobibliography
*
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Appendix A Appendix
A.1 More Related Work
Multimodal Large Language Models (MLLMs)
The advancement of artificial intelligence has advanced the development of Multimodal Large Language Models (MLLMs). MLLMs extend the capabilities of language-centric models by integrating multiple sensory inputs, primarily visual and auditory, with text. Unlike traditional Large Language Models (LLMs) which process solely textual data, MLLMs are designed to perceive and reason across modalities such as vision and language, thereby achieving the fusion and interaction of multimodal information. The development of MLLMs has been driven by extensive efforts, including enhancements across model structure and data curation. In terms of model structure, multiple studies (Bai et al. 2025; Liu et al. 2023; Yao et al. 2024) achieve notable performance through a method that utilizing connectors to align the embeddings of vision from Vision Foundation Models (VFMs) (Chen et al. 2024) with the latent space of LLMs (Bai et al. 2023; Touvron et al. 2023a, b). Alternatively, another line of research (Grattafiori et al. 2024; Tian et al. 2024) enhances pre-trained LLMs by adding supplementary layers to integrate visual features, which reduces the number of visual tokens but incurs additional training costs. Regarding dataset curation, recent research has achieved substantial advancements. Specifically, MultimodalC4 (Zhu et al. 2023) extends C4 corpus containing only text with images and constructs a corpus that supports pre-training for MLLMs. Furthermore, OmniCorpus (Li et al. 2024) delivers a large-scale yet noisy multimodal dataset suitable for pre-training, and MMInstruct (Liu et al. 2024b) presents an open-source collection of high-quality data designed for instruction tuning. The majority of research efforts have been concentrated on the training processes of MLLMs, leaving significant room for exploration in Test-Time Scaling (TTS) technique. In our work, we investigate the potential of enhancing the performance of MLLMs by incorporating Process Reward Model (PRM) into the TTS framework.
A.2 Benchmark
We provide more details about the Refined-BoN test benchmarks in Table 3:
| DynaMath | Plane Geometry | 770 |
| --- | --- | --- |
| MathVerse | Vision-Only | 788 |
| MathVista | Testmini | 1000 |
| WeMath | Testmini | 1740 |
| MathVision | Full | 3040 |
Table 3: More details about the Refined-BoN test benchmarks.
A.3 Dataset
To ensure a balanced distribution of process labels, we carefully construct the training dataset. The final dataset used to train GM-PRM contains 19,614 samples in total, comprising 9,061 solutions that contain incorrect stepsâas jointly identified by GPT-4o and Monte Carlo (MC) estimationâand 10,553 solutions in which all steps are judged to be correct.
A.4 Prompt
In this section, we introduce the prompts used to construct the training dataset and generate the reasoning processes and final answers. The prompt we guide the policy models to generate reasoning processes and final answers of multi-choice problems is represented in Figure 6.
<details>
<summary>figures/Prompt_multi_choice.png Details</summary>
### Visual Description
## Text Block: Prompt for Reasoning
### Overview
The image contains a block of text that serves as a prompt for generating reasoning for multi-choice problems. It instructs the user to act as an expert in solving multimodal mathematical problems and to provide step-by-step reasoning before selecting the correct option.
### Components/Axes
The text is divided into two main sections:
1. **Prompt:** This section provides instructions and context for the task.
2. **Problem:** This section indicates where the mathematical problem will be placed.
### Detailed Analysis or ### Content Details
The text content is as follows:
"Prompt for generating reasoning of multi-choice problems:
You are an expert in solving multimodal mathematical problems. I will provide a mathematical problem along with its corresponding image. According to the problem and the image, please first conduct step-by-step reasoning, and after your reasoning, please provide the correct option letter (e.g., A, B, C, D, E) using the format: "Final answer: ..."
Problem:
<Question>"
### Key Observations
The prompt emphasizes the importance of step-by-step reasoning before selecting an answer. The format for the final answer is explicitly specified.
### Interpretation
The prompt is designed to guide users in solving complex mathematical problems by encouraging a structured and logical approach. It aims to improve the accuracy and understanding of the solutions by requiring detailed reasoning. The `<Question>` placeholder indicates where the specific mathematical problem will be inserted.
</details>
Figure 6: Prompt for policy models to generate reasoning and final answers of multi-choice problem.
The prompt we guide the policy models to generate reasoning processes and final answers of free-form problems is represented in Figure 7.
<details>
<summary>figures/Prompt_free_form.png Details</summary>
### Visual Description
## Text Block: Prompt for Generating Reasoning
### Overview
The image contains a block of text that outlines a prompt for generating reasoning for free-form problems. It describes the role of the user as an expert in solving multimodal mathematical problems and the expected format for providing the final answer.
### Components/Axes
The text is divided into the following sections:
1. **Title:** "Prompt for generating reasoning of free-form problems:"
2. **Instructions:** A paragraph explaining the user's role and the expected process.
3. **Problem:** A label indicating where the mathematical problem will be stated.
4. **Question:** Placeholder for the actual question.
### Detailed Analysis or ### Content Details
The text content is as follows:
"Prompt for generating reasoning of free-form problems:
You are an expert in solving multimodal mathematical
problems. I will provide a mathematical problem along with its
corresponding image. According to the problem and the image,
please first conduct step-by-step reasoning, and after your
reasoning, please provide your final answer using the format:
"Final answer: ..."
Problem:
<Question>"
### Key Observations
The prompt is designed for a user who is an expert in multimodal mathematical problems. The user is expected to provide step-by-step reasoning before stating the final answer in a specific format.
### Interpretation
The text provides instructions for a user who is expected to solve mathematical problems using both textual and visual information. The prompt emphasizes the importance of step-by-step reasoning and a standardized format for the final answer. The `<Question>` placeholder indicates where the specific mathematical problem will be presented.
</details>
Figure 7: Prompt for policy models to generate reasoning and final answers of free-form problem.
The prompt we use to employ GPT-4o to generate the training dataset is shown in Figure 8.
<details>
<summary>figures/Prompt.png Details</summary>
### Visual Description
## Prompt: GPT-4o Training Dataset Construction
### Overview
The image presents a prompt designed for GPT-4o to construct a training dataset. The prompt outlines the task of analyzing and evaluating multi-step solutions to multimodal mathematical problems, focusing on image alignment and reasoning logic. It also specifies the output format, including step-by-step analysis and correction of the first incorrect step.
### Components/Axes
The prompt is structured into several key sections:
- **Task**: Defines the objectives, including analyzing the purpose and correctness of each step in a solution.
- **Question**: Placeholder for the multimodal mathematical problem.
- **Solution Steps**: Placeholder for the multiple-step solution.
- **Output Format**: Specifies the structure for the output, including step intent analysis, image alignment analysis, reasoning logic analysis, and final judgement.
### Detailed Analysis or ### Content Details
The prompt provides detailed instructions for evaluating each step of a solution:
1. **Task**:
* Analyze the purpose of each step and the specific actions taken.
* Analyze each step's correctness in terms of image alignment and reasoning logic.
* Image alignment: Check if the information and reasoning used are consistent with the provided image.
* Reasoning logic: Check if the reasoning is logically sound, calculations are correct, and information used matches previous steps and the question.
* Output "Correct" or "Incorrect" for each judgement.
* Correct the first incorrect step based on error analysis and output the corrected step.
2. **Question**:
* The multimodal mathematical problem is as follows: `<Question>`
3. **Solution Steps**:
* The multiple-step solution is as follows: `<Solution Steps>`
4. **Output Format**:
* Step 1:
* Step intent analysis: \[Describe what the step aims to do and the specific actions]
* Image alignment analysis: \[Analyze the consistency of image alignment]
* Judgement of image alignment: \[Correct/Incorrect]
* Reasoning logic analysis: \[Analyze the rationality of logic, correctness of calculations and consistency with prior step]
* Judgement of reasoning logic: \[Correct/Incorrect]
* Final judgement of the current step: \[Correct/Incorrect]
* Step 2: ...
* Corrected step of the first incorrected step in solution:
* Step n: \[assume that the first incorrect step is step n, and fill in the corrected step n in the square bracket]
### Key Observations
- The prompt emphasizes the importance of both image alignment and reasoning logic in evaluating the correctness of each step.
- The output format is highly structured, requiring detailed analysis and judgement for each step.
- The prompt includes placeholders for the question and solution steps, indicating that these will be provided separately.
### Interpretation
The prompt is designed to guide GPT-4o in creating a training dataset for evaluating multi-step solutions to multimodal mathematical problems. By focusing on image alignment and reasoning logic, the prompt aims to ensure that the model can accurately assess the correctness of each step and provide appropriate corrections. The structured output format facilitates the creation of a high-quality training dataset that can be used to improve the model's performance on similar tasks.
</details>
Figure 8: Prompt for GPT-4o to construct training dataset.
A.5 More Cases
We collect 2 more cases of GM-PRM and the Refined-BoN process illustrated in Figure 9 and 10.
<details>
<summary>figures/More_Cases_1.png Details</summary>
### Visual Description
## Problem Solving: Area Calculation
### Overview
The image presents a problem involving area calculation, with a visual representation and textual explanations of both an original and a refined solution. It includes a diagram showing areas to be subtracted and step-by-step instructions for solving the problem.
### Components/Axes
* **Header:** Divided into "Question (Visual Part)" and "Question (Textual Part)".
* **Visual Part:** A diagram showing a rectangle divided into three sections, representing areas.
* Areas labeled: 6000, 60, and an unlabeled area.
* Dimensions labeled: 6, 10, 1000.
* Total area labeled: 6096.
* **Textual Part:** The question "How much area is remaining?".
* **Solutions:** Presented in two columns: "Original Solution" and "Refined Solution".
* **GM-PRM: Analysis & Judgement:** A section providing analysis of the original solution.
* **Checkmarks and Crossmarks:** Green checkmarks indicate correct steps, while a red crossmark indicates an error.
### Detailed Analysis or Content Details
**Visual Part:**
* The rectangle is divided into three sections.
* The first section has an area of 6000.
* The second section has an area of 60.
* The third section's area is to be determined.
* The total area is given as 6096.
**Original Solution:**
* **Step 1:** To determine the remaining area, we start with the total area and subtract the areas that have been allocated. (Green Checkmark)
* **Step 2:** Total Area: The total area is given as 6096 square units. (Green Checkmark)
* **Step 3:** Allocated Areas: The first allocated area is 6000 square units. The second allocated area is 60 square units. (Green Checkmark)
* **Step 4:** Subtract Allocated Areas from Total Area: (Red Crossmark)
* First subtraction: (6096 - 6000 = 96)
* Second subtraction: (96 - 60 = 6)
**Refined Solution:**
* **Step 4:** Subtract Allocated Areas from Total Area: (Green Checkmark)
* First subtraction: (6096 - 6000 = 96)
* Second subtraction: (96 - 60 = 36)
* **Step 5:** Final answer: The remaining area is 36 square units. (Green Checkmark)
**GM-PRM: Analysis & Judgement:**
* **Step intent analysis:** This step performs the subtraction of the allocated areas from the total area to find the remaining area.
* **Image alignment analysis:** The image supports the subtraction process by showing the areas that need to be subtracted.
* **Reasoning logic analysis:** The calculations are performed correctly: (6096 - 6000 = 96) and (96 - 60 = 36), but the final result should be 36 instead of 6.
### Key Observations
* The original solution incorrectly calculates the final subtraction, resulting in an incorrect remaining area of 6.
* The refined solution correctly calculates the final subtraction, resulting in a remaining area of 36.
* The analysis section identifies the error in the original solution and confirms the correct calculations in both solutions up to the final step of the original solution.
### Interpretation
The image demonstrates a problem-solving process where an initial attempt contains an arithmetic error, which is then corrected in a refined solution. The analysis section highlights the importance of verifying each step in a calculation to ensure accuracy. The visual representation aids in understanding the concept of subtracting areas to find the remaining area. The problem highlights a common mistake in arithmetic and emphasizes the need for careful calculation and verification.
</details>
Figure 9: The illustration of a representative example before and after applying GM-PRM. In particular, GM-PRM first judges the steps of the original solution generated by the policy model. Subsequently, GM-PRM finds that the forth step is incorrect regarding reasoning logic and refines the forth step to generate the correct version. The correct steps are input to the policy model to generate the refined solution, and finally the correct answer is obtained.
<details>
<summary>figures/More_Cases_2.png Details</summary>
### Visual Description
## Problem Solving: Mosquitoes and Bats
### Overview
The image presents a mathematical problem involving the population of mosquitoes and bats, modeled by two quadratic equations. It includes a visual representation of the equations as graphs, a textual problem statement, and two attempted solutions: an "Original Solution" deemed incorrect and a "Refined Solution" presented as correct. The image also contains an analysis and judgement section highlighting errors in the original solution.
### Components/Axes
* **Titles:** "Question (Visual Part)", "Question (Textual Part)", "Original Solution", "Refined Solution", "GM-PRM: Analysis & Judgement"
* **Graph Axes:**
* x-axis: Labeled "x" with tick marks at integers from 1 to 6. Represents rainfall in centimeters.
* y-axis: Labeled "y" with tick marks at integers from 1 to 6. Represents the number of mosquitoes/bats.
* **Graph Equations:**
* Blue curve: y = 5x - x² (Mosquitoes)
* Pink curve: y = 3x - 0.5x² (Bats)
* **Solution Steps:** Numbered steps outlining the process of solving the problem in both the "Original Solution" and "Refined Solution" sections.
* **Checkmarks/Cross:** Green checkmarks indicate correct steps, while a red cross indicates an incorrect step.
### Detailed Analysis or ### Content Details
**1. Question (Textual Part):**
* The problem states that 'x' represents rainfall in centimeters.
* When rainfall is 0cm, the number of mosquitoes and bats is the same.
* The question asks for another rainfall amount where the number of mosquitoes equals the number of bats.
* The answer should be rounded to the nearest half centimeter.
**2. Question (Visual Part):**
* The graph shows two curves representing the equations for mosquito and bat populations.
* The blue curve (y = 5x - x²) represents the mosquito population. It is a parabola opening downwards, with a vertex around x=2.5 and y=6.
* The pink curve (y = 3x - 0.5x²) represents the bat population. It is a parabola opening downwards, with a vertex around x=3 and y=4.5.
* The curves intersect at x=0 and approximately x=4.
**3. Original Solution:**
* **Step 1:** To solve the problem, we need to find the point where the number of bats equals the number of mosquitos, other than the known 0 cm of rainfall. (Green Checkmark)
* **Step 2:** 1. **Identify the equations for mosquitoes and bats**: - Mosquitoes: ( y = 5x - x² ) - Bats: ( y = 3x - 0.5x2) (Green Checkmark)
* **Step 3:** 2. **Set the equations equal to each other to find the points of intersection**: [ 5x - x² = 3x - 0.5 x²] (Green Checkmark)
* **Step 4:** 3 **Simplify the equation**: [ 5x - x2 = 3x - 0.5 x²] Rearrange the equation to collect like terms: [ 5x - =x - x² + 0.5x2 = 0] [5x - =x+0.5x2 - x2 = 0] [5x - =x - 0.5x2=0] [5x - =x - 0.5x2 = 0] [5x -=x-0.5x2 = 0 ] (Red Cross)
**4. Refined Solution:**
* **Step 4:** 3. **Simplify the equation**:** [ 5x - x2 = 3x - 0.5x2] Rearrange the equation to collect like terms: [ 5x - x² - = x + 0.5x2 = 0] [2x -0.5x2 = 0] [0.5x2 - 2x = 0 ] [x(0.5x - 2) = 0 ] This corrected simplification should lead to the correct roots for further analysis. (Green Checkmark)
* **Step 5:** **Solve for ( x )**: [ x(0.5x - 2) = 0 ] This gives us two possible solutions for ( x ): 1. ( x = 0 ) 2. (0.5x - 2 = 0) [ 0.5x = 2 ] [ x = 4 ] (Green Checkmark)
* **Step 8:** Final answer: (4) cm. (Green Checkmark)
**5. GM-PRM: Analysis & Judgement:**
* **Step intent analysis:** This step simplifies the equation set in Step 3 to prepare it for solving.
* **Image alignment analysis:** The simplification process should reflect the intersection points seen in the image.
* **Reasoning logic analysis:** There is a mistake in the simplification process. The rearrangement and simplification steps contain errors in algebraic manipulation.
### Key Observations
* The "Original Solution" makes an error in the algebraic manipulation during the simplification process (Step 4).
* The "Refined Solution" correctly simplifies and solves the equation, leading to the correct answer of x = 4.
* The graph visually confirms the solution, showing an intersection point at approximately x = 4.
### Interpretation
The problem demonstrates solving a system of equations to find the intersection points, representing the rainfall amount where the mosquito and bat populations are equal. The "Original Solution" highlights the importance of accurate algebraic manipulation, as a single error can lead to an incorrect result. The "Refined Solution" provides the correct steps and solution, which is visually supported by the graph. The GM-PRM analysis correctly identifies the error in the original solution, emphasizing the need for careful reasoning and verification in mathematical problem-solving. The intersection point at x=4 suggests that 4 cm of rainfall is the other point where the number of mosquitoes and bats is the same.
</details>
Figure 10: The illustration of a representative example before and after applying GM-PRM. In particular, GM-PRM first judges the steps of the original solution generated by the policy model. Subsequently, GM-PRM finds that the forth step is incorrect regarding reasoning logic and refines the forth step to generate the correct version. The correct steps are input to the policy model to generate the refined solution, and finally the correct answer is obtained.