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arxiv: 2503.16549 · v2 · submitted 2025-03-19 · 💻 cs.CV

MathFlow: Enhancing the Perceptual Flow of MLLMs for Visual Mathematical Problems

Shuhang Chen , Hangjie Yuan , Yunqiu Xu , Pengwei Liu , Tao Feng , Jun Cen , Zeying Huang , Yi Yang This is my paper

Pith reviewed 2026-05-22 22:49 UTC · model grok-4.3

classification 💻 cs.CV
keywords multimodal large language modelsvisual mathematical reasoningdiagram perceptionperception-inference decouplingFlowVerse benchmark
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The pith

Decoupling perception into a separate trained stage improves MLLM accuracy on diagram-based math problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that multimodal large language models still fail to reliably extract information from diagrams in math problems, even when they can reason once the facts are given. It introduces the FlowVerse benchmark to measure perception and reasoning separately and reveals clear gaps in current models. In response it proposes MathFlow, a pipeline that splits the task into an independent perception stage followed by inference. Training MathFlow-P-7B specifically for perception produces measurable gains when the model is paired with existing closed- and open-source reasoners. A reader would care because the separation lets each component be improved without retraining the entire system.

Core claim

MathFlow is a modular pipeline that decouples perception of diagrams from subsequent inference. Training a dedicated perception model, MathFlow-P-7B, and feeding its output to various inference models produces substantial performance gains on visual mathematical tasks.

What carries the argument

The MathFlow pipeline that isolates diagram perception as an independent, trainable stage before handing structured information to an inference model.

If this is right

  • Perception can be optimized independently of the reasoning component.
  • The same perception model works with multiple closed-source and open-source inference systems.
  • FlowVerse supplies separate scores for perception accuracy and reasoning accuracy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same separation might help other visual domains where diagrams or charts are central.
  • Specialized perception models could be developed for different diagram styles or subjects.
  • The benchmark could be used to diagnose exactly which visual features current models miss.

Load-bearing premise

That training perception separately will reliably improve end-to-end results without introducing new interface errors between the two stages.

What would settle it

An experiment in which MathFlow-P-7B is paired with the same inference models on FlowVerse and yields no accuracy increase over the baseline MLLM.

Figures

Figures reproduced from arXiv: 2503.16549 by Hangjie Yuan, Jun Cen, Pengwei Liu, Shuhang Chen, Tao Feng, Yi Yang, Yunqiu Xu, Zeying Huang.

Figure 1
Figure 1. Figure 1: The Typical Process of Humans Solving Visual Math￾ematical Problems. We can summarize two key capabilities ob￾served in the typical human problem-solving process: perception and inference. The perception capability involves extracting rel￾evant information from both visual and textual inputs, ensuring accurate reasoning, which inspired the development of FlowVerse and MathFlow. els (MLLMs) [10, 30, 51] are… view at source ↗
Figure 2
Figure 2. Figure 2: Six Versions of Problems in FlowVerse. FlowVerse begins by categorizing the original problem information into four distinct components: Descriptive Information (DI), Essential Information (EI), Only Question (OQ), and Reasoned Property (RP). The first three components are derived directly from the original problem statement, while RP is extracted from the solution and represents the inferences needed to so… view at source ↗
Figure 4
Figure 4. Figure 4: The FlowVerse-CoT-E Strategy. abilities beyond end-to-end performance. In contrast, Math￾Vista [40] integrates diverse mathematical and visual tasks to challenge models with fine-grained visual understand￾ing and compositional reasoning across various contexts. To provide a more comprehensive and in-depth evaluation of MLLMs, MathVerse [79] builds upon MathVista, focus￾ing on eliminating textual redundancy… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of Two Different CoT Evaluation Per￾formances on FlowVerse† . Visual Perception Error Question: Answer: 1 To �ind the area of the shaded region in the square ABCD, We �irst note that the square has a side length of 2. The area of the entire square is therefore (22= 4) square units. The line through O (the center of the square) that intersects AD and BC at points E and F respectively, divides the… view at source ↗
Figure 6
Figure 6. Figure 6: Problem-solving Comparison of MathFlow⋆ GPT-4V and GPT-4V. GPT-4V. In terms of CoT-based evaluation (CoT-E), MathFlow⋆ GPT−4V also demonstrates consistent superior￾ity. On the other hand, Tab. 3 shows the performance comparison of MLLMs on the FlowVerse† and MathVerse datasets, where FlowVerse† refers to the raw, unmodified version of FlowVerse. Notably, MathFlow⋆ Gemini1.5−pro achieves the highest accurac… view at source ↗
Figure 8
Figure 8. Figure 8: Manual Modification for EI in FlowVerse. For the original problems shown, we transfer some of the EI from diagrams to question texts (highlighted in green) to mark the Vision Centric version. A.2. Subject and Subfield Definition Plane Geometry. This foundational area studies the prop￾erties and relationships of points, lines, and surfaces within a two-dimensional plane. It covers key concepts such as cir￾c… view at source ↗
Figure 9
Figure 9. Figure 9: Subject Distribution of FlowVerse. 0 20 40 60 80 100 0.00 0.05 0.10 0.15 0.20 Percentage Distribution of Question Lengths (Word) Text Dominant Text Lite Vision Dominant Text Dominant Average Text Lite Average Vision Dominant Average [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Distribution of Question Length for Four Problem Versions. We present the distribution of question length for the four problem versions, with the horizontal axis representing ques￾tion length in characters and the vertical axis depicting the corre￾sponding probability distribution. we observe a clear downward trend in both the distribution of question lengths and their average values. A.4. Details of Eval… view at source ↗
Figure 11
Figure 11. Figure 11: Visualization of Different Error Type across Dif￾ferent Versions using GPT-4 on FlowVerse. The horizontal axis represents different problem versions, while the vertical axis indi￾cates the error types. The radius of each bubble corresponds to the number of visual perception errors, with smaller radii indicating fewer visual perception errors. rors, underscoring the importance of a balanced informa￾tion re… view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of Six Problem Versions in FlowVerse. Descriptive Information Essential Information Only Question Reasoned Property Text Centric Text Limited Text Plus Vision Dense Vision Centric Vision Primary 班长对捐款情况进行了统计 ， 并绘制成了统计图 ．根据统计 图可知 。6人捐款 10元，13人 捐款20元，20人捐款30元，8 人捐款50元，3人捐款100元。 捐款超过30元的有8+3=11(人) 捐款总数为 6×10 + 13×20 + 20×30 + 8×50 + 3×100 = 1620(元). 则下列说法中，不正 确的是（） choices: A. "'捐款30元的是20 人’”, B. … view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of Six Problem Versions in FlowVerse [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of Six Problem Versions in FlowVerse. According to the properties of congruent triangles, we obtain ∠ACB = ∠DFE, and then, by the criterion for parallel lines, we can reach the conclusion.：∵ △ABC ≌ △DEF, ∴ ∠ACB = ∠DFE, ∴ AC ∥ DF. Solving-problem Solution Step Extraction Question: As shown in the �igure, △ABC ≌ △DEF, what is the relationship between AC and DF? Visual math problems Answer: AC ∥ D… view at source ↗
Figure 15
Figure 15. Figure 15: Data Annotation of the MathFlow-RP We first employ Qwen2.5-72B to extract the corresponding steps from the solving￾problem solution, then select step N as the target for prediction. Subsequently, the preceding N-1 steps are provided as input within the prompt, enabling the MLLM to predict the next step based on this sequential context. other hand, correctly identifies and matches corresponding sides and a… view at source ↗
Figure 16
Figure 16. Figure 16: Data Annotation of the MathFlow-SFT.We manually extract the corresponding EI and RP from the solving-problem solution and associated diagram. In this representation, the red-highlighted portions indicate EI, while the blue-highlighted sections represent RP [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Response Comparison of GPT-4V and MathFlow⋆ GPT−4V Visual Perception Error Question: Answer: 8 From the given information, we can determine the length of chord AC in the circle. 1. We know that△𝑂𝐴𝐵 is a right triangle with ∠OAB = 90° , OB = 13 (radius of the circle), and AB = 12 . Using the Pythagorean theorem, 𝐴𝑂 = 𝑂𝐵2 − 𝐴𝐵2 = 132 − 122 = 169 − 144 = 25 = 5 . 2. It is given that AC = 2AH, 𝐴𝐻 = 𝐴𝑂2 − 𝐻𝑂2 … view at source ↗
Figure 18
Figure 18. Figure 18: Response Comparison of GPT-4V and MathFlow⋆ GPT−4V [PITH_FULL_IMAGE:figures/full_fig_p022_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Response Comparison of GPT-4V and MathFlow⋆ GPT−4V [PITH_FULL_IMAGE:figures/full_fig_p023_19.png] view at source ↗
read the original abstract

Despite strong results on many tasks, multimodal large language models (MLLMs) still underperform on visual mathematical problem solving, especially in reliably perceiving and interpreting diagrams. Inspired by human problem-solving, we hypothesize that the ability to extract meaningful information from diagrams is pivotal, as it directly conditions subsequent inference. Hence, we introduce FlowVerse, a comprehensive benchmark that provides a fine-grained evaluation of MLLMs' perception and reasoning capabilities. Our preliminary results on FlowVerse reveal that existing MLLMs exhibit substantial limitations when extracting essential information and reasoned properties from diagrams and performing complex reasoning based on these visual inputs. In response, we introduce MathFlow, a modular problem-solving pipeline that decouples perception and inference into distinct stages, thereby optimizing each independently. Given the perceptual limitations observed in current MLLMs, we trained MathFlow-P-7B as a dedicated perception model. Experimental results indicate that MathFlow-P-7B yields substantial performance gains when integrated with various closed-source and open-source inference models. This demonstrates the effectiveness of the MathFlow pipeline and its compatibility with diverse inference frameworks. Project page: https://github.com/MathFlow-zju/MathFlow.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces FlowVerse, a benchmark providing fine-grained evaluation of MLLMs' perception and reasoning on visual mathematical problems, and MathFlow, a modular pipeline decoupling perception from inference. It trains MathFlow-P-7B as a dedicated perception model and reports that this yields substantial performance gains when paired with various closed- and open-source inference models.

Significance. If the claimed gains hold under scrutiny, the work would provide evidence that separating and specializing the perception stage can address a key bottleneck in MLLM visual math solving, while also supplying a new benchmark for the community.

major comments (2)
  1. [Abstract] Abstract: the central claim of 'substantial performance gains' from MathFlow-P-7B is stated without any quantitative metrics, dataset sizes, error bars, or ablation results, preventing verification of the effect size or statistical reliability.
  2. [Abstract] The hypothesis that decoupling perception into a separate trained stage will reliably improve end-to-end performance is presented as pivotal, yet the abstract supplies no evidence on whether the perception model was trained on data disjoint from the inference models or benchmarks.
minor comments (1)
  1. [Abstract] The project page link is provided, which supports potential reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the two major comments point-by-point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of 'substantial performance gains' from MathFlow-P-7B is stated without any quantitative metrics, dataset sizes, error bars, or ablation results, preventing verification of the effect size or statistical reliability.

    Authors: We agree that the abstract would be strengthened by including quantitative support for the central claim. In the revision we will add specific performance deltas on FlowVerse (with dataset sizes), while retaining the note that full ablations, error bars, and statistical details appear in the experimental sections. revision: yes

  2. Referee: [Abstract] The hypothesis that decoupling perception into a separate trained stage will reliably improve end-to-end performance is presented as pivotal, yet the abstract supplies no evidence on whether the perception model was trained on data disjoint from the inference models or benchmarks.

    Authors: We acknowledge the abstract does not explicitly state data disjointness. We will insert a concise clause confirming that MathFlow-P-7B was trained on data disjoint from both the FlowVerse benchmark and the inference models used in the paired experiments. The full training-data description and separation protocol are already detailed in Sections 3 and 4. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper describes an empirical pipeline: creation of FlowVerse benchmark, observation of MLLM perception limits, introduction of modular MathFlow decoupling perception/inference, training of MathFlow-P-7B, and reporting of integration gains. No equations, fitted parameters, derivations, or uniqueness theorems appear. No self-citation chains or ansatzes are invoked as load-bearing. The performance claims rest on external experimental results rather than any quantity that reduces to its own inputs by construction. This is a standard empirical contribution with self-contained evaluation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no equations, fitted constants, or explicit assumptions beyond the high-level hypothesis; therefore no free parameters, axioms, or invented entities can be identified.

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