PAPER_TITLE

FIRST_AUTHOR_LAST, FIRST_AUTHOR_FIRST; SECOND_AUTHOR_LAST, SECOND_AUTHOR_FIRST

CogFlow: Bridging Perception and Reasoning through Knowledge Internalization for Visual Mathematical Problem Solving

Shuhang Chen¹, Yunqiu Xu¹, Junjie Xie¹, Aojun Lu⁴, Tao Feng³, Zeying Huang², Ning Zhang², Yi Sun², Yi Yang¹, Hangjie Yuan^1*
¹Zhejiang University ²Intelligent Learning ³Tsinghua University ⁴Sichuan University

Paper Code (comming soon) Dataset (comming soon) arXiv

Overview of the proposed visual mathematical reasoning framework CogFlow. Inspired by the canonical three-stage human reasoning flow, CogFlow adopts a hierarchical pipeline that integrates Synergistic Visual Rewards (SynVRs) for enhanced perception, a Knowledge Internalization Reward (IntlzR) to bridge perception and reasoning, and Visual-Gated Policy Optimization (VGPO) with Inference Reward (InfR) to anchor multi-step reasoning in perceptual accuracy.

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Abstract

Despite recent advances, multimodal large language models continue to struggle with visual mathematical problem solving. Some recent works recognize that visual perception is a bottleneck in visual mathematical reasoning, but their solutions are limited to improving the extraction and interpretation of visual inputs. Notably, they all ignore the key issue of whether the extracted visual cues are faithfully integrated and properly utilized in subsequent reasoning. Motivated by this, we present CogFlow, a novel cognitive-inspired three-stage framework that incorporates a knowledge internalization stage, explicitly simulating the hierarchical flow of human reasoning: perception ⇒ internalization ⇒ reasoning. In line with this hierarchical flow, we holistically enhance all its stages. We devise synergistic visual rewards to boost perception capabilities in parametric and semantic spaces, jointly improving visual information extraction from symbols and diagrams. To guarantee faithful integration of extracted visual cues into subsequent reasoning, we introduce a visual-anchored reward model in the internalization stage, bridging perception and reasoning. Moreover, we design a visual-gated policy optimization algorithm to further enforce the reasoning is grounded with the visual knowledge, preventing models seeking shortcuts that appear coherent but are visually ungrounded reasoning chains. Moreover, we contribute a new dataset MathCog for model training, which contains samples with over 120K high-quality perception-reasoning aligned annotations. Comprehensive experiments and analysis on three commonly used visual mathematical reasoning benchmarks validate the superiority of the proposed CogFlow.

CogFlow Framework

One-step framework yields unstructured reasoning, while decoupled pipeline modularly disentangles the flow. We adopt a cognitive-inspired three-stage framework with knowledge internalization.

Workflow of SynVRs. SynVRs consist of a Visual Semantic Reward and a Visual Parametric Reward, ensuring local geometric fidelity and global perceptual coherence respectively. Together, these two complementary visual rewards provide a unified supervision mechanism for training robust and accurate visual perception.

Experiments

Main results

Accuracy (%) and FlowVerse-style CoT-E (%) results on FlowVerse.
Model	All		Text Centric		Text Limited		Text Plus		Vision Dense		Vision Centric		Vision Primary
Model	CoT-E	Acc	CoT-E	Acc	CoT-E	Acc	CoT-E	Acc	CoT-E	Acc	CoT-E	Acc	CoT-E	Acc
Claude-3.5-Sonnet	55.5	45.1	60.8	52.6	58.7	50.3	64.0	58.3	45.0	25.4	56.5	48.0	48.1	45.2
GPT-4o	56.9	49.7	61.0	56.8	58.7	54.4	62.2	58.2	45.2	30.0	58.6	52.6	54.1	51.0
GPT-4V	64.2	58.7	69.1	57.1	65.0	55.0	72.0	61.4	48.1	30.3	61.8	46.3	42.0	36.7
MathFlow^★_GPT-4V	64.2	59.5	69.5	58.2	67.2	57.4	71.1	64.1	52.7	47.5	62.1	57.1	60.4	57.0
Gemini-2.5-pro	64.5	56.2	68.3	61.9	66.1	60.8	68.9	64.1	52.1	37.1	65.7	57.9	57.0	54.6
GPT-5	68.2	59.3	74.3	68.1	73.5	66.7	77.0	69.2	53.8	44.7	67.1	61.7	60.3	57.5
InfiMM-Math-7B	37.8	29.5	43.8	38.1	40.6	36.7	46.1	40.1	28.8	15.4	39.6	30.3	26.1	23.2
InternVL2.5-8B	46.3	40.1	49.2	41.3	40.5	38.4	49.6	42.7	38.4	20.2	41.0	35.9	35.8	33.9
Math-LLaVA-13B	39.3	30.8	45.1	39.3	44.4	37.4	-	-	36.2	18.6	41.7	35.9	37.0	34.2
MultiMath-7B	45.2	35.3	50.6	44.8	49.9	42.9	-	-	41.7	22.1	47.2	40.4	39.7	38.8
SVE-Math-Qwen2.5-7B	47.9	38.7	53.1	47.3	53.4	45.8	-	-	44.2	28.6	48.9	44.2	45.8	42.0
VLM-R1-7B	50.7	41.2	59.0	54.2	57.9	49.8	65.5	58.9	36.2	24.5	46.1	37.8	30.6	26.1
CogFlow-7B	66.0	56.2	67.9	58.6	67.3	58.3	68.1	60.9	57.8	42.7	68.2	61.1	66.7	63.5

Accuracy (%) and MathVerse-style CoT-E (%) results on *testmini* set of MathVerse.
Model	All		Text Dominant		Text Lite		Text Only		Vision Intensive		Vision Dominant		Vision Only
Model	CoT-E	Acc	CoT-E	Acc	CoT-E	Acc	CoT-E	Acc	CoT-E	Acc	CoT-E	Acc	CoT-E	Acc
Qwen-VL-Plus	21.3	11.8	26.0	15.7	21.2	11.1	25.2	14.5	18.5	9.0	19.1	13.0	21.8	10.0
Gemini-Pro	35.3	23.5	39.8	26.3	34.7	23.5	44.5	27.3	32.0	23.0	36.8	22.3	33.3	22.2
Qwen-VL-Max	37.2	25.3	42.8	30.7	37.7	26.1	47.9	28.9	33.6	24.1	35.9	24.1	35.9	21.4
GPT-4V	54.4	39.4	63.1	54.7	56.6	41.4	60.3	48.7	51.4	34.9	50.8	34.4	50.3	31.6
MathFlow^★_GPT-4V	56.7	43.8	65.2	51.1	58.9	46.4	62.1	48.5	53.7	40.3	52.1	37.4	52.5	39.0
SPHINX-MoE-56B	25.8	15.6	33.3	22.2	21.9	16.4	40.7	18.3	21.1	14.8	19.6	12.6	18.3	9.1
InternLM-XC2-7B	25.9	16.5	36.9	22.3	28.3	17.0	42.5	16.5	20.1	15.7	24.4	16.4	19.8	11.0
Math-LLaVA-13B	-	20.1	-	22.8	-	21.8	-	-	-	21.1	-	19.2	-	15.4
MultiMath-7B	-	26.9	-	34.8	-	30.8	-	-	-	28.1	-	25.9	-	15.0
SVE-Math-Qwen2.5-7B	-	31.4	-	37.6	-	36.8	-	-	-	34.9	-	31.5	-	16.0
DVLR-14B	48.1	-	54.3	-	49.0	-	-	-	46.3	-	47.2	-	43.8	-
SophiaVL-R1-7B	48.8	-	45.4	-	43.9	-	-	-	45.1	-	58.5	-	51.3	-
CogFlow-7B	53.9	39.5	60.7	41.9	51.2	37.0	52.3	40.1	55.0	42.4	58.7	44.8	44.2	26.3

**Accuracy (%) results on MathVista.** CogFlow demonstrates consistent superiority.
Model	All	FQA	GPS	MWP	TQA	VQA
GPT-4V	49.9	43.1	50.5	57.5	65.2	38.0
Claude-3.5-Sonnet	67.7	-	-	-	-	-
Doubao-pro-1.5	79.5	77.7	88.9	86.0	82.3	62.0
G-LLaVA-7B	25.1	19.1	48.7	3.6	25.0	28.7
VCAR-7B	33.7	30.9	34.6	38.7	37.3	28.5
SPHINX-Plus-56B	36.7	54.6	16.4	23.1	41.8	43.0
SVE-Math-7B	37.4	31.9	53.9	29.0	41.4	30.8
MultiMath-7B	50.0	40.1	66.8	61.8	50.0	33.0
SophiaVL-R1-7B	71.3	-	-	-	73.4	-
ThinkLite-VL-7B	71.6	-	-	-	-	-
VL-Rethinker-7B	73.7	-	-	-	-	-
CogFlow-7B	76.8	70.4	93.1	73.7	86.9	59.3

CogFlow shows competitive accuracy (%) results on more visual math benchmarks.
Models	WeMath	LogicVista	DynaMath
Claude-3.7-Sonnet	49.3	58.2	39.7
GLM-4.5V	68.8	62.4	53.9
Doubao-1.5-Pro	65.7	64.2	44.9
GPT-5	71.1	70.0	60.9
Gemini-2.5-Pro	78.0	73.8	56.3
Ovis-8B	27.2	39.4	20.4
Qwen2.5-VL-8B	35.2	44.1	21.0
InternVL3-8B	37.1	44.1	25.5
Keye-VL-8B	60.7	54.8	37.3
InternVL3.5-8B	57.0	57.3	37.7
GLM-4.1V-9B	63.8	60.4	42.5
CogFlow-7B	64.1	58.1	46.2

More analysis

Ablation analysis of SynVRs. Variants exhibit consistent improvements.

Impact of different error types — Ablation analysis of SynVRs. Variants exhibit consistent improvements.

Distribution of visual reward values among different post-training methods — The distribution of visual reward values among different post-training methods. A higher concentration of values indicates stronger perceptual grounding achieved by the corresponding training strategy.

Ablation analysis of visual gate — The distribution of visual reward values among different post-training methods. A higher concentration of values indicates stronger perceptual grounding achieved by the corresponding training strategy.

Case study

BibTeX

@article{chen2026cogflow,
  title   = {CogFlow: Bridging Perception and Reasoning through Knowledge Internalization for Visual Mathematical Problem Solving},
  author  = {Chen, Shuhang and Xu, Yunqiu and Xie, Junjie and Lu, Aojun and Feng, Tao and Huang, Zeying and Zhang, Ning and Sun, Yi and Yang, Yi and Yuan, Hangjie},
  journal = {arXiv preprint arXiv:2601.01874},
  year    = {2026}
}

SynVRs	IntlzR	VGPO	FlowVerse		MathVerse
SynVRs	IntlzR	VGPO	CoT-E	Acc	CoT-E	Acc
✗	✗	✗	57.4	48.7	48.2	35.6
✓	✗	✗	63.2	54.7	50.5	36.9
✗	✓	✗	62.7	53.5	49.9	36.2
✗	✗	✓	63.4	54.8	50.8	37.3
✓	✓	✗	64.4	55.1	52.1	38.0
✓	✓	✓	66.0	56.2	53.9	39.5

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