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

TL;DR

CogFlow addresses the persistent challenge of reasoning drift in visual mathematical problem solving by modeling a human-like perception → internalization → reasoning flow. It introduces Synergistic Visual Rewards (SynVRs) to enhance perception, a Knowledge Internalization Reward (IntlzR) to ground internalized knowledge, and Visual-Gated Policy Optimization (VGPO) to stabilize multi-step reasoning, all trained on the newly created MathCog dataset (~120K perception–reasoning aligned samples). Through two-phase training (SFT and RL) and a multi-component reward structure, CogFlow achieves state-of-the-art results across multiple visual math benchmarks, with strong gains in both final accuracy and the groundedness of intermediate reasoning. The approach demonstrates robust, scalable improvements and offers a pathway toward general visually grounded reasoning in multimodal models.

Abstract

Despite significant progress, 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$\Rightarrow$internalization$\Rightarrow$reasoning. Inline 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 Knowledge Internalization 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 commonly used visual mathematical reasoning benchmarks validate the superiority of the proposed CogFlow.

Figures (23)

• Figure 1: The one-step reasoning framework (e.g., VLM-R1 VLM-R1) often yields suboptimal results, while the decoupled reasoning pipeline (e.g., MathFlow MathFlow) enhances perception yet still yields illogical reasoning steps that disregard visual evidence. In contrast, CogFlow adopts a cognitive-inspired three-stage framework to effectively mitigate reasoning drift.
• Figure 2: 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.
• Figure 3: 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.
• Figure 3: Accuracy (%) results on MathVista.CogFlow demonstrates consistent superiority.
• Figure 4: Pipeline of VGPO. VGPO introduces visual gate and multiple rewards to strengthen multi-step visual reasoning. Coupling perceptual quality control with outcome-based optimization promotes stability.