Title:Func-R1: Incentivizing Mathematical Function Reasoning in Multimodal Large Language Models

Abstract:Performing deliberate mathematical reasoning in visual contexts is a hallmark of advanced Multimodal Large Language Models (MLLMs) and requires a sophisticated synthesis of perceptual grounding and symbolic logic. However, in the realm of mathematical functions, our investigation reveals a critical modality interference phenomenon: even advanced models, while performing textual computational reasoning, tend to disregard or misinterpret essential visual cues. To address this challenge, we propose Func-R1, which synergistically harmonizes precise visual perception and rigorous logical reasoning. Concretely, built upon an explicitly decoupled architecture, we employ a hierarchical post-training framework to progressively identify critical visual evidence and conduct in-depth theoretical reasoning. Furthermore, the Perception-Aligned Theoretic Optimization (PATO) strategy is proposed to steer policy updating towards internalizing fundamental theoretical properties while dynamically rectifying heterogeneous visual information throughout the reasoning process. Extensive experiments across diverse benchmarks demonstrate that Func-R1 delivers the optimal performance among open-source MLLMs, even surpassing GPT-5 with an 8.4% improvement on MathVerse's function-oriented tasks.

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