We introduce NAGO, a noise-aware generative operator for efficient sampling from invariant measures of stochastic partial differential equations (SPDEs) in function space. Our central idea is to treat the source measure as a design variable rather than a fixed Gaussian default. We establish admissibility conditions for well-defined infinite-dimensional flows and show that the finite-band source--target covariance ratio controls the spectral stiffness of the marginal velocity field. This analysis yields an admissibility-constrained spectral-matching rule: choose a source that is sufficiently rough for well-posedness, then minimize covariance mismatch over the resolved Fourier modes. On three representative stochastic-field benchmarks, this source design improves few-step spectral accuracy and several complementary distributional and physical statistics. In a controlled Matern-field ablation, it reduces the accurate RK4 budget from about 30 steps for Gaussian functional flow matching to 3--5 steps, a $6$--$10\times$ reduction in neural ODE evaluations. These gains are strongest for stationary multiscale fields and depend on the metric, resolution, and guidance protocol.
Shuhang Chen, Hangjie Yuan DAMO Academy, Alibaba Group, Yunqiu Xu National University of Singapore, Pengwei Liu, Tao Feng Tsinghua University, Tsinghua University, Jun Cen Alibaba Group, Zeying Huang Intelligent Learning Tech(Zhejiang), Yi Yang Zhejiang University
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.
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cs.CL · cs.CV
Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers) · 2026 · 正式发表