用流匹配方法从稀疏观测中同时实现偏微分方程正演与反演，理论可保证精度。
Guided Flow Matching for Forward and Inverse PDE Problems with Sparse Observations: Algorithm and Theory
- 基于流匹配学习系数与解的联合分布，支持正演与反演
- 稀疏观测下误差有理论保障，推理速度优于扩散模型
- 适配确定性、随机及混合采样，适合科学计算中的数据稀缺场景
从稀疏观测中重构偏微分方程（PDE）解是科学计算的核心挑战。本文提出FM4PDE，一种基于流匹配的生成框架，可学习PDE系数（或初态）与解（或终态）的联合分布，从而在有限成对数据下实现正演模拟与反演恢复。推理时，通过复合损失引导采样，强制与稀疏观测一致并最小化PDE残差；支持确定性、随机及混合采样器。本文提供这些引导过程的误差保证：对确定性优化器，共焦条件确保轨迹有界，分阶段压缩带来对数级复杂度；对随机采样器，引入自适应引导，并假设速度场耗散性，获得独立于噪声底限的统一矩界，推导出多项式时间误差界；下界表明恒定引导会引入不可避免的正偏差，促使自适应设计。还提供了确定性-随机混合分析。在静态与时变基准PDE上的实验显示，该方法精度具竞争力，推理速度显著快于基于扩散的生成模型。
原文摘要 · Abstract (English)
Reconstructing PDE solutions from sparse observations is a core challenge in scientific computing. We present FM4PDE, a flow-matching generative framework that learns the joint distribution of PDE coefficients (or initial states) and solutions (or final states), enabling both forward simulation and inverse recovery with limited paired data. At inference, sampling is guided by a composite loss that enforces agreement with sparse measurements and reduces the PDE residual; we support deterministic, stochastic, and hybrid samplers. We provide error guarantees for these guided procedures. For the deterministic optimizer, a coercivity condition ensures trajectory boundedness and a phase-wise contraction yields logarithmic complexity in the target accuracy. For the stochastic sampler, we introduce adaptive guidance and assume dissipativity of the velocity field to obtain uniform moment bounds independent of the noise-floor parameter. This leads to polynomial-time error bounds, and a matching lower bound shows constant guidance induces an unavoidable positive bias, motivating adaptivity. A hybrid deterministic-stochastic analysis is also provided. Experiments on static and time-dependent benchmark PDEs demonstrate competitive accuracy and faster inference than diffusion-based generative models.
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