利用核最优传输改进函数空间流匹配
Improving Function Space Flow Matching with Kernel Optimal Transport
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中文总结 AI 辅助
针对函数空间流匹配中先验与数据样本任意配对的问题，提出核函数流匹配（kFFM），利用核诱导代价下的熵最优传输改进配对，在时间序列和PDE基准上显著提升分布匹配性能。
中文摘要 AI 辅助
针对函数值数据（如时间序列和偏微分方程解）的生成模型必须学习无限维空间上的分布。函数流匹配（FFM）将流匹配扩展到该设置，学习一个速度场，其流将高斯先验传输到数据分布，但它继承了标准流匹配的独立端点配对：在每个批次中，先验样本和数据样本被任意匹配，因此条件桥必须同时穿越数据集的共享全局结构和实例特定的残差。在函数空间中，这比在有限维中更难修复，因为函数空间上的最优传输（OT）在表述上很微妙，而平坦的欧几里得替代方案忽略了区分函数值数据的几何结构。我们提出了核函数流匹配（kFFM），它通过核诱导代价下的熵最优传输（即希尔伯特Sinkhorn散度（HSD）背后的耦合）取代独立配对，保持FFM神经算子架构不变。我们证明了核代价和HSD目标在Banach环境空间上是一致有界且适定的，推导了针对紧度量空间上二次代价OT的误差分解，该分解隔离了一个不可约的核代价失配项，并证明了一个离散化不变性界，其速率由Sobolev正则性控制。实验上，kFFM在时间序列和PDE基准上优于FFM、扩散、对抗和有限维OT基线，在配对种子下相比FFM有显著提升，且改进在非核和基于物理的诊断（包括湍流Navier-Stokes基准）中持续存在。有界核代价已优于原始$L^2$ Sinkhorn，而函数空间感知核（签名核、Sobolev RBF）在粗糙或路径值数据上带来进一步增益。
英文摘要
Generative models for function-valued data, such as time series and solutions of partial differential equations, must learn distributions over infinite-dimensional spaces. Functional Flow Matching (FFM) extends Flow Matching to this setting, learning a velocity field whose flow transports a Gaussian prior to the data distribution, but it inherits the independent endpoint pairing of standard Flow Matching: in each batch, prior and data samples are matched arbitrarily, so the conditional bridge must traverse both the shared global structure of the dataset and instance-specific residuals. In function space this is harder to fix than in finite dimensions, since optimal transport (OT) on function spaces is delicate to formulate and a flat Euclidean surrogate ignores the geometry that distinguishes function-valued data. We propose kernel Functional Flow Matching (kFFM), which replaces the independent pairing by entropic OT under a kernel-induced cost, the coupling underlying the Hilbert Sinkhorn Divergence (HSD), leaving the FFM neural-operator architecture unchanged. We prove that the kernel cost and the HSD objective are uniformly bounded and well-posed on Banach ambient spaces, derive an error decomposition against quadratic-cost OT on compact metric spaces that isolates an irreducible kernel-cost mismatch term, and prove a discretization-invariance bound whose rate is governed by Sobolev regularity. Empirically, kFFM improves distributional matching over FFM, diffusion, adversarial, and finite-dimensional OT baselines on time-series and PDE benchmarks, with significant paired-seed gains over FFM and improvements that persist under non-kernel and physics-based diagnostics, including a turbulent Navier-Stokes benchmark. Bounded kernel costs already outperform raw $L^2$ Sinkhorn, and function-space-aware kernels (signature, Sobolev RBF) give further gains on rough or path-valued data.
发表机构
- University of California, Los Angeles（加州大学洛杉矶分校）
- Block, Inc.（Block公司）
- Imperial College London（伦敦帝国学院）
机构由 AI 辅助整理，请以论文原文为准。
