条件均值算子的多任务学习：在动力系统与不确定性量化中的应用

中文摘要 AI 辅助

在许多数据驱动的应用中，包括不确定性量化和动力系统分析，估计条件统计量以及学习条件分布总体的表示是核心问题。条件均值算子（CMO）是一类函数空间之间的线性算子，通过提供对广泛条件统计量的访问来解决这些目标。然而，现有方法通常独立估计每个CMO，或将其限制在预先指定的函数空间中，从而无法利用相关分布之间的共享结构。在本工作中，我们假设相关的CMO共享有限维的输入和输出函数空间，并通过映射这些空间的线性算子针对每个任务进行特化。基于这一假设，我们提出了MTL-CMO，一个多任务框架，它在多个数据集上联合学习共享的函数空间和任务特定的算子。我们进一步提出了T-CMO，一种迁移学习方法，它重用共享空间以闭式形式估计新条件分布的算子。我们建立了统计保证，量化了联合学习共享函数空间的好处。我们的实验表明，学习共享函数空间在广泛的条件分布上改善了不确定性量化，并且当应用于Langevin和等离子体动力学时，能够生成保留物理有意义信息并支持参数识别的复杂动力学的紧凑表示。

英文摘要

Estimating conditional statistics and learning representations of a population of conditional distributions are central problems in many data-driven applications, including uncertainty quantification and dynamical systems analysis. Conditional mean operators (CMOs), a class of linear operators between function spaces, resolve these objectives by providing access to a broad class of conditional statistics. However, existing methods typically estimate each CMO independently or constrain it to prespecified function spaces, thereby preventing the exploitation of shared structure across related distributions. In this work, we posit that related CMOs share finite-dimensional input and output function spaces, and are specialized for each task with a linear operator mapping these spaces. Based on this hypothesis, we introduce MTL-CMO, a multi-task framework that jointly learns shared function spaces and task-specific operators across multiple datasets. We further introduce T-CMO, a transfer learning method that reuses the shared spaces to estimate, in closed form, the operator of a new conditional distribution. We establish statistical guarantees quantifying the benefits of jointly learning the shared function spaces. Our experiments demonstrate that learning shared function spaces improves uncertainty quantification across a broad range of conditional distributions and, when applied to Langevin and plasma dynamics, yields compact representations of complex dynamics that retain physically meaningful information and enable parameter identification.

发表机构

• Ecole Polytechnique（巴黎综合理工学院）
• University of Novi Sad（诺维萨德大学）

机构由 AI 辅助整理，请以论文原文为准。