Multi-Task Learning of Conditional Mean Operators: applications to dynamical systems and uncertainty quantification

2026-09-28

AI Summary

This study addresses the limitation of existing conditional mean operator (CMO) estimation methods, which perform independent estimation and neglect shared structures across distributions. To this end, we propose MTL-CMO, a multi-task learning framework that assumes related CMOs share a finite-dimensional function space. By jointly learning this shared space alongside task-specific operators, the method enables efficient modeling across multiple datasets. Furthermore, we introduce T-CMO, a transfer learning approach that derives closed-form solutions and establishes statistical error bounds under novel distributions. Integrating functional analysis with linear operator theory, our framework effectively enhances uncertainty quantification accuracy. It generates compact, physically meaningful representations for complex dynamical systems while supporting parameter identification tasks.

Abstract

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.

Problem

• Conditional Mean Operators
• Multi-Task Learning
• Uncertainty Quantification
• Dynamical Systems

Innovation

• Conditional Mean Operators
• Multi-Task Learning
• Transfer Learning
• Uncertainty Quantification
• Dynamical Systems