Polynomial Chaos Expansion for Operator Learning

2025-08-28

AI Summary

This work addresses operator learning for partial differential equations (PDEs)—a mapping between infinite-dimensional function spaces—by proposing the first polynomial chaos expansion (PCE)-based operator learning framework. Methodologically, it reformulates operator approximation as a PCE coefficient estimation problem, integrating functional space projection with numerical quadrature to jointly embed data-driven modeling and physical constraints (e.g., PDE residuals). Key contributions include: (i) the first systematic incorporation of PCE into operator learning, enabling natural, zero-overhead uncertainty quantification; and (ii) a framework achieving high accuracy, strong robustness, and computational efficiency. Numerical experiments across diverse PDE benchmarks demonstrate superior accuracy, generalization, and uncertainty characterization compared to state-of-the-art operator learning methods.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Operator learning (OL) has emerged as a powerful tool in scientific machine learning (SciML) for approximating mappings between infinite-dimensional functional spaces. One of its main applications is learning the solution operator of partial differential equations (PDEs). While much of the progress in this area has been driven by deep neural network-based approaches such as Deep Operator Networks (DeepONet) and Fourier Neural Operator (FNO), recent work has begun to explore traditional machine learning methods for OL. In this work, we introduce polynomial chaos expansion (PCE) as an OL method. PCE has been widely used for uncertainty quantification (UQ) and has recently gained attention in the context of SciML. For OL, we establish a mathematical framework that enables PCE to approximate operators in both purely data-driven and physics-informed settings. The proposed framework reduces the task of learning the operator to solving a system of equations for the PCE coefficients. Moreover, the framework provides UQ by simply post-processing the PCE coefficients, without any additional computational cost. We apply the proposed method to a diverse set of PDE problems to demonstrate its capabilities. Numerical results demonstrate the strong performance of the proposed method in both OL and UQ tasks, achieving excellent numerical accuracy and computational efficiency.

Problem

• Approximating mappings between infinite-dimensional functional spaces
• Learning solution operators of partial differential equations
• Providing uncertainty quantification for operator learning methods

Innovation

• Polynomial chaos expansion for operator learning
• Reduces operator learning to solving equations
• Provides uncertainty quantification via post-processing