# Polynomial Chaos Expansion for Operator Learning

## TL;DR

This paper introduces polynomial chaos expansion as a novel operator learning method in scientific machine learning, capable of approximating PDE solution operators with high accuracy and providing uncertainty quantification without extra cost.

## Contribution

It establishes a mathematical framework for PCE in operator learning, enabling data-driven and physics-informed approximations, and demonstrates its effectiveness on various PDE problems.

## Key findings

- High numerical accuracy in operator approximation
- Efficient uncertainty quantification via PCE coefficients
- Strong performance on diverse PDE problems

## 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.