Learning Operators through Coefficient Mappings in Fixed Basis Spaces
Chuqi Chen, Yang Xiang, Weihong Zhang
TL;DR
The paper tackles operator learning for PDE mappings by proposing FB-C2CNet, which learns operators in the coefficient space induced by fixed bases, thereby decoupling basis selection from network training and enabling efficient, scalable learning. By projecting both input and output functions onto fixed basis spaces (e.g., random features or FEM), a neural network maps input coefficients to output coefficients, with reconstruction performed in the fixed output basis; the training uses a relative loss and regularized least-squares coefficient encodings. Theoretical analysis shows the ultimate accuracy is bounded by the output-basis approximation error, and extensive experiments across 1D/2D Darcy flow, Poisson equations on complex domains, elasticity, and high-dimensional Poisson problems demonstrate high accuracy, stability, and resolution-generalization, with multi-input/output extensions showing favorable performance for vector-valued bases. The work provides a practical, principled framework for operator learning that reduces training cost, clarifies the role of basis choice, and offers robust performance across varying geometries and dimensions, making it attractive for fast, generalizable PDE solvers in engineering and physics applications.
Abstract
Operator learning has emerged as a powerful paradigm for approximating solution operators of partial differential equations (PDEs) and other functional mappings. \textcolor{red}{}{Classical approaches} typically adopt a pointwise-to-pointwise framework, where input functions are sampled at prescribed locations and mapped directly to solution values. We propose the Fixed-Basis Coefficient to Coefficient Operator Network (FB-C2CNet), which learns operators in the coefficient space induced by prescribed basis functions. In this framework, the input function is projected onto a fixed set of basis functions (e.g., random features or finite element bases), and the neural operator predicts the coefficients of the solution function in the same or another basis. By decoupling basis selection from network training, FB-C2CNet reduces training complexity, enables systematic analysis of how basis choice affects approximation accuracy, and clarifies what properties of coefficient spaces (such as effective rank and coefficient variations) are critical for generalization. Numerical experiments on Darcy flow, Poisson equations in regular, complex, and high-dimensional domains, and elasticity problems demonstrate that FB-C2CNet achieves high accuracy and computational efficiency, showing its strong potential for practical operator learning tasks.