Quantitative approximation theorem for neural operators for semilinear elliptic PDEs

抄録

We study the approximation of nonlinear partial differential equations (PDEs) using neural operators. While traditional numerical methods suffer from high computational costs due to nonlinearities and high dimensionality, neural operators provide a promising alternative by learning mappings between infinite-dimensional function spaces. In this work, we establish a quantitative approximation theorem of neural operators for the solution operator of semilinear elliptic PDEs. Our construction is based on Banach’s fixed point theorem and Picard iteration, and demonstrates that the depth and width of the neural operator grow at most logarithmically with respect to the approximation accuracy. This result shows that exponential growth in model complexity can be avoided. Our framework is generalizable to a broader class of PDEs and offers a constructive foundation for both theoretical analysis and future empirical studies.