Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equations

TL;DR

This paper establishes a theoretical framework for neural operators approximating solutions to nonlinear parabolic PDEs, demonstrating their efficiency and potential for broader classes of equations.

Contribution

It provides the first quantitative approximation rate for neural operators solving nonlinear PDEs, linking neural operators to classical iterative methods like Picard's iteration.

Findings

Neural operators can approximate solution operators efficiently without exponential complexity.

The approach is based on transferring PDEs into integral equations and leveraging Picard's iteration.

Results suggest potential generalization to other nonlinear PDEs such as Navier-Stokes and nonlinear Schrödinger equations.

Abstract

Neural operators serve as universal approximators for general continuous operators. In this paper, we derive the approximation rate of solution operators for the nonlinear parabolic partial differential equations (PDEs), contributing to the quantitative approximation theorem for solution operators of nonlinear PDEs. Our results show that neural operators can efficiently approximate these solution operators without the exponential growth in model complexity, thus strengthening the theoretical foundation of neural operators. A key insight in our proof is to transfer PDEs into the corresponding integral equations via Duahamel's principle, and to leverage the similarity between neural operators and Picard's iteration, a classical algorithm for solving PDEs. This approach is potentially generalizable beyond parabolic PDEs to a range of other equations, including the Navier-Stokes equation, nonlinear Schr\"odinger equations and nonlinear wave equations, which can be solved by Picard's iteration.