Single-step Neural Operator Solver for Semilinear Evolution Equations

Abstract

The neural operator theory offers a promising framework for efficiently solving complex systems governed by partial differential equations (PDEs for short). However, existing neural operators still face significant challenges when applied to spatiotemporal systems that evolve over large time scales, particularly those described by evolution PDEs with time-derivative terms. This paper introduces a novel neural operator designed explicitly for solving evolution equations based on the theory of operator semigroups. The proposed approach is an iterative algorithm where each computational unit, termed the single-step neural operator solver (SSNOS for short), approximates the solution operator for the initial-boundary value problem of semilinear evolution equations over a single time step. The SSNOS consists of both linear and nonlinear components: The linear part approximates the linear operator in the solution map; in contrast, the nonlinear part captures deviations in the solution function caused by the equations nonlinearities. To evaluate the performance of the algorithm, the authors conducted numerical experiments by solving the initial-boundary value problem for a two-dimensional semilinear hyperbolic equation. The experimental results demonstrate that their neural operator can efficiently and accurately approximate the true solution operator. Moreover, the model can achieve a relatively high approximation accuracy with simple pre-training.

Keywords

Neural operator / Evolution equations / Semigroup of operators / Predictor-corrector method

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