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| Single-step Neural Operator Solver for Semilinear Evolution Equations |
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Citation： |
Zhen LEI,Lei SHI,Xiyuan WANG.Single-step Neural Operator Solver for Semilinear Evolution Equations[J].Chinese Annals of Mathematics B,2025,46(3):321~340 |
| Page view： 119
Net amount： 51 |
Authors： |
Zhen LEI; Lei SHI;Xiyuan WANG |
Foundation： |
the National Natural Science Foundation Major Program of China(No. 12494544), the National Natural Science Foundation General Program of China (No. 12171039), theNew Cornerstone Science Foundation through the XPLORER PRIZE and Sino-German Center MobilityProgramme (No. M-0548) and the Shanghai Science and Technology Program (No. 21JC1400600). |
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| Abstract： |
The neural operator theory o?ers a promising framework for efciently solvingcomplex systems governed by partial di?erential equations (PDEs for short). However,existing neural operators still face signifcant challenges when applied to spatiotemporalsystems that evolve over large time scales, particularly those described by evolution PDEswith time-derivative terms. This paper introduces a novel neural operator designedexplicitly for solving evolution equations based on the theory of operator semigroups. Theproposed approach is an iterative algorithm where each computational unit, termed thesingle-step neural operator solver (SSNOS for short), approximates the solution operatorfor the initial-boundary value problem of semilinear evolution equations over a single timestep. 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 capturesdeviations in the solution function caused by the equations nonlinearities. To evaluate theperformance of the algorithm, the authors conducted numerical experiments by solving theinitial-boundary value problem for a two-dimensional semilinear hyperbolic equation. Theexperimental results demonstrate that their neural operator can efciently and accuratelyapproximate the true solution operator. Moreover, the model can achieve a relatively highapproximation accuracy with simple pre-training. |
Keywords： |
Neural operator Evolution equations Semigroup of operators,Predictor-corrector method |
Classification： |
68T07, 65M22, 65J08 |
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