四川大学学报（自然科学版）2026，Vol.63Issue(1)：100-110,11.DOI:10.19907/j.0490-6756.250138
融合数学物理知识的神经网络算子方法
Neural operator methods integrating mathematical and physical knowledge
摘要
Abstract
Traditional deep operator networks often suffer from insufficient physical consistency and limited prediction accuracy when solving Partial Differential Equations(PDEs).To address these issues,this paper proposes an improved neural operator architecture that incorporates the energy method,the maximum prin-ciple,and concepts from Green′s function.The energy method is employed to ensure that the energy evolu-tion of the solution adheres to physical laws,while the maximum principle is applied to avoid the emergence of non-physical extrema in the numerical solution.Furthermore,principles derived from Green′s function are leveraged to guide the learning of the response characteristics of the underlying operator.Specifically,the en-ergy method and the maximum principle are integrated as loss functions,and the Green′s function replaces the branch network in the traditional deep operator network with a Fourier layer.We systematically evaluated the efficacy of these strategies on several benchmark problems,including the 1D and 2D heat conduction equations,the 1D Allen-Cahn equation,and the1D Burgers equation,by assessing their impact on the solu-tion accuracy and physical credibility.The experimental results show that the proposed neural network opera-tor integrating mathematical and physical knowledge achieves lower Mean squared error(MSE)and the L2 relative error(L2RE)compared to the baseline model.For instance,the L2RE was reduced by up to 30.5%for the 1D heat conduction equation,12.79%for the 1D Burgers equation,6.28%for the 1D Allen-Cahn equation,and 57.47%for the 2D heat conduction equation.For the MSE,reductions reached a maximum of 47.06%for the 1D heat equation,20.58%for the 1D Burgers equation,15.78%for the 1D Allen-Cahn equation,and 81.95%for the 2D heat conduction equation.Our resutls demonstrate that the Green′s func-tion method yielded the most significant accuracy improvement,the energy method excelled in energy-conservative equations,and the maximum principle effectively suppresses non-physical extrema.The result-ing solutions strictly adhere to the laws of energy conservation or energy dissipation and satisfy the maximum boundary,showing a significant improvement in physical consistency.关键词
深度算子网络/物理约束/能量方法/最大值原理/格林函数Key words
DeepONet/physical constraints/energy method/maximum principle/green's function分类
数理科学引用本文复制引用
刘泳成,黄睿,俞靓文,胡明,冯文韬,周吉喆,叶庆,黄树东,吕建成..融合数学物理知识的神经网络算子方法[J].四川大学学报（自然科学版）,2026,63(1):100-110,11.基金项目
国家自然科学基金青年科学基金项目(62306199) （62306199）
