Neural Evolutionary Kernel Method: A Knowledge-Guided Framework for Solving Evolutionary PDEs
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914328341381120 |
|---|---|
| author | Ling, Shuo Ying, Wenjun Zhang, Zhen |
| author_facet | Ling, Shuo Ying, Wenjun Zhang, Zhen |
| contents | Numerical solution of partial differential equations (PDEs) plays a vital role in various fields of science and engineering. In recent years, deep neural networks (DNNs) have emerged as a powerful tool for solving PDEs, leveraging their approximation capabilities to handle complex domains and high-dimensional problems. Among these, operator learning has gained increasing attention by learning mappings between function spaces using DNNs. This paper proposes a novel approach, termed the Neural Evolutionary Kernel Method (NEKM), for solving a class of time-dependent partial differential equations (PDEs) via deep neural network (DNN)-based kernel representations. By integrating boundary integral techniques with operator learning, prior mathematical information of time-dependent partial differential equations (PDEs) is embedded into the design of neural network architectures for predicting their solutions, enhancing both computational efficiency and solution accuracy. Numerical experiments on the heat, wave, and Schrödinger equations demonstrate that the Neural Evolutionary Kernel Method (NEKM) achieves high accuracy and favorable computational efficiency. Furthermore, the operator learning framework inherently supports the simultaneous prediction of solutions to multiple PDEs with different coefficients, rendering its capability for solving random PDEs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12872 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Neural Evolutionary Kernel Method: A Knowledge-Guided Framework for Solving Evolutionary PDEs Ling, Shuo Ying, Wenjun Zhang, Zhen Numerical Analysis Numerical solution of partial differential equations (PDEs) plays a vital role in various fields of science and engineering. In recent years, deep neural networks (DNNs) have emerged as a powerful tool for solving PDEs, leveraging their approximation capabilities to handle complex domains and high-dimensional problems. Among these, operator learning has gained increasing attention by learning mappings between function spaces using DNNs. This paper proposes a novel approach, termed the Neural Evolutionary Kernel Method (NEKM), for solving a class of time-dependent partial differential equations (PDEs) via deep neural network (DNN)-based kernel representations. By integrating boundary integral techniques with operator learning, prior mathematical information of time-dependent partial differential equations (PDEs) is embedded into the design of neural network architectures for predicting their solutions, enhancing both computational efficiency and solution accuracy. Numerical experiments on the heat, wave, and Schrödinger equations demonstrate that the Neural Evolutionary Kernel Method (NEKM) achieves high accuracy and favorable computational efficiency. Furthermore, the operator learning framework inherently supports the simultaneous prediction of solutions to multiple PDEs with different coefficients, rendering its capability for solving random PDEs. |
| title | Neural Evolutionary Kernel Method: A Knowledge-Guided Framework for Solving Evolutionary PDEs |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2602.12872 |