Neural Evolutionary Kernel Method: A Knowledge-Guided Framework for Solving Evolutionary PDEs

TL;DR

The paper tackles time-dependent PDEs by learning operators that map boundary data, source terms, and PDE parameters to interior solutions. It fuses neural operator learning with boundary integral methods to construct two specialized operators: a volumetric source-term operator and a boundary-density operator governed by a second-kind integral equation, enabling stable, efficient time stepping. NEKM demonstrates high-accuracy solutions for the heat, wave, and Schrödinger equations on both simple and complex domains, including random PDEs, and supports parallel evaluation for uncertainty quantification. This approach offers a scalable surrogate framework with potential impact on UQ, inverse problems, and simulations across parameter families.

Abstract

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.

Figures (8)

• Figure 1: Schematic diagram of the network architecture described in \ref{['eq0608_1']}. In this framework, the sub-networks $\mathcal{NN}_f$, $\mathcal{NN}_k$, and $\mathcal{NN}_G$ can be instantiated using architectures such as multi-layer perceptrons (MLPs), residual neural networks (ResNets), or other function-approximating models.
• Figure 2: Numerical results for Model 1 with $\kappa = 0.067$ which is not in the training set.
• Figure 3: Numerical results for Model 2 with $\kappa = \frac{1}{72}$ which is not in the training set.
• Figure 6: Numerical results at time $T = 4$ for wave equation \ref{['eq0220_2']} with $a = 0.6, \tau = \frac{1}{4}$.
• Figure 7: Numerical results at time $T = 4$ for wave equation \ref{['eq0220_2']} with $a = 0.6, \tau = \frac{1}{6}$.

Theorems & Definitions (8)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Definition A.1
• Definition A.2
• Definition A.3