Neural Evolutionary Kernel Method: A Knowledge-Based Learning Architechture for Evolutionary PDEs
Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)
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Keywords: Numerical PDE, structure preserving neural network, operator learning, boundary integral
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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. DNN-based methods exploit the approximation capabilities of neural networks to obtain solutions to PDEs in general domains or high-dimensional spaces. However, many of these methods lack the use of mathematical prior knowledge, and DNN-based methods usually require a large number of sample points and parameters, making them computationally expensive and challenging to train. This paper aims to introduce a novel method named the Neural Evolutionary Kernel Method (NEKM) for solving a class of evolutionary PDEs through DNNs based kernels. By using operator splitting and boundary integral techniques, we propose particular neural network architectures which approximate evolutionary kernels of solutions and preserve structures of time-dependent PDEs. Mathematical prior knowledge are naturally built into these DNNs based kernels through convolutional representation with pre-trained Green functions, leading to serious reduction in the number of parameters in the NEKM and very efficient training processes. Experimental results demonstrate the efficiency and accuracy of the NEKM in solving heat equations and Allen-Cahn equations in complex domains and on manifolds, showcasing its promising potential for applications in data driven scientific computing.
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Submission Number: 9498
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