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Integrating Classical Numerical Methods into Deep Operator Networks

  • 정경진 (아주대학교 일반대학원)

초록/요약

We propose an operator learning framework for Poisson’s equation using convolutional neural networks trained in an unsupervised manner. The loss functions are derived from classical numerical methods such as the finite difference method and the finite element method. Our approach then enforces the partial differential equation and exactly imposes Dirichlet boundary condition in an interpolation sense. Similar to traditional numerical methods, we generate a mesh for the domain and map each node to the corresponding image pixel to construct input images; each pixel in the output image corresponds to the numerical solution at a node on the mesh. We introduce a strategy that decomposes the linear partial differential equation into two subproblems, which are learned independently. The solutions from these submodels are then combined to reconstruct the approximate solution, achieving efficient training and improved accuracy. Numerical experiments show that even with a very small number of training data for each subproblem, the decomposition strategy yields more accurate approximations than training a model directly on the original problem using a large dataset. This method not only demonstrates the training efficiency resulting from the decomposition strategy but also offers advantages in terms of modularity and scalability in operator learning. Furthermore, the model size and inference time remain small even as the number of degrees of freedom increases, making it more efficient in both memory and computing time than classical numerical methods.

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목차

1 Introduction 1
2 Preliminaries 4
3 Methodology 7
3.1 Finite Difference Operator Network 9
3.2 Finite Element Operator Network 14
4 Numerical Experiments 17
4.1 Performance Evaluation 19
4.2 Data Efficiency via Decomposition Strategy 22
4.3 Inference Efficiency and Scalability 23
5 Conclusion 26
References 28

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