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In recent years, there has been a growing interest in leveraging deeplearning and neural networks to address scientific problems, particularly insolving partial differential equations (PDEs). However, current neuralnetwork-based PDE solvers often rely on extensive training data or labeledinput-output pairs, making them prone to challenges in generalizing toout-of-distribution examples. To mitigate the generalization gap encountered byconventional neural network-based methods in estimating PDE solutions, weformulate a fully unsupervised approach, requiring no training data, toestimate finite difference solutions for PDEs directly via small convolutionalneural networks. Our proposed algorithms demonstrate a comparable accuracy tothe true solution for several selected elliptic and parabolic problems comparedto the finite difference method.

Solutions to Elliptic and Parabolic Problems via Finite Difference Based Unsupervised Small Linear Convolutional Neural Networks