Neural network as a function approximator and its application in solving differential equations
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A neural network (NN) is a powerful tool for approximating bounded continuous functions in machine learning. The NN provides a framework for numerically solving ordinary differential equations (ODEs) and partial differential equations (PDEs) combined with the automatic differentiation (AD) technique. In this work, we explore the use of NN for the function approximation and propose a universal solver for ODEs and PDEs. The solver is tested for initial value problems and boundary value problems of ODEs, and the results exhibit high accuracy for not only the unknown functions but also their derivatives. The same strategy can be used to construct a PDE solver based on collocation points instead of a mesh, which is tested with the Burgers equation and the heat equation (i.e., the Laplace equation).
Key wordsneural network (NN) function approximation ordinary differential equation (ODE) solver partial differential equation (PDE) solver
Chinese Library ClassificationO241
2010 Mathematics Subject Classification65D15
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- HAYKIN, S. S. Neural Networks and Learning Machines, 3rd ed., Prentice Hall, Englewood, New Jersey (2008)Google Scholar
- KRIZHEVSKY, A., SUTSKEVER, I., and HINTON, G. E. Imagenet classification with deep convolutional neural networks. Proceedings of the 25th International Conference on Neural Information Processing Systems, 1, 1097–1105 (2012)Google Scholar
- WERBOS, P. Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences, Ph.D. dissertation, Harvard University (1974)Google Scholar
- RAISSI, M., PERDIKARIS, P., and KARNIADAKIS, G. E. Physics informed deep learning (part i): data-driven solutions of nonlinear partial differential equations. arXiv, arXiv: 1711.10561 (2017) https://arxiv.org/abs/1711.10561Google Scholar
- BERG, J. and NYSTRÖM, K. A unified deep artificial neural network approach to partial differential equations in complex geometries. arXiv, arXiv: 1711.06464 (2017) https://arxiv.org/abs/ 1711.06464Google Scholar
- ABADI, M., BARHAM, P., CHEN, J., CHEN, Z., DAVIS, A., DEAN, J., DEVIN, M., GHEMAWAT, S., IRVING, G., ISARD, M., and KUDLUR, M. Tensorflow: a system for large-scale machine learning. The 12th USENIX Symposium on Operating Systems Design and Implementation, 16, 265–283 (2016)Google Scholar