Abstract
We present a discontinuity detector constructed by deep neural networks. Using convolutional neural network (CNN) structure, we design a comprehensive set of synthetic training data. The data consist of randomly generated piecewise smooth functions evaluated at equidistance grids, with labels denoting troubled cells where discontinuities are present. Upon successful training of the network, the CNN based detection network is capable of accurately identifying discontinuities in newly given function data by correctly labeling the troubled cells. Even though all of our training data have fixed size, the constructed detector can be applied to function data of arbitrary size, so long as they are on equidistance grids. To increase the detection efficiency in two- and three-dimensional cases, we propose a two-level detection procedure, where the detector is applied to a coarsened grid first and then to the fine grids only at the troubled cells identified at the coarse level. Through an extensive set of numerical tests, we demonstrate that the developed detectors possess strong generalization capabilities, in the sense that they are able to accurately detect discontinuity with structures much more complex than those in the training data.
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Notes
For the CNN above, \(\mathcal {N}(\tilde{v}_{f})=\text{ FC }\circ \text{ conv5 }\circ \text{ conv4 }\circ \text{ conv3 }\circ \text{ conv2 }\circ \text{ conv1 }(\tilde{v}_{f})\), where conv1, conv2, conv3, conv4, conv5 are the functions of the five convolutional layers and FC is the function of the fully connected layer. See the description of those functions in Sect. 2.2.
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This work was partially supported by AFOSR FA9550-18-1-0102.
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Wang, S., Zhou, Z., Chang, LB. et al. Construction of discontinuity detectors using convolutional neural networks. J Sci Comput 91, 40 (2022). https://doi.org/10.1007/s10915-022-01804-z
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DOI: https://doi.org/10.1007/s10915-022-01804-z