Self-adaptive deep neural network: Numerical approximation to functions and PDEs

• URL: http://arxiv.org/abs/2109.02839v1
• Date: Tue, 7 Sep 2021 03:16:57 GMT
• Title: Self-adaptive deep neural network: Numerical approximation to functions and PDEs
• Authors: Zhiqiang Cai, Jingshuang Chen, Min Liu
• Abstract summary: We introduce a self-adaptive algorithm for designing an optimal deep neural network for a given task. The ANE method is written as loops of the form train, estimate and enhance. We demonstrate that the ANE method can automatically design a nearly minimal NN for learning functions exhibiting sharp transitional layers.
• Score: 3.6525914200522656
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Designing an optimal deep neural network for a given task is important and challenging in many machine learning applications. To address this issue, we introduce a self-adaptive algorithm: the adaptive network enhancement (ANE) method, written as loops of the form train, estimate and enhance. Starting with a small two-layer neural network (NN), the step train is to solve the optimization problem at the current NN; the step estimate is to compute a posteriori estimator/indicators using the solution at the current NN; the step enhance is to add new neurons to the current NN. Novel network enhancement strategies based on the computed estimator/indicators are developed in this paper to determine how many new neurons and when a new layer should be added to the current NN. The ANE method provides a natural process for obtaining a good initialization in training the current NN; in addition, we introduce an advanced procedure on how to initialize newly added neurons for a better approximation. We demonstrate that the ANE method can automatically design a nearly minimal NN for learning functions exhibiting sharp transitional layers as well as discontinuous solutions of hyperbolic partial differential equations.

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