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Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks

by K. Lye and S. Mishra and D. Ray and P. Chandrashekar

(Report number 2020-53)

Abstract
We present a novel active learning algorithm, termed a iiterative surrogate model optimization (ISMO), for robust and efficient numerical approximation of PDE constrained optimization problems. This algorithm is based on deep neural networks and its key feature is the iterative selection of training data through a feedback loop between deep neural networks and any underlying standard optimization algorithm. Under suitable hypotheses, we show that the resulting optimizers converge exponentially fast (and with exponentially decaying variance), with respect to increasing number of training samples. Numerical examples for optimal control, parameter identification and shape optimization problems for PDEs are provided to validate the proposed theory and to illustrate that ISMO significantly outperforms a standard deep neural network based surrogate optimization algorithm.

Keywords: Deep Neural Networks, Active Learning, Surrogates, PDE constrained optimization

@Techreport{LMRC20_926,
  author = {K. Lye and S. Mishra and D. Ray and P. Chandrashekar},
  title = {Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-53},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-53.pdf },
  year = {2020}
}

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