Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks
This work addresses robust and efficient optimization for PDE problems like control and shape optimization, offering a novel active learning approach that is incremental in improving upon existing deep learning methods.
The authors tackled PDE constrained optimization by introducing ISMO, an active learning algorithm that iteratively selects training data via a feedback loop between deep neural networks and standard optimization algorithms, achieving exponential convergence with exponentially decaying variance in optimizers and significantly outperforming standard deep neural network surrogate methods in numerical examples.
We present a novel active learning algorithm, termed as iterative 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.