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Optimization with learning-informed differential equation constraints and its applications
Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.
Prediction of Short Fiber Composite Properties by an Artificial Neural Network Trained on an RVE Database
In this study, an artificial neural network is designed and trained to predict the elastic properties of short fiber reinforced plastics. The results of finite element simulations of three-dimensional representative volume elements are used as a data basis for the neural network. The fiber volume fraction, fiber length, matrix-phase properties, and fiber orientation are varied so that the neural network can be used within a very wide range of parameters. A comparison of the predictions of the neural network with additional finite element simulations shows that the stiffnesses of short fiber reinforced plastics can be predicted very well by the neural network. The average prediction accuracy is equal or better than by a two-step homogenization using the classical method of Mori and Tanaka. Moreover, it is shown that the training of the neural network on an extended data set works well and that particularly calculation-intensive data points can be avoided without loss of prediction quality.