DeepSet and their derivative networks for solving symmetric problems
Abstract
Machine learning methods represent a breakthrough for solving nonlinear partial differential equations (PDEs) and control problems in very high dimension, and have been the subject of intense research over the last five years. In this talk, we consider a widespread class of problems that are invariant to permutations of their inputs (state variables or model parameters). This occurs for example in multi- asset models for option pricing with exchangeable payoff, or for optimal trading portfolio with respect to the market price of covariance risk. Our main application comes actually from mean-field control problems and the corresponding PDEs in the Wasserstein space of probability measures. Their particle approximations, for which we provide a rate of convergence, lead to symmetric PDEs that are solved by deep learning algorithms based on certain types of neural networks, named DeepSet. We illustrate the performance and accuracy of the DeepSet networks compared to classical feedforward ones, and provide several numerical results of our algorithm for the examples of a mean-field systemic risk, and mean-variance problem. Finally, we show how the combination of DeepSet and DeepOnet, a network architecture recently proposed for learning operators, provides an efficient approximation for a family of optimal trading strategies in terms of market price of covariance risk coefficients.
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