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Mean-field neural networks-based algorithms for McKean-Vlasov control problems *

Huyên Pham, Xavier Warin

TL;DR

This paper proposes several algorithms either based on dynamic programming with control learning by policy or value iteration, or backward SDE from stochastic maximum principle with global or local loss functions with global or local loss functions.

Abstract

This paper is devoted to the numerical resolution of McKean-Vlasov control problems via the class of mean-field neural networks introduced in our companion paper [25] in order to learn the solution on the Wasserstein space. We propose several algorithms either based on dynamic programming with control learning by policy or value iteration, or backward SDE from stochastic maximum principle with global or local loss functions. Extensive numerical results on different examples are presented to illustrate the accuracy of each of our eight algorithms. We discuss and compare the pros and cons of all the tested methods.

Mean-field neural networks-based algorithms for McKean-Vlasov control problems *

TL;DR

This paper proposes several algorithms either based on dynamic programming with control learning by policy or value iteration, or backward SDE from stochastic maximum principle with global or local loss functions with global or local loss functions.

Abstract

This paper is devoted to the numerical resolution of McKean-Vlasov control problems via the class of mean-field neural networks introduced in our companion paper [25] in order to learn the solution on the Wasserstein space. We propose several algorithms either based on dynamic programming with control learning by policy or value iteration, or backward SDE from stochastic maximum principle with global or local loss functions. Extensive numerical results on different examples are presented to illustrate the accuracy of each of our eight algorithms. We discuss and compare the pros and cons of all the tested methods.
Paper Structure (28 sections, 72 equations, 23 figures, 29 tables, 8 algorithms)

This paper contains 28 sections, 72 equations, 23 figures, 29 tables, 8 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characterization of McKean-Vlasov control
  4. Mean-field neural networks
  5. Dynamic programming-based algorithms
  6. Global learning on control
  7. Control learning by policy iteration
  8. Control learning by value iteration
  9. Backward SDE-based algorithms
  10. Local algorithms
  11. Global algorithms
  12. Numerical examples
  13. The test examples
  14. Systemic risk model
  15. Min/max linear quadratic MKV control
  16. ...and 13 more sections

Figures (23)

  • Figure 1: Bin approximation of a Gaussian distribution.
  • Figure : Case 1
  • Figure : Case 1
  • Figure : Case 1
  • Figure : Case 1
  • ...and 18 more figures