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This paper proposes a scalable learning framework to solve a system of coupled forward–backward partial differential equations (PDEs) arising from mean field games (MFGs). The MFG system incorporates a forward PDE to model the propagation of population dynamics and a backward PDE for a representative agent’s optimal control. Existing work mainly focus on solving the mean field game equilibrium (MFE) of the MFG system when given fixed boundary conditions, including the initial population state and terminal cost. To obtain MFE efficiently, particularly when the initial population density and terminal cost vary, we utilize a physics-informed neural operator (PINO) to tackle the forward–backward PDEs. A learning algorithm is devised and its performance is evaluated on one application domain, which is the autonomous driving velocity control. Numerical experiments show that our method can obtain the MFE accurately when given different initial distributions of vehicles. The PINO exhibits both memory efficiency and generalization capabilities compared to physics-informed neural networks (PINNs).
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Optimal control of a discrete-time plant–herbivore/pest model with bistability in fluctuating environments
Motivated by regulating/eliminating the population of herbivorous pests, we investigate a discrete-time plant–herbivore model with two different constant control strategies (removal versus reduction), and formulate the corresponding optimal control problems when its dynamics exhibits varied types of bi-stability and fluctuating environments. We provide basic analysis and identify the critical factors to characterize the optimal controls and the corresponding plant–herbivore dynamics such as the control upper bound (the effectiveness level of the implementation of control measures) and the initial conditions of the plant and herbivore. Our results show that optimal control could be easier when the model has simple dynamics such as stable equilibrium dynamics under constant environment or the model exhibits chaotic dynamics under fluctuating environments. Due to bistability, initial conditions are important for optimal controls. Regardless of with or without fluctuating environments, initial conditions taken from the near the boundary makes optimal control easier. In general, the pest is hard to be eliminated when the control upper bound is not large enough. However, as the control upper bound is increased or the initial conditions are chosen from near the boundary of the basin of attractions, the pest can be manageable regardless of the fluctuating environments.
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- PAR ID:
- 10383882
- Date Published:
- Journal Name:
- Mathematical Biosciences and Engineering
- Volume:
- 19
- Issue:
- 5
- ISSN:
- 1551-0018
- Page Range / eLocation ID:
- 5075 to 5103
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3934/mbe.2022237
