A mean field game inverse problem

Ding, Lisang; Li, Wuchen; Osher, Stanley; Yin, Wotao

doi:10.48550/arxiv.2007.11551

Cited by 4 publications

(6 citation statements)

References 28 publications

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“…The probability measure constraint was not treated [28] The running cost F F belongs to an analytic class It is difficult to recover more than one unknowns with the MFG system [16] The running cost F and Hamiltonian…”

Section: Unknowns Belong To Certain Analytic Classesmentioning

confidence: 99%

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Simultaneously recovering running cost and Hamiltonian in Mean Field Games system

Liu¹,

Zhang²

2023

Preprint

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We propose and study several inverse problems for the mean field games (MFG) system in a bounded domain. Our focus is on simultaneously recovering the running cost and the Hamiltonian within the MFG system by the associated boundary observation. There are several technical novelties that make our study highly intriguing and challenging. First, the MFG system couples two nonlinear parabolic PDEs within one moving forward and the other one moving backward in time. Second, there is a probability measure constraint on the population distribution of the agents. Third, the simultaneous recovery of two coupling factors within the MFG system is far from being trivial. We present two different techniques that can ensure the probability constraint as well as effectively tackle the inverse problems, which are respectively termed as high-order variation and successive linearisation. In particular, the high-order variation method is new to the literature, which demonstrates a novel concept to examine the inverse problems by positive inputs only. Both cases when the running cost depends on the measure locally and non-locally are investigated.

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Section: Unknowns Belong To Certain Analytic Classesmentioning

confidence: 99%

“…On the other hand, the inverse problems for MFGs are far less studied in the literature. To our best knowledge, there are only several numerical results available in [13,16]. In [27], the authors derived unique identifiability results for an MFG system with unknown running cost and total cost.…”

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confidence: 99%

Simultaneously recovering running cost and Hamiltonian in Mean Field Games system

Liu¹,

Zhang²

2023

Preprint

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“…Despite of the large body of work on theory, numerical methods, and applications [2], inverse problems arisen from MFG is still quite an unexplored terrain. To the best of our knowledge, only [4,9,16] study such problems. The work in [9] is the closest to our objective but considers the case with a full space-time measurement of data in the sampling domain.…”

Section: Related Workmentioning

confidence: 99%

“…To the best of our knowledge, only [4,9,16] study such problems. The work in [9] is the closest to our objective but considers the case with a full space-time measurement of data in the sampling domain. However, most inverse problems in practice only have partial boundary measurements available, either obtained via non-invasive measurement methods or because of the limited access to the sampling domain.…”

Section: Related Workmentioning

confidence: 99%

A numerical algorithm for inverse problem from partial boundary measurement arising from mean field game problem

Chow¹,

Fung²,

Liu³

et al. 2022

Inverse Problems

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In this work, we consider a novel inverse problem in mean-field games (MFG). We aim to recover the MFG model parameters that govern the underlying interactions among the population based on a limited set of noisy partial observations of the population dynamics under the limited aperture. Due to its severe ill-posedness, obtaining a good quality reconstruction is very difficult. Nonetheless, it is vital to recover the model parameters stably and efficiently in order to uncover the underlying causes for population dynamics for practical needs. Our work focuses on the simultaneous recovery of running cost and interaction energy in the MFG equations from a finite number of boundary measurements of population profile and boundary movement. To achieve this goal, we formalize the inverse problem as a constrained optimization problem of a least squares residual functional under suitable norms. We then develop a fast and robust operator splitting algorithm to solve the optimization using techniques including harmonic extensions, three-operator splitting scheme, and primal-dual hybrid gradient method. Numerical experiments illustrate the effectiveness and robustness of the algorithm.

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“…To the best of our knowledge, only some numerical studies have been conducted to the inverse problem of MFGs. It starts from the recent work Ding-Li-Osher-Yin [15]. The authors reconstructed the running cost from the observation of the distribution of the population and the agents' strategy.…”

Section: Introductionmentioning

confidence: 99%

Inverse problems for mean field games

Liu¹,

Mou²,

Zhang³

2022

Preprint

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The theory of mean field games studies the limiting behaviors of large systems where the agents interact with each other in a certain symmetric way. The running and terminal costs are critical for the agents to decide the strategies. However, in practice they are often partially known or totally unknown for the agents, while the total cost is known at the end of the game. To address this challenging issue, we propose and study several inverse problems on mean field games. When the Lagrangian is a kinetic energy, we first establish unique identifiability results, showing that one can recover either the running cost or the terminal cost from observation of the total cost. If the running cost is limited to the time-independent class, we can further prove that one can simultaneously recover both the running and the terminal costs. Finally, we extend the results to the setup with general Lagrangians. Contents 1. Introduction 1 2. Statement of Main Results 3 2.1. Notations and Basic Setting 3 2.2. Mean Field Game 4 2.3. Inverse Problem 5 3. Well-posedness of the forward problems 7 4. Non-uniqueness and necessity of admissibility conditions 8 5. Proofs of Theorems 2.1, 2.2 and 2.3 9 5.1. Higher-order linearization 9 5.2. Unique determination of single unknown function 11 5.3. Simultaneous recovery results for inverse problems 14 6. General Lagrangians 17 6.1. Well-posedness of the general system 18 6.2. Proof of Theorem 6.1 and 6.2 18 Acknowledgment 22 References 22

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A mean field game inverse problem

Cited by 4 publications

References 28 publications

Simultaneously recovering running cost and Hamiltonian in Mean Field Games system

Simultaneously recovering running cost and Hamiltonian in Mean Field Games system

A numerical algorithm for inverse problem from partial boundary measurement arising from mean field game problem

Inverse problems for mean field games

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