pith. sign in

arxiv: 2504.13228 · v4 · submitted 2025-04-17 · 💻 cs.LG · cs.GT

Neural Mean-Field Games: Extending Mean-Field Game Theory with Neural Stochastic Differential Equations

Anna C.M. Th\"oni , Yoram Bachrach , Tal Kachman This is my paper

Pith reviewed 2026-05-22 19:09 UTC · model grok-4.3

classification 💻 cs.LG cs.GT
keywords mean-field gamesneural stochastic differential equationsdata-driven modelingstrategic interactionsepidemic simulationautomatic differentiation
0
0 comments X

The pith

Neural stochastic differential equations extend mean-field game theory to learn strategic interactions directly from data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a new modeling approach that merges mean-field game theory with neural stochastic differential equations to create data-driven representations of large-scale strategic interactions. Traditional mean-field methods require analytical solutions to systems of partial differential equations, which can introduce modeling bias and fail to guarantee existence or uniqueness of solutions. By training neural SDEs via automatic differentiation on observations, the approach learns player distributions and behaviors in games that vary in noise, observability, and complexity. A sympathetic reader would care because the method handles real-world scenarios like epidemic spread from limited data, reducing reliance on hand-crafted models.

Core claim

Neural mean-field games combine mean-field game theory with neural stochastic differential equations to produce a data-driven, lightweight model that learns extensive strategic interactions from observations, using automatic differentiation to improve robustness over finite-difference methods while solving games of varying complexity and simulating viral dynamics on real data.

What carries the argument

Neural stochastic differential equations that parameterize the dynamics of player distributions in mean-field games and are trained end-to-end via automatic differentiation on observed trajectories.

If this is right

  • The model solves mean-field games that differ in complexity, observability, and noise levels.
  • It accurately reproduces epidemic outbreak evolution when trained on real-world viral data.
  • It learns the data distribution using few observations.
  • Automatic differentiation makes the method more robust and objective than finite-difference alternatives.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same neural-SDE construction could be tested on other large-population systems such as traffic or financial markets where analytical mean-field solutions are unavailable.
  • If the learned models preserve solution existence and uniqueness in practice, they could serve as a practical workaround for cases where classical mean-field theory breaks down.
  • A direct comparison on synthetic data with injected noise would quantify how much the neural approach reduces modeling bias relative to hand-specified interaction functions.

Load-bearing premise

That neural SDEs trained on limited observations can recover the true underlying strategic interactions and distributions without introducing new modeling bias or losing the existence and uniqueness properties of classical mean-field solutions.

What would settle it

Run the trained neural model on a simple mean-field game with a known closed-form analytical solution and check whether the learned distribution and interaction terms match the analytical result within a small error margin.

Figures

Figures reproduced from arXiv: 2504.13228 by Anna C.M. Th\"oni, Tal Kachman, Yoram Bachrach.

Figure 1
Figure 1. Figure 1: An overview of modelling MFGs with neural SDEs. The resulting neural mean-field game combines the MFG mechanics with the neural network output to create more informed strategies. this paper are fourfold: (1) We introduce neural MFGs, providing the theoretical framework for combining MFG theory with neural 1 arXiv:2504.13228v3 [cs.LG] 17 Oct 2025 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The distribution of meeting arrival times for the standard version of the meeting arrival times game. The randomly initialized distribution of arrival times quickly converges to a narrow distribution centred at ˜s. The value of ˜s is subject to the Brownian noise of the SDE. The behaviour of all other agents is summarized in the actual start￾ing time ˜s, which is the mean-field of this game that considers … view at source ↗
Figure 4
Figure 4. Figure 4: The distribution of probabilities of going to the bar for the standard version of the El Farol Bar problem. In the standard ver￾sion, the distribution of probabilities converges towards the crowding threshold of c = 0.9 (blue). The density histograms show the normal￾ized distributions at turn 15. 2 4 6 8 10 12 14 Turn index (t) 0.00 0.25 0.50 0.75 1.00 0 0.5 1 Density at t = 15 [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 6
Figure 6. Figure 6: The state transition diagram of a single player in the SIR model. The figure describes the player’s possible states: susceptible (S), infected (I), and recovered (R). The transitions between states depend on the transition parameters γ, ρ, and π, and the distribution of the population m at time t. The player’s state does not change once they have recovered from an infection or have been vaccinated. This ob… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the noisy, observed data (orange) and the predictions made with the neural MFG (blue) for the proportions of the Japanese population that are susceptible, infected, or removed. The results are normalized with respect to the total population to the domain [0, 1] and consider the period between October 1st, 2020, and October 3rd, 2021. with Gaussian noise (where ε ∼ N (0, 0.05), scaled to the d… view at source ↗
Figure 8
Figure 8. Figure 8: Predictions made with a neural SDE (blue) with a predefined, deterministic drift based on Equation 9 and a neural diffusion. The results are normalized with respect to the total population to the domain [0, 1] and consider the period between October 1st, 2020, and October 3rd, 2021. highlight a biological application, we expect the model to find ap￾plications in any system associated with non-atomic, anony… view at source ↗
Figure 9
Figure 9. Figure 9: a shows the influence of the initial bluff strategy on the game length. The results from Figure 9a suggest that the strategy from the neural ODE players (orange) tends to yield shorter games for λ0 < 3 compared to the strategy based on the MFG dynamics (blue). This effect is caused by the larger drift of λ in the neural ODE, where the neural ODE-based players are more likely to learn larger values for λt f… view at source ↗
Figure 10
Figure 10. Figure 10: The KL divergence between the true (θ) and estimated (θˆT ) dice odds as a function of the number of training dice considered. The KL divergence was averaged for all players in 100 games. The error bars indicate a standard deviation from this average. C.3 The ratio of successful challenges As mentioned in Section C.1, a more deceitful playing style of the neural ODE-based players does not seem to increase… view at source ↗
Figure 11
Figure 11. Figure 11: A comparison between the bluffing strategies of the agents that are purely based on the MFG mechanics (blue) and the agents that are based on a neural ODE (orange). All agents have a correct belief about the dice odds. (a) presents the learned versus initial bluff strategy, and the ratio of successful challenges as a function of the initial bluff strategy is shown in (b). The results are aggregated over 1… view at source ↗
Figure 12
Figure 12. Figure 12: A comparison between bluffing strategies within a game with unfair dice. The strategies of the players that are purely based on the MFG dynamics are shown in blue, and the ones of the neural ODE-based players are shown in orange. (a), the game length as a function of the initial bluff strategy. (b), the learned bluff strategy versus the initial bluff strategy. (c) the relation between the initial bluff st… view at source ↗
read the original abstract

Mean-field game theory relies on approximating games that are intractable to model due to a very large to infinite population of players. While these kinds of games can be solved analytically via the associated system of partial derivatives, this approach is not model-free, can lead to the loss of the existence or uniqueness of solutions, and may suffer from modelling bias. To reduce the dependency between the model and the game, we introduce neural mean-field games: a combination of mean-field game theory and deep learning in the form of neural stochastic differential equations. The resulting model is data-driven, lightweight, and can learn extensive strategic interactions that are hard to capture using mean-field theory alone. In addition, the model is based on automatic differentiation, making it more robust and objective than approaches based on finite differences. We highlight the efficiency and flexibility of our approach by solving two mean-field games that vary in their complexity, observability, and the presence of noise. Lastly, we illustrate the model's robustness by simulating viral dynamics based on real-world data. Here, we demonstrate that the model's ability to learn from real-world data helps to accurately model the evolution of an epidemic outbreak. Using these results, we show that the model is flexible, generalizable, and requires few observations to learn the distribution underlying the data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces neural mean-field games by combining classical mean-field game (MFG) theory with neural stochastic differential equations (SDEs). The central claim is that this data-driven framework, trained via automatic differentiation on observations, yields a lightweight model that learns strategic interactions while avoiding the modeling bias, loss of existence/uniqueness, and analytical intractability of the classical coupled HJB-Fokker-Planck system. The approach is illustrated on two synthetic MFGs of varying complexity and on real epidemic data, with claims of robustness, generalizability, and the ability to learn from few observations.

Significance. If the neural SDE construction can be shown to recover a valid MFG equilibrium (i.e., a measure flow consistent with best-response optimality), the method would offer a practical route to data-driven MFG modeling in settings where analytical solutions are unavailable. The use of automatic differentiation and limited-data training are potentially useful strengths, but the significance hinges on whether the learned dynamics preserve the game-theoretic fixed-point condition rather than merely matching observed marginals.

major comments (3)
  1. [Abstract and §3 (method description)] The manuscript replaces the classical HJB-Fokker-Planck fixed-point system with a generic neural SDE trained by auto-diff on trajectories, yet provides no derivation or verification that the learned drift and interaction terms satisfy the best-response optimality condition required for an MFG equilibrium. Without this link, the model may fit observed distributions without solving the underlying game.
  2. [§4 (synthetic experiments) and §5 (epidemic application)] The claims of robustness and avoidance of existence/uniqueness issues rest on the neural SDE being a faithful proxy for the MFG; however, no analysis (e.g., via fixed-point residual, optimality gap, or comparison to a known analytical equilibrium) is presented to confirm that the trained model satisfies the MFG equilibrium definition rather than producing a non-equilibrium trajectory fit.
  3. [§5] The epidemic example uses real-world data to demonstrate learning from few observations, but lacks controls (e.g., comparison to a classical MFG fit or ablation on observation density) that would isolate whether the neural component recovers strategic interaction parameters or simply interpolates marginal statistics.
minor comments (2)
  1. [§3] Notation for the neural SDE drift and diffusion terms should be explicitly related to the classical MFG interaction kernel to clarify the modeling assumptions.
  2. [§4] Figure captions for the synthetic games should include the ground-truth equilibrium measure or value function for visual comparison.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and insightful comments. We address each major point below and outline revisions to strengthen the manuscript's connection to mean-field game equilibria while preserving the data-driven focus of the work.

read point-by-point responses

  1. Referee: [Abstract and §3 (method description)] The manuscript replaces the classical HJB-Fokker-Planck fixed-point system with a generic neural SDE trained by auto-diff on trajectories, yet provides no derivation or verification that the learned drift and interaction terms satisfy the best-response optimality condition required for an MFG equilibrium. Without this link, the model may fit observed distributions without solving the underlying game.

    Authors: We agree that explicit verification of best-response optimality is valuable. Our framework is intentionally data-driven: observed trajectories are treated as samples from an underlying MFG equilibrium, and the neural SDE learns the effective drift and interaction kernel that reproduce the empirical measure flow. In the revised version we will add a dedicated subsection in §3 deriving the link under the assumption that training data arise from equilibrium play, together with a numerical check of the fixed-point residual on the synthetic examples where analytical equilibria are available. revision: yes

  2. Referee: [§4 (synthetic experiments) and §5 (epidemic application)] The claims of robustness and avoidance of existence/uniqueness issues rest on the neural SDE being a faithful proxy for the MFG; however, no analysis (e.g., via fixed-point residual, optimality gap, or comparison to a known analytical equilibrium) is presented to confirm that the trained model satisfies the MFG equilibrium definition rather than producing a non-equilibrium trajectory fit.

    Authors: We acknowledge the absence of these diagnostics. For the synthetic games in §4 we will report both the fixed-point residual and an optimality-gap metric obtained by comparing the learned policy against the known best-response operator. These additions will directly test whether the trained neural SDE recovers an equilibrium rather than an arbitrary trajectory fit, thereby supporting the robustness claims. revision: yes

  3. Referee: [§5] The epidemic example uses real-world data to demonstrate learning from few observations, but lacks controls (e.g., comparison to a classical MFG fit or ablation on observation density) that would isolate whether the neural component recovers strategic interaction parameters or simply interpolates marginal statistics.

    Authors: We will add an ablation study varying the number of observations and a comparison against a non-strategic neural-SDE baseline (interaction kernel set to zero) to isolate the contribution of the learned strategic terms. A direct comparison to a classical analytic MFG is infeasible on real epidemic data precisely because of the modeling bias and intractability that motivate the neural approach; we will clarify this limitation in the revised text. revision: partial

Circularity Check

0 steps flagged

No circularity: neural SDE framework is a distinct modeling proposal, not a reduction to inputs

full rationale

The paper proposes neural mean-field games as a data-driven alternative that replaces the classical coupled HJB-Fokker-Planck system with a neural SDE trained by automatic differentiation on observations. No derivation step is shown to reduce by construction to the same inputs (e.g., no fitted parameter is relabeled as a prediction of equilibrium, and no uniqueness theorem is imported from self-citation). The abstract and described approach emphasize empirical flexibility and robustness on synthetic games and epidemic data rather than a closed mathematical loop. The central claim therefore remains self-contained as an extension rather than an equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The approach rests on standard assumptions from neural SDE training and mean-field approximations; no explicit free parameters or invented entities are named beyond the neural network itself.

axioms (1)
  • domain assumption Neural SDEs can represent the mean-field interaction dynamics without loss of solution existence or uniqueness.
    Invoked when claiming the model avoids the problems of classical PDE-based MFG.
invented entities (1)
  • Neural mean-field game no independent evidence
    purpose: Data-driven replacement for analytical MFG models
    New modeling construct introduced to combine MFG theory with neural SDEs.

pith-pipeline@v0.9.0 · 5769 in / 1273 out tokens · 29984 ms · 2026-05-22T19:09:27.741067+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

84 extracted references · 84 canonical work pages · 6 internal anchors

  1. [1]

    Mean field games: Numerical methods.SIAM Journal on Numerical Analysis, 48(3):1136–1162, 2010

    Yves Achdou and Italo Capuzzo-Dolcetta. Mean field games: Numerical methods.SIAM Journal on Numerical Analysis, 48(3):1136–1162, 2010. doi:10.1137/090758477

    work page doi:10.1137/090758477 2010

  2. [2]

    Mean Field Games and Applications: Numerical Aspects, 2020

    Yves Achdou and Mathieu Lauri` ere. Mean Field Games and Applications: Numerical Aspects, 2020. URLhttps://arxiv. org/abs/2003.04444

    work page arXiv 2020

  3. [3]

    Mean field games: convergence of a finite difference method,

    Yves Achdou, Fabio Camilli, and Italo Capuzzo Dolcetta. Mean field games: convergence of a finite difference method,

    work page

  4. [4]

    URLhttps://arxiv.org/abs/1207.2982

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    Springer, Heidelberg, Germany, 2019

    Yves Achdou, Pierre Cardaliaguet, Fran¸ cois Delarue, Alessio Porretta, and Filippo Santambrogio.Mean Field Games, volume 2281. Springer, Heidelberg, Germany, 2019. doi:10.1007/978-3-030-59837-2

    work page doi:10.1007/978-3-030-59837-2 2019

  6. [6]

    Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach.The Review of Economic Studies, 89(1):45–86, 04 2021

    Yves Achdou, Jiequn Han, Jean-Michel Lasry, Pierre-Louis Lions, and Benjamin Moll. Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach.The Review of Economic Studies, 89(1):45–86, 04 2021. ISSN 0034-6527. doi:10.1093/restud/rdab002

    work page doi:10.1093/restud/rdab002 2021

  7. [7]

    Adams and Christopher Essex.Calculus: A Com- plete Course

    Robert A. Adams and Christopher Essex.Calculus: A Com- plete Course. Pearson, Ontario, 8 edition, 2013. ISBN 978-0- 32-178107-9

    work page 2013

  8. [8]

    McKenzie Alexander

    J. McKenzie Alexander. Evolutionary Game Theory. In Edward N. Zalta, editor,The Stanford Encyclope- dia of Philosophy. Metaphysics Research Lab, Stanford University, Stanford, CA, Summer 2021 edition, 2021. doi:10.1017/9781108582063

    work page doi:10.1017/9781108582063 2021

  9. [9]

    Q- Learning in Regularized Mean-field Games.Dynamic Games and Applications, 13(1):89–117, 2023

    Berkay Anahtarci, Can Deha Kariksiz, and Naci Saldi. Q- Learning in Regularized Mean-field Games.Dynamic Games and Applications, 13(1):89–117, 2023. doi:10.1007/s13235- 022-00450-2

    work page doi:10.1007/s13235- 2023

  10. [10]

    Brian W. Arthur. Inductive Reasoning and Bounded Ratio- nality.The American Economic Review, 84(2):406–411, 1994. URLhttps://www.jstor.org/stable/2117868

    work page arXiv 1994

  11. [11]

    Deep learning for Mean Field Games with non-separable Hamiltonians.Chaos, Solitons & Fractals, 174:113802, 2023

    Mouhcine Assouli and Badr Missaoui. Deep learning for Mean Field Games with non-separable Hamiltonians.Chaos, Solitons & Fractals, 174:113802, 2023. ISSN 0960-0779. doi:10.1016/j.chaos.2023.113802

    work page doi:10.1016/j.chaos.2023.113802 2023

  12. [12]

    Deep Policy Iteration for high-dimensional mean field games.Applied Mathemat- ics and Computation, 481:128923, 2024

    Mouhcine Assouli and Badr Missaoui. Deep Policy Iteration for high-dimensional mean field games.Applied Mathemat- ics and Computation, 481:128923, 2024. ISSN 0096-3003. doi:10.1016/j.amc.2024.128923

    work page doi:10.1016/j.amc.2024.128923 2024

  13. [13]

    Aumann and Lloyd S

    Robert J. Aumann and Lloyd S. Shapley.Values of non- atomic games. Princeton University Press, Princeton, NJ,

    work page

  14. [14]

    doi:10.1515/9781400867080

    work page doi:10.1515/9781400867080

  15. [15]

    Modeling tagged pedestrian motion: A mean-field type game approach.Trans- portation Research Part B: Methodological, 121:168–183,

    Alexander Aurell and Boualem Djehiche. Modeling tagged pedestrian motion: A mean-field type game approach.Trans- portation Research Part B: Methodological, 121:168–183,

    work page

  16. [16]

    doi:10.1016/j.trb.2019.01.011

    ISSN 0191-2615. doi:10.1016/j.trb.2019.01.011

    work page doi:10.1016/j.trb.2019.01.011 2019

  17. [17]

    The DeepMind JAX Ecosystem, 2020

    Igor Babuschkin, Kate Baumli, Alison Bell, Surya Bhupati- raju, Jake Bruce, Peter Buchlovsky, David Budden, Trevor Cai, Aidan Clark, Ivo Danihelka, Antoine Dedieu, Claudio Fantacci, Jonathan Godwin, Chris Jones, Ross Hemsley, Tom Hennigan, Matteo Hessel, Shaobo Hou, Steven Kapturowski, Thomas Keck, Iurii Kemaev, Michael King, Markus Kunesch, Lena Martens,...

    work page 2020

  18. [18]

    An epsilon-Nash equilibrium for non-linear Markov games of mean-field-type on finite spaces.Communications on Stochas- tic Analysis, 8(4):449–468, 2014

    Rani Basna, Astrid Hilbert, and Vassili N Kolokoltsov. An epsilon-Nash equilibrium for non-linear Markov games of mean-field-type on finite spaces.Communications on Stochas- tic Analysis, 8(4):449–468, 2014. doi:10.31390/cosa.8.4.02

    work page doi:10.31390/cosa.8.4.02 2014

  19. [19]

    A Markovian decision process.Jour- nal of mathematics and mechanics, 6(5):679–684, 1957

    Richard Bellman. A Markovian decision process.Jour- nal of mathematics and mechanics, 6(5):679–684, 1957. doi:10.1512/IUMJ.1957.6.56038

    work page doi:10.1512/iumj.1957.6.56038 1957

  20. [20]

    On carath´ eodory’s conditions for the initial value problem.Proceedings of the American Mathematical Society, 125, 01 1997

    Daniel Biles and Paul Binding. On carath´ eodory’s conditions for the initial value problem.Proceedings of the American Mathematical Society, 125, 01 1997. doi:10.1090/S0002-9939- 97-03942-7

    work page doi:10.1090/s0002-9939- 1997

  21. [21]

    On the implementation of a primal-dual algorithm for second order time-dependent Mean Field Games with local cou- plings.ESAIM: Proceedings and Surveys, 65:330–348, 01

    Luis Brice˜ no-Arias, Dante Kalise, Ziad Kobeissi, Mathieu Lauri` ere,´Alvaro Mateos Gonz´ alez, and Francisco Silva. On the implementation of a primal-dual algorithm for second order time-dependent Mean Field Games with local cou- plings.ESAIM: Proceedings and Surveys, 65:330–348, 01

    work page

  22. [22]

    doi:10.1051/proc/201965330

    work page doi:10.1051/proc/201965330

  23. [23]

    PhD thesis, School of Com- puter Science, Carnegie Mellon University, 2020

    Noam Brown.Equilibrium Finding for Large Adversarial Imperfect-Information Games. PhD thesis, School of Com- puter Science, Carnegie Mellon University, 2020

    work page 2020

  24. [24]

    John Wiley & Sons, Chich- ester, United Kingdom, 2016

    John Charles Butcher.Numerical Methods for Ordi- nary Differential Equations. John Wiley & Sons, Chich- ester, United Kingdom, 2016. ISBN 9781119121534. doi:10.1002/9781119121534

    work page doi:10.1002/9781119121534 2016

  25. [25]

    Notes on Mean Field Games, 09 2013

    Pierre Cardaliaguet. Notes on Mean Field Games, 09 2013

    work page 2013

  26. [26]

    An introduction to Mean Field Game theory

    Pierre Cardaliaguet and Alessio Porretta. An introduction to Mean Field Game theory. InMean Field Games: Cetraro, Italy 2019, pages 1–158. Springer, Heidelberg, Germany, 2021. 7

    work page 2019

  27. [27]

    Probabilistic analysis of mean-field games.SIAM Journal on Control and Optimiza- tion, 51(4):2705–2734, 2013

    Ren´ e Carmona and Fran¸ cois Delarue. Probabilistic analysis of mean-field games.SIAM Journal on Control and Optimiza- tion, 51(4):2705–2734, 2013. doi:10.1137/120883499

    work page doi:10.1137/120883499 2013

  28. [28]

    Convergence Analy- sis of Machine Learning Algorithms for the Numerical Solu- tion of Mean Field Control and Games I: The Ergodic Case

    Ren´ e Carmona and Mathieu Lauri` ere. Convergence Analy- sis of Machine Learning Algorithms for the Numerical Solu- tion of Mean Field Control and Games I: The Ergodic Case. SIAM Journal on Numerical Analysis, 59(3):1455–1485, 2021. doi:10.1137/19M1274377

    work page doi:10.1137/19m1274377 2021

  29. [29]

    Rene Carmona and Mathieu Lauri` ere. Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: II—the finite horizon case.The Annals of Applied Probability, 32, 12 2022. doi:10.1214/21- AAP1715

    work page doi:10.1214/21- 2022

  30. [30]

    Springer Nature, Heidelberg, Germany, 3 2018

    Ren´ e Carmona, Fran¸ cois Delarue, et al.Probabilistic Theory of Mean Field Games with Applications I-II. Springer Nature, Heidelberg, Germany, 3 2018. ISBN 978-3-319-56437-1

    work page 2018

  31. [31]

    Probabilistic Approach to Finite State Mean Field Games.Applied Mathemat- ics & Optimization, 81(2):253–300, 2020

    Alekos Cecchin and Markus Fischer. Probabilistic Approach to Finite State Mean Field Games.Applied Mathemat- ics & Optimization, 81(2):253–300, 2020. ISSN 1432-0606. doi:10.1007/s00245-018-9488-7

    work page doi:10.1007/s00245-018-9488-7 2020

  32. [32]

    Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David K. Duvenaud. Neural ordinary differential equations. Advances in Neural Information Processing Systems, 31,

    work page

  33. [33]

    doi:10.48550/arXiv.1806.07366

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1806.07366

  34. [34]

    Mean-Field Games with Explicit Interactions.Preprint hal-01277098,

    Josu Doncel, Nicolas Gast, and Bruno Gaujal. Mean-Field Games with Explicit Interactions.Preprint hal-01277098,

    work page

  35. [35]

    working paper or preprint

    URLhttps://inria.hal.science/hal-01277098v1. working paper or preprint

    work page

  36. [36]

    Discrete mean field games: Existence of equilibria and convergence.Journal of Dynamics and Games, 6(3):221–239, 2019

    Josu Doncel, Nicolas Gast, and Bruno Gaujal. Discrete mean field games: Existence of equilibria and convergence.Journal of Dynamics and Games, 6(3):221–239, 2019. ISSN 2164-

    work page 2019

  37. [37]

    doi:10.3934/jdg.2019016

    work page doi:10.3934/jdg.2019016

  38. [38]

    A Mean Field Game analysis of SIR dynamics with Vaccination.Probabil- ity in the Engineering and Informational Sciences, 36(2):482– 499, 2022

    Josu Doncel, Nicolas Gast, and Bruno Gaujal. A Mean Field Game analysis of SIR dynamics with Vaccination.Probabil- ity in the Engineering and Informational Sciences, 36(2):482– 499, 2022

    work page 2022

  39. [39]

    Cambridge university press, Cambridge, UK, 2010

    David Easley, Jon Kleinberg, et al.Networks, crowds, and markets: Reasoning about a highly connected world, vol- ume 1. Cambridge university press, Cambridge, UK, 2010. doi:10.1017/CBO9780511761942

    work page doi:10.1017/cbo9780511761942 2010

  40. [40]

    Actor-critic learning for mean-field control in continuous time.Journal of Machine Learning Re- search, 26(127):1–42, 2025

    Noufel Frikha, Maximilien Germain, Mathieu Lauri` ere, Huyˆ en Pham, and Xuanye Song. Actor-critic learning for mean-field control in continuous time.Journal of Machine Learning Re- search, 26(127):1–42, 2025

    work page 2025

  41. [41]

    Implementing the nelder-mead simplex algorithm with adaptive parameters.Comput

    Fuchang Gao and Lixing Han. Implementing the nelder- mead simplex algorithm with adaptive parameters.Compu- tational Optimization and Applications, 51(1):259–277, 2012. ISSN 1573-2894. doi:10.1007/s10589-010-9329-3. URLhttps: //doi.org/10.1007/s10589-010-9329-3

    work page doi:10.1007/s10589-010-9329-3 2012

  42. [42]

    Princeton University Press, Princeton, NJ, 1992

    Robert Gibbons.Game Theory for Applied Economists. Princeton University Press, Princeton, NJ, 1992. ISBN 9780691003955

    work page 1992

  43. [43]

    Gomes, Joana Mohr, and Rafael Rig˜ ao Souza

    Diogo A. Gomes, Joana Mohr, and Rafael Rig˜ ao Souza. Con- tinuous Time Finite State Mean Field Games.Applied Mathe- matics & Optimization, 68(1):99–143, 2013. ISSN 1432-0606. doi:10.1007/s00245-013-9202-8

    work page doi:10.1007/s00245-013-9202-8 2013

  44. [44]

    Mean Field Games and Applications, pages 205–266

    Olivier Gu´ eant, Jean-Michel Lasry, and Pierre-Louis Lions. Mean Field Games and Applications, pages 205–266. Springer Berlin Heidelberg, Berlin, Heidelberg, 2011. ISBN 978-3-642- 14660-2. doi:10.1007/978-3-642-14660-2 3

    work page doi:10.1007/978-3-642-14660-2 2011

  45. [45]

    A cross-country database of covid-19 testing.Scientific Data, 7(1):345, 2020

    Joe Hasell, Edouard Mathieu, Diana Beltekian, Bobbie Mac- donald, Charlie Giattino, Esteban Ortiz-Ospina, Max Roser, and Hannah Ritchie. A cross-country database of covid-19 testing.Scientific Data, 7(1):345, 2020. ISSN 2052-4463. doi:10.1038/s41597-020-00688-8. URLhttps://doi.org/10. 1038/s41597-020-00688-8

    work page doi:10.1038/s41597-020-00688-8 2020

  46. [46]

    Deep residual learning for image recognition,

    Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep Residual Learning for Image Recognition. In2016 IEEE Conference on Computer Vision and Pattern Recogni- tion (CVPR), pages 770–778, Cambridge, Boston, MA, 2016. IEEE. doi:10.1109/CVPR.2016.90

    work page doi:10.1109/cvpr.2016.90 2016

  47. [47]

    Malham´ e, and Peter E

    Minyi Huang, Roland P. Malham´ e, and Peter E. Caines. Large population stochastic dynamic games: closed-loop McKean- Vlasov systems and the Nash certainty equivalence principle. Communications in Information & Systems, 6(3):221 – 252,

    work page

  48. [48]

    doi:10.4310/CIS.2006.v6.n3.a5

    work page doi:10.4310/cis.2006.v6.n3.a5 2006

  49. [49]

    Neural Jump Stochastic Differential Equations.Advances in Neural Information Pro- cessing Systems, 32, 2019

    Junteng Jia and Austin R Benson. Neural Jump Stochastic Differential Equations.Advances in Neural Information Pro- cessing Systems, 32, 2019

    work page 2019

  50. [50]

    Nonlinear sdes driven by l´ evy processes and related pdes,

    Benjamin Jourdain, Sylvie M´ el´ eard, and Wojbor Woyczyn- ski. Nonlinear sdes driven by l´ evy processes and related pdes,

    work page

  51. [51]

    URLhttps://arxiv.org/abs/0707.2723

    work page internal anchor Pith review Pith/arXiv arXiv

  52. [52]

    William Ogilvy Kermack, A. G. McKendrick, and Gilbert Thomas Walker. A contribution to the math- ematical theory of epidemics.Proceedings of the Royal Society of London. Series A, Containing Pa- pers of a Mathematical and Physical Character, 115 (772):700–721, 1927. doi:10.1098/rspa.1927.0118. URL https://royalsocietypublishing.org/doi/abs/10.1098/ rspa.1927.0118

    work page doi:10.1098/rspa.1927.0118 1927

  53. [53]

    PhD thesis, University of Oxford, 2021

    Patrick Kidger.On Neural Differential Equations. PhD thesis, University of Oxford, 2021

    work page 2021

  54. [54]

    Equinox: neural net- works in JAX via callable PyTrees and filtered transforma- tions.Differentiable Programming workshop at Neural Infor- mation Processing Systems 2021, 2021

    Patrick Kidger and Cristian Garcia. Equinox: neural net- works in JAX via callable PyTrees and filtered transforma- tions.Differentiable Programming workshop at Neural Infor- mation Processing Systems 2021, 2021

    work page 2021

  55. [55]

    Efficient and Accurate Gradients for Neural SDEs

    Patrick Kidger, James Foster, Xuechen Li, and Terry Lyons. Efficient and Accurate Gradients for Neural SDEs. InAd- vances in Neural Information Processing Systems 34. Curran Associates, Inc., 2021

    work page 2021

  56. [56]

    Kolokoltsov.Nonlinear Markov Processes and Kinetic Equations

    Vassili N. Kolokoltsov.Nonlinear Markov Processes and Kinetic Equations. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2010. ISBN 9780521111843. doi:10.1017/CBO9780511760303

    work page doi:10.1017/cbo9780511760303 2010

  57. [57]

    Individual Vaccination as Nash Equilibrium in a SIR Model with Application to the 2009–2010 Influenza A (H1N1) Epidemic in France

    Laetitia Laguzet and Gabriel Turinici. Individual Vaccination as Nash Equilibrium in a SIR Model with Application to the 2009–2010 Influenza A (H1N1) Epidemic in France. Bulletin of Mathematical Biology, 77(10):1955–1984, 2015. ISSN 1522-9602. doi:10.1007/s11538-015-0111-7. URL https://doi.org/10.1007/s11538-015-0111-7https:// link.springer.com/article/10...

    work page doi:10.1007/s11538-015-0111-7 2009

  58. [58]

    Jeux ` a champ moyen

    Jean-Michel Lasry and Pierre-Louis Lions. Jeux ` a champ moyen. i – le cas stationnaire.Comptes Rendus Mathematique, 343(9):619–625, 2006. ISSN 1631-073X. doi:https://doi.org/10.1016/j.crma.2006.09.019. URL https://www.sciencedirect.com/science/article/pii/ S1631073X06003682

    work page doi:10.1016/j.crma.2006.09.019 2006

  59. [59]

    Jeux ` a champ moyen

    Jean-Michel Lasry and Pierre-Louis Lions. Jeux ` a champ moyen. ii – horizon fini et contrˆ ole optimal.Comptes Rendus Mathematique, 343(10):679–684, 2006. ISSN 1631- 073X. doi:https://doi.org/10.1016/j.crma.2006.09.018. URL https://www.sciencedirect.com/science/article/pii/ S1631073X06003670

    work page doi:10.1016/j.crma.2006.09.018 2006

  60. [60]

    Mean Field Games.Japanese Journal of Mathematics, 2:229–260, 03

    Jean-Michel Lasry and Pierre-Louis Lions. Mean Field Games.Japanese Journal of Mathematics, 2:229–260, 03

    work page

  61. [61]

    doi:10.1007/s11537-007-0657-8

    work page doi:10.1007/s11537-007-0657-8

  62. [62]

    Learning Mean Field Games: A Survey.arXiv, 05

    Mathieu Lauri` ere, Sarah Perrin, Matthieu Geist, and Olivier Pietquin. Learning Mean Field Games: A Survey.arXiv, 05

    work page

  63. [63]

    doi:10.48550/arXiv.2205.12944

    work page doi:10.48550/arxiv.2205.12944

  64. [64]

    Continuous control with deep reinforcement learning

    Timothy Lillicrap, Jonathan Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wier- stra. Continuous Control with Deep Reinforcement Learning. CoRR, 09 2015. doi:10.48550/arXiv.1509.02971

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1509.02971 2015

  65. [65]

    Alex Tong Lin, Samy Wu Fung, Wuchen Li, Levon Nurbekyan, and Stanley J. Osher. Alternating the population and control neural networks to solve high-dimensional stochastic mean- field games.Proceedings of the National Academy of Sciences, 118(31):e2024713118, 2021. doi:10.1073/pnas.2024713118

    work page doi:10.1073/pnas.2024713118 2021

  66. [66]

    Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise

    Xuanqing Liu, Tesi Xiao, Si Si, Qin Cao, Sanjiv Kumar, and Cho-Jui Hsieh. Neural SDE: Stabilizing Neural ODE Net- works with Stochastic Noise, 2019. URLhttps://arxiv.org/ abs/1906.02355

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  67. [67]

    Marinov and Rossitza S

    Tchavdar T. Marinov and Rossitza S. Marinova. Adap- tive sir model with vaccination: simultaneous identifi- cation of rates and functions illustrated with covid-19. Scientific Reports, 12(1):15688, 2022. ISSN 2045-2322. doi:10.1038/s41598-022-20276-7. URLhttps://doi.org/10. 1038/s41598-022-20276-7

    work page doi:10.1038/s41598-022-20276-7 2022

  68. [68]

    Covid-19 pandemic.Our World in Data, 2020

    Edouard Mathieu, Hannah Ritchie, Lucas Rod´ es- Guirao, Cameron Appel, Daniel Gavrilov, Charlie Giattino, Joe Hasell, Bobbie Macdonald, Saloni Dat- tani, Diana Beltekian, Esteban Ortiz-Ospina, and Max Roser. Covid-19 pandemic.Our World in Data, 2020. https://ourworldindata.org/coronavirus. 8

    work page 2020

  69. [69]

    Model-Free Reinforcement Learning for Mean Field Games

    Rajesh Mishra, Deepanshu Vasal, and Sriram Vishwanath. Model-Free Reinforcement Learning for Mean Field Games. IEEE Transactions on Control of Network Systems, 10(4): 2141–2151, 2023. doi:10.1109/TCNS.2023.3264934

    work page doi:10.1109/tcns.2023.3264934 2023

  70. [70]

    Infinitely deep neural networks as diffusion processes, 2020

    Stefano Peluchetti and Stefano Favaro. Infinitely deep neural networks as diffusion processes, 2020. URLhttps://arxiv. org/abs/1905.11065

    work page arXiv 2020

  71. [71]

    Mean Field Games Flock! The Reinforcement Learning Way

    Sarah Perrin, Mathieu Lauri` ere, P´ erolat Julien, Matthieu Geist, Romuald Elie, and Olivier Pietquin. Mean Field Games Flock! The Reinforcement Learning Way. InPro- ceedings of the Thirtieth International Joint Conference on Artificial Intelligence (IJCAI-21), pages 356–362, 08 2021. doi:10.24963/ijcai.2021/50

    work page doi:10.24963/ijcai.2021/50 2021

  72. [72]

    Springer, Heidelberg, Germany, 2008

    Hans Peters.Game Theory: A Multi-Leveled Approach. Springer, Heidelberg, Germany, 2008. ISBN 978-3-540-69290-

    work page 2008

  73. [73]

    Pontryagin, E.F

    L.S. Pontryagin, E.F. Mishchenko, G.V. Boltyanskii, and R.V. Gamkrelidze.Mathematical Theory of Optimal Pro- cesses. Routledge, London, UK, 1962. ISBN 9782881240775. doi:10.1201/9780203749319

    work page doi:10.1201/9780203749319 1962

  74. [74]

    Universal Differential Equations for Scientific Machine Learning

    Christopher Rackauckas, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Supekar, Dominic Skinner, Ali Ramadhan, and Alan Edelman. Universal differential equations for scientific machine learning.arXiv preprint arXiv:2001.04385, 2020. doi:10.48550/arXiv.2001.04385

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2001.04385 2001

  75. [75]

    Equilibrium Points of Non-Atomic Games.Journal of Statistical Physics, 7:295–300, May 1973

    David Schmeidler. Equilibrium Points of Non-Atomic Games.Journal of Statistical Physics, 7:295–300, May 1973. doi:10.1007/BF01014905

    work page doi:10.1007/bf01014905 1973

  76. [76]

    Su and J

    J. Su and J. E. Renaud. Automatic Differentiation in Robust Optimization.AIAA Journal, 35(6):1072–1079, June 1997. doi:10.2514/2.196

    work page doi:10.2514/2.196 1997

  77. [77]

    Taylor, Mark Crowley, and Pascal Poupart

    Sriram Ganapathi Subramanian, Matthew E. Taylor, Mark Crowley, and Pascal Poupart. Partially Observ- able Mean Field Reinforcement Learning. InProceed- ings of the 20th International Conference on Autonomous Agents and MultiAgent Systems, page 537–545, 2021. doi:10.48550/arXiv.2012.15791

    work page doi:10.48550/arxiv.2012.15791 2021

  78. [78]

    Regarding the number of vaccinations of the new coronavirus vaccine during the special temporary vaccination period

    The Japanese Ministry of Health, Labour and Welfare. Regarding the number of vaccinations of the new coronavirus vaccine during the special temporary vaccination period. https://www.mhlw.go.jp/stf/seisakunitsuite/bunya/ kenkou_iryou/kenkou/kekkaku-kansenshou/yobou-sesshu/ syukeihou_00002.html. Accessed on 12-08-2025

    work page 2025

  79. [79]

    Neural Stochastic Differ- ential Equations: Deep Latent Gaussian Models in the Diffu- sion Limit, 2019

    Belinda Tzen and Maxim Raginsky. Neural Stochastic Differ- ential Equations: Deep Latent Gaussian Models in the Diffu- sion Limit, 2019. URLhttps://arxiv.org/abs/1905.09883

    work page arXiv 2019

  80. [80]

    Sami Fadali and Hao Xu

    Zejian Zhou and M. Sami Fadali and Hao Xu. Biomimetic Optimal Tracking Control using Mean Field Games and Spik- ing Neural Networks.IFAC-PapersOnLine, 53(2):8112–8117,

    work page

Showing first 80 references.