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A Globally Convergent Flow for Time-Dependent Mean Field Games and a Solver-Agnostic Framework for Inverse Problems

T0 review · 2 major / 5 minor · reviewed 2026-07-14 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A positivity-preserving discrete flow solves time-dependent mean field games from any start, and inverse recovery no longer depends on which forward solver you use.

desk verdict Clean discretize-then-flow extension of stationary HRF to time-dependent MFGs with full discrete global-convergence proofs, plus a usable solver-agnostic inverse layer; the only real soft spot is the unanalyzed adjoint well-posedness. read the letter →

arxiv 2603.10336 v2 pith:JIGQ5SR2 submitted 2026-03-11 math.OC

Hanwei Yan , Xianjin Yang , Jingguo Zhang This is my paper · ORCID
classification math.OC MSC 49N8091A1665K1035Q89
keywords meanfieldgamesHessian-Riemannianflowmonotoneflowsinverseproblemssolver-agnosticGauss-NewtonLasry-Lionsmonotonicity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Mean field games model huge crowds of strategic agents, but solving them numerically is hard: many solvers only work near a good guess, and inverse recovery of unknown costs often has to be rewritten whenever the forward code changes. This paper builds a discrete Hessian–Riemannian flow that freezes the prescribed initial density and terminal value, evolves only the interior unknowns, and stays on the positive-mass manifold by construction. Under the usual convexity and Lasry–Lions monotonicity assumptions it converges globally to the unique solution of the fully discretized system. For inverse problems the same paper treats the discrete MFG equations as an implicit constraint: outer gradients and Gauss–Newton steps are obtained by differentiating those equations at the converged state rather than unrolling any particular solver. The result is a plug-and-play pipeline that works for both stationary and time-dependent models, potential and non-potential couplings alike.

What carries the argument

The discrete Hessian–Riemannian flow (2.19): density blocks evolve by componentwise multiplication with the residual (minus its mass-average), while value blocks evolve by the Fokker–Planck residual; the metric is the Hessian of the discrete entropy, so positivity and mass are automatic.

What would settle it

Construct a discretized time-dependent MFG that satisfies the paper’s convexity and monotonicity assumptions, start the flow from a feasible positive density far from the solution, and check whether the residual fails to go to zero or the density leaves the positive-mass set.

Watch

Extended reading notes

Core claim

After space-time discretization of a time-dependent mean-field game, a Hessian–Riemannian flow on the interior density and value unknowns preserves positivity and mass, freezes the mixed end-point data, and converges globally to the unique discrete solution under standard Lasry–Lions monotonicity and Hamiltonian convexity. The same discrete residual, once solved, can be differentiated implicitly so that outer parameter updates never depend on the internal iterations of the forward solver.

Load-bearing premise

The outer adjoint system is treated as reliably solvable even though the discrete residual is only strictly monotone, not strongly monotone; the paper leaves a full coercivity proof for later.

Editorial extensions

If this is right

  • Any accurate discrete MFG residual can be dropped into the outer inverse loop without rewriting adjoint or Gauss–Newton code.
  • Forward solves for time-dependent MFGs no longer require careful initialization once the discrete HRF is used.
  • Non-potential monotone couplings become accessible to the same inverse pipeline that previously needed potential structure.
  • Gauss–Newton outer steps consistently need fewer iterations than pure gradient descent on the same bilevel objective.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same residual-as-constraint interface could recover the Hamiltonian or coupling function themselves once they are parametrized, not only the spatial cost.
  • If the discrete residual can be made strongly monotone by a cheap regularizer, the adjoint solve would become unconditionally well-posed and the inverse theory would close.
  • The discretize-then-flow idea may transfer to other evolution systems that mix positivity constraints with two-point boundary data.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops two contributions for mean field games. First, it extends the Hessian–Riemannian flow (HRF) from stationary MFGs to fully discretized time-dependent MFGs: after space-time finite-difference discretization (Achdou–Capuzzo-Dolcetta type), endpoint data are frozen and an entropy-induced Riemannian metric is used to produce the explicit flow (2.19) on the positive-mass manifold. Under Assumptions 2.1–2.3 (strict convexity of the numerical Hamiltonian, strict Lasry–Lions monotonicity of the coupling, local Lipschitz regularity), Propositions 2.1–2.3 and Theorem 2.4 establish strict monotonicity of the discrete residual, global existence with positivity and mass preservation, and global convergence of the flow to the unique discrete solution, independent of initialization. Second, inverse problems for recovering the spatial cost (and the associated equilibrium) from partial noisy observations of density and cost are cast as state-eliminated bilevel optimizations; outer gradients and Gauss–Newton steps are obtained by implicit differentiation of the discrete MFG residual at the converged inner solution (Propositions 3.2–3.3), yielding a solver-agnostic adjoint/GN framework. Numerical experiments on stationary (including non-potential) and time-dependent examples illustrate reconstruction quality and the advantage of GN over gradient descent.

Significance. If the discrete analysis holds, the work supplies a practically usable, positivity-preserving, globally convergent forward solver for time-dependent MFGs that does not require potential structure or careful initialization—an advance over ambient monotone flows and local Newton-type methods. The inverse framework is genuinely modular: any accurate forward solver can be swapped without redesigning the outer loop, and the paper demonstrates this with HRF, Newton, and policy iteration on a non-potential example. The appendix proofs of the forward claims are complete and standard; the numerical suite covers both stationary and evolutionary settings and includes a non-potential test. These are concrete, citable contributions for the numerical MFG community.

major comments (2)
  1. Section 3.2 (paragraph after Proposition 3.2) and the adjoint system (3.8): the outer gradient formula is derived under the assumption that (3.8) admits a solution, yet the paper only has strict (not strong) monotonicity of F and explicitly defers a coercivity analysis. While the numerical experiments report stable adjoint solves, the inverse claims rest on this unproved well-posedness. A short local strong-monotonicity argument on compact subsets of the positive-mass manifold (or a regularized adjoint with a clear error bound) would close the gap between the formal derivation and the claimed solver-agnostic method.
  2. Theorem 2.4 and the numerical section: global convergence is proved only for continuous artificial time; the experiments discretize (2.19) by implicit Euler without a discrete-in-flow convergence statement or step-size restriction. Because the central forward claim is “a globally convergent flow,” a brief remark (or a discrete Lyapunov argument under a CFL-type condition) is needed to justify that the implemented scheme inherits the continuous-time guarantee.
minor comments (5)
  1. Notation for the discrete residual switches between F_h(Y) (Section 2) and F_h(Y;V_h) (Section 3); a single consistent signature would improve readability.
  2. Figures 1, 4, 6, 9, 10 pack many panels; the captions are dense and the error color scales are not always stated. Slightly larger labels or a shared colorbar would help.
  3. The RKHS reconstruction (3.2) and the choice of Matérn kernels are well motivated, but the concrete kernel hyperparameters used in the experiments are not listed; a short table or sentence would aid reproducibility.
  4. Assumption 2.3 requires local Lipschitz continuity of ∇_q g; a one-line reference to the standard Godunov Hamiltonian used in the numerics would make the assumption checkable.
  5. In the inverse experiments the observation counts (e.g., 8/10, 128/320, 72/180) appear without a clear sampling protocol; stating whether they are fixed or redrawn across runs would clarify the reported errors.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: forward convergence is proved from external Lasry–Lions/convexity hypotheses; inverse recoveries are synthetic reconstructions, not self-defined predictions.

full rationale

The load-bearing forward claim is Theorem 2.4: under Assumptions 2.1–2.3 the discrete HRF (2.19) exists globally on the positive-mass manifold and converges to the unique discrete MFG solution. The appendix proofs of Propositions 2.1–2.3 and Theorem 2.4 are self-contained: discrete monotonicity of Fh follows from Godunov convexity of g plus strict Lasry–Lions monotonicity of f; positivity/mass follow from the multiplicative structure of (2.19); global convergence uses a Bregman Lyapunov argument plus Barbalat. These are standard external hypotheses (Lasry–Lions, Achdou–Capuzzo-Dolcetta discrete well-posedness), not quantities fitted inside the paper. Citations to the authors’ prior stationary HRF [28] supply only the algorithmic starting point; the time-dependent construction and its proofs are new and independent. On the inverse side, outer gradients come from implicit differentiation of the discrete residual at a converged state (Proposition 3.2); numerical experiments recover known synthetic spatial costs from noisy observations and do not rename fitted free parameters as discoveries. The acknowledged open point—possible singularity of the adjoint system (3.8)—is left for future work and is not used to force any claimed prediction. No self-definitional loop, fitted-input-as-prediction, or load-bearing self-citation chain is present.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard MFG structural hypotheses (Lasry-Lions monotonicity, convexity of H, existence of a classical or discrete solution) plus a handful of free regularization weights chosen for the inverse experiments. No new physical entities are postulated.

free parameters (3)
  • inverse regularization weights (α, β, γ) = experiment-dependent (0.002–0.04, 1–2)
    Hand-tuned per experiment (e.g., α=0.002/0.04, β=γ=1 or 2) to balance RKHS smoothness against data fidelity; different values alter recovered fields.
  • observation noise level η = 1e-3
    Fixed at 10^{-3} for all synthetic tests; controls the difficulty of the inverse problem.
  • spatial/temporal grid sizes (h, Δt) = 1/25–1/100
    Chosen by hand (e.g., 1/40, 1/30, 1/25) and affect both forward accuracy and inverse conditioning.
assumptions (4)
  • domain assumption Lasry-Lions strict monotonicity of the coupling f (Assumption 2.2)
    Used to obtain strict monotonicity of the discrete residual F_h (Prop. 2.1) and hence global convergence of the flow (Thm. 2.4).
  • domain assumption Strict convexity and consistency of the numerical Hamiltonian g (Assumption 2.1)
    Guarantees the discrete transport operator is the adjoint of the Hamiltonian linearization (Lemma A.1) and supplies the strict inequality needed for uniqueness.
  • domain assumption Existence of at least one feasible discrete equilibrium Y* with F_h(Y*)=0
    Invoked in Prop. 2.3 and Thm. 2.4 to obtain a Lyapunov function and identify the limit; existence itself is cited from Achdou-Capuzzo-Dolcetta (2010).
  • standard math Local Lipschitz continuity of the residual map (Assumption 2.3)
    Standard ODE well-posedness hypothesis used for local existence of the flow.

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Cite this review

Pith. "Pith review of A Globally Convergent Flow for Time-Dependent Mean Field Games and a Solver-Agnostic Framework for Inverse Problems." pith.science (2026). https://pith.science/paper/JIGQ5SR2

@misc{pith2026260310336,
  author       = {Pith},
  title        = {Pith review of: A Globally Convergent Flow for Time-Dependent Mean Field Games and a Solver-Agnostic Framework for Inverse Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JIGQ5SR2}},
  note         = {Machine review of arXiv:2603.10336}
}
read the original abstract

Mean field games (MFGs) describe the limiting behavior of large populations of strategically interacting agents. This paper addresses two numerical challenges for MFGs: globally convergent forward solvers and solver-agnostic methods for inverse problems. For the forward problem, we extend the Hessian--Riemannian flow (HRF), previously developed for stationary MFGs, to time-dependent MFGs. We first discretize the system in space and time and then construct the flow directly on the resulting finite-dimensional problem. The proposed flow exploits Lasry--Lions monotonicity, preserves the initial density and terminal value function, and maintains positivity and mass of the density. Under standard assumptions, we prove global convergence of the HRF and show how to recover a solution of the full discretized time-dependent MFG system from its limit. For the inverse problem, we formulate parameter estimation as a bilevel problem in which the outer problem updates unknown coefficients and the inner problem solves the discretized MFG system. Gradients of the outer objective are obtained by differentiating the discretized MFG system at the inner solution, rather than differentiating through the iterations of a particular forward solver. This yields a solver-agnostic framework with adjoint-based gradient descent and Gauss--Newton acceleration. Numerical experiments on stationary and time-dependent MFGs demonstrate the effectiveness of the proposed methods.

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Reference graph

Works this paper leans on

61 extracted references · 5 linked inside Pith

  1. [1]

    Achdou, F

    Y. Achdou, F. Camilli, and I. Capuzzo-Dolcetta. Mean field games: Convergence of a finite difference method.SIAM Journal on Numerical Analysis, 51(5):2585–2612, 2013

    2013

  2. [2]

    Achdou and I

    Y. Achdou and I. Capuzzo-Dolcetta. Mean field games: Numerical methods.SIAM Journal on Numerical Analysis, 48(3):1136–1162, 2010

    2010

  3. [3]

    Achdou and V

    Y. Achdou and V. Perez. Iterative strategies for solving linearized discrete mean field games systems.Networks and Heterogeneous Media, 7(2):197, 2012

    2012

  4. [4]

    Agrawal, B

    A. Agrawal, B. Amos, S. Barratt, S. Boyd, S. Diamond, and J. Z. Kolter. Differentiable convex optimization layers.Advances in Neural Information Processing Systems, 32, 2019

    2019

  5. [5]

    Almulla, R

    N. Almulla, R. Ferreira, and D. A. Gomes. Two numerical approaches to stationary mean-field games.Dynamic Games and Applications, 7(4):657–682, 2017

    2017

  6. [6]

    Amos and J

    B. Amos and J. Z. Kolter. Optnet: Differentiable optimization as a layer in neural networks. InInternational Conference on Machine Learning, pages 136–145. PMLR, 2017

    2017

  7. [7]

    Bakaryan, C

    T. Bakaryan, C. Aoun, R. D. L. Ribeiro, N. Hovakimyan, and D. A. Gomes. Hessian Riemannian flow for multi-population Wardrop equilibrium.arXiv preprint arXiv:2504.16028, 2025

    arXiv 2025

  8. [8]

    Blondel, Q

    M. Blondel, Q. Berthet, M. Cuturi, R. Frostig, S. Hoyer, F. Llinares-L´ opez, F. Pedregosa, and J.-P. Vert. Efficient and modular implicit differentiation.Advances in Neural Information Processing Systems, 35:5230–5242, 2022

    2022

Show all 61 references
  1. [9]

    L. M. Brice˜ no-Arias, D. Kalise, Z. Kobeissi, M. Lauri` ere, A. M. Gonz´ alez, and F. Silva. On the implementation of a primal-dual algorithm for second order time-dependent mean field games with local couplings.ESAIM: Proceedings and Surveys, 65:330–348, 2019

    2019

  2. [10]

    L. M. Briceno-Arias, D. Kalise, and F. Silva. Proximal methods for stationary mean field games with local couplings.SIAM Journal on Control and Optimization, 56(2):801–836, 2018

    2018

  3. [11]

    L. M. Briceno-Arias, F. J. Silva, and X. Yang. Forward-backward algorithm for functions with locally Lipschitz gradient: applications to mean field games.Set-Valued and Variational Analysis, 32(2):16, 2024

    2024

  4. [12]

    Cacace, F

    S. Cacace, F. Camilli, and A. Goffi. A policy iteration method for mean field games.ESAIM: Control, Optimisation and Calculus of Variations, 27:85, 2021

    2021

  5. [13]

    Cardaliaguet

    P. Cardaliaguet. Notes on mean field games. Technical report, Technical report Technical report, 2010

    2010

  6. [14]

    Cardaliaguet

    P. Cardaliaguet. Long time average of first order mean field games and weak KAM theory.Dynamic Games and Applications, 3(4):473–488, 2013

    2013

  7. [15]

    Cardaliaguet, J.-M

    P. Cardaliaguet, J.-M. Lasry, P.-L. Lions, and A. Porretta. Long time average of mean field games.Networks&Heterogeneous Media, 7(2), 2012

    2012

  8. [16]

    Cardaliaguet and C.-A

    P. Cardaliaguet and C.-A. Lehalle. Mean field game of controls and an application to trade crowding.Mathematics and Financial Economics, 12(3):335–363, 2018

    2018

  9. [17]

    Cardaliaguet and A

    P. Cardaliaguet and A. Porretta. An introduction to mean field game theory. InMean Field Games: Cetraro, Italy 2019, pages 1–158. Springer, 2021

    2019

  10. [18]

    Carlini and F

    E. Carlini and F. J. Silva. A semi-Lagrangian scheme for a degenerate second order mean field game system.Discrete and Continuous Dynamical Systems, 35(9):4269–4292, 2015

    2015

  11. [19]

    Carlini, F

    E. Carlini, F. J. Silva, and A. Zorkot. A Lagrange–Galerkin scheme for first order mean field game systems.SIAM Journal on Numerical Analysis, 62(1):167–198, 2024

    2024

  12. [20]

    Carmona and M

    R. Carmona and M. Lauri` ere. Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games I: The ergodic case.SIAM Journal on Numerical Analysis, 59(3):1455–1485, 2021

    2021

  13. [21]

    Carmona and M

    R. Carmona and M. 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(6):4065–4105, 2022. HESSIAN RIEMANNIAN FLOWS AND A SOLVER-AGNOSTIC FRA...

    2022

  14. [22]

    Y. Chen, B. Hosseini, H. Owhadi, and A. M. Stuart. Solving and learning nonlinear PDEs with Gaussian processes.Journal of Computational Physics, 447:110668, 2021

    2021

  15. [23]

    L. Ding, W. Li, S. Osher, and W. Yin. A mean field game inverse problem.Journal of Scientific Computing, 92(1):7, 2022

    2022

  16. [24]

    D. A. Gomes, E. A. Pimentel, and V. Voskanyan.Regularity theory for mean-field game systems. Springer, 2016

    2016

  17. [25]

    D. A. Gomes and J. Sa´ ude. Mean field games models—a brief survey.Dynamic Games and Applications, 4(2):110–154, 2014

    2014

  18. [26]

    D. A. Gomes and J. Sa´ ude. Monotone numerical methods for finite-state mean-field games.arXiv preprint arXiv:1705.00174, 2017

    2017 arXiv

  19. [27]

    D. A. Gomes and J. Sa´ ude. Numerical methods for finite-state mean-field games satisfying a monotonicity condition.Applied Mathematics&Optimization, 83(1):51–82, 2021

    2021

  20. [28]

    D. A. Gomes and X. Yang. The Hessian Riemannian flow and Newton’s method for effective Hamiltonians and Mather measures.ESAIM: Mathematical Modelling and Numerical Analysis, 54(6):1883–1915, 2020

    1915

  21. [29]

    J. Guo, C. Mou, X. Yang, and C. Zhou. Decoding mean field games from population and environment observations by Gaussian processes.Journal of Computational Physics, 508:112978, 2024

    2024

  22. [30]

    Huang, J

    H. Huang, J. Yu, T. Chen, and R. Lai. Joint inference of trajectory and obstacle in mean-field games via bilevel optimization. arXiv preprint arXiv:2507.19344, 2025

    2025 arXiv

  23. [31]

    Huang, P

    M. Huang, P. E. Caines, and R. P. Malham´ e. Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized ε-Nash equilibria.IEEE Transactions on Automatic Control, 52(9):1560–1571, 2007

    2007

  24. [32]

    Huang, R

    M. Huang, R. P. Malham´ e, and P. E. Caines. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle.Communications in Information and Systems, 6(3):221–252, 2006

    2006

  25. [33]

    Imanuvilov, H

    O. Imanuvilov, H. Liu, and M. Yamamoto. Lipschitz stability for determination of states and inverse source problem for the mean field game equations.Inverse Problems and Imaging, 18(4):824–859, 2024

    2024

  26. [34]

    H. K. Khalil and J. W. Grizzle.Nonlinear systems, volume 3. Prentice hall Upper Saddle River, NJ, 2002

    2002

  27. [35]

    M. V. Klibanov, J. Li, and H. Liu. H¨ older stability and uniqueness for the mean field games system via Carleman estimates. Studies in Applied Mathematics, 151(4):1447–1470, 2023

    2023

  28. [36]

    M. V. Klibanov, J. Li, and Z. Yang. Convexification for a coefficient inverse problem for a system of two coupled nonlinear parabolic equations.Computers&Mathematics with Applications, 179:41–58, 2025

    2025

  29. [37]

    Lachapelle, J.-M

    A. Lachapelle, J.-M. Lasry, C.-A. Lehalle, and P.-L. Lions. Efficiency of the price formation process in presence of high frequency participants: a mean field game analysis.Mathematics and Financial Economics, 10(3):223–262, 2016

    2016

  30. [38]

    Lasry and P.-L

    J.-M. Lasry and P.-L. Lions. Jeux ` a champ moyen. i–le cas stationnaire.Comptes Rendus Math´ ematique, 343(9):619–625, 2006

    2006

  31. [39]

    Lasry and P.-L

    J.-M. Lasry and P.-L. Lions. Jeux ` a champ moyen. ii–horizon fini et contrˆ ole optimal.Comptes Rendus. Math´ ematique, 343(10):679–684, 2006

    2006

  32. [40]

    Lasry and P.-L

    J.-M. Lasry and P.-L. Lions. Mean field games.Japanese Journal of Mathematics, 2(1):229–260, 2007

    2007

  33. [41]

    Lauri` ere, J

    M. Lauri` ere, J. Song, and Q. Tang. Policy iteration method for time-dependent mean field games systems with non-separable Hamiltonians.Applied Mathematics&Optimization, 87(2):17, 2023

    2023

  34. [42]

    A. T. Lin, S. W. Fung, W. Li, L. Nurbekyan, and S. 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

    2021

  35. [43]

    H. Liu, C. Mou, and S. Zhang. Inverse problems for mean field games.Inverse Problems, 39(8):085003, 2023

    2023

  36. [44]

    S. Liu, M. Jacobs, W. Li, L. Nurbekyan, and S. J. Osher. Computational methods for first-order nonlocal mean field games with applications.SIAM Journal on Numerical Analysis, 59(5):2639–2668, 2021

    2021

  37. [45]

    Liu and L

    S. Liu and L. Nurbekyan. Splitting methods for a class of non-potential mean field games.Journal of Dynamics and Games, 8(4):467–486, 2021

    2021

  38. [46]

    Meng and X

    R. Meng and X. Yang. Sparse Gaussian processes for solving nonlinear PDEs.Journal of Computational Physics, 490:112340, 2023

    2023

  39. [47]

    C. Mou, X. Yang, and C. Zhou. Numerical methods for mean field games based on Gaussian processes and Fourier features. Journal of Computational Physics, 460:111188, 2022

    2022

  40. [48]

    Nurbekyan and J

    L. Nurbekyan and J. Sa´ ude. Fourier approximation methods for first-order nonlocal mean-field games.Portugaliae Mathematica, 75(3):367–396, 2019

    2019

  41. [49]

    Y. A. Osborne and I. Smears. Finite element approximation of time-dependent mean field games with nondifferentiable hamiltonians.Numerische Mathematik, 157(1):165–211, 2025

    2025

  42. [50]

    Owhadi and C

    H. Owhadi and C. Scovel.Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization: From a Game Theoretic Approach to Numerical Approximation and Algorithm Design, volume 35. Cambridge University Press, 2019

    2019

  43. [51]

    K. Ren, N. Soedjak, and S. Tong. A policy iteration method for inverse mean field games.arXiv preprint arXiv:2409.06184, 2024

    2024

  44. [52]

    K. Ren, N. Soedjak, K. Wang, and H. Zhai. Reconstructing a state-independent cost function in a mean-field game model. Inverse Problems, 40(10):105010, 2024

    2024

  45. [53]

    Ruthotto, S

    L. Ruthotto, S. J. Osher, W. Li, L. Nurbekyan, and S. W. Fung. A machine learning framework for solving high-dimensional mean field game and mean field control problems.Proceedings of the National Academy of Sciences, 117(17):9183–9193, 2020

    2020

  46. [54]

    Wendland.Scattered data approximation, volume 17

    H. Wendland.Scattered data approximation, volume 17. Cambridge University Press, 2004. 36 HESSIAN RIEMANNIAN FLOWS AND A SOLVER-AGNOSTIC FRAMEWORK FOR INVERSE PROBLEMS

    2004

  47. [55]

    L. Yang, S. Liu, T. Meng, and S. J. Osher. In-context operator learning with data prompts for differential equation problems. Proceedings of the National Academy of Sciences, 120(39):e2310142120, 2023

    2023

  48. [56]

    L. Yang, X. Sun, B. Hamzi, H. Owhadi, and N. Xie. Learning dynamical systems from data: A simple cross-validation perspective, part V: Sparse kernel flows for 132 chaotic dynamical systems.Physica D: Nonlinear Phenomena, 460:134070, 2024

    2024

  49. [57]

    Yang and H

    X. Yang and H. Owhadi. A mini-batch method for solving nonlinear PDEs with Gaussian processes.arXiv preprint arXiv:2306.00307, 2023

    2023 arXiv

  50. [58]

    Yang and J

    X. Yang and J. Zhang. Gaussian process policy iteration with additive Schwarz acceleration for forward and inverse HJB and mean field game problems.arXiv preprint arXiv:2505.00909, 2025

    2025 arXiv

  51. [59]

    J. Yu, J. Liu, and H. Zhao. Equilibrium correction iteration for a class of mean-field game inverse problems.Inverse Problems, 41(12):125009, 2025

    2025

  52. [60]

    J. Yu, Q. Xiao, T. Chen, and R. Lai. A bilevel optimization method for inverse mean-field games.Inverse Problems, 40(10):105016, 2024

    2024

  53. [61]

    Zhang, X

    J. Zhang, X. Yang, C. Mou, and C. Zhou. Learning surrogate potential mean field games via Gaussian processes: A data-driven approach to ill-posed inverse problems.Journal of Computational Physics, 543:114412, 2025

    2025

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