# Simultaneously Solving Infinitely Many LQ Mean Field Games In Hilbert Spaces: The Power of Neural Operators

arXiv: https://arxiv.org/abs/2510.20017 · HTML: https://mlsanity.com/paper/2510.20017/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 58 references that are in the map.
- First posted on arXiv 22 Oct 2025.

## Builds on (most cited first, 30 of 31)

- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Optimal Approximation Rate of ReLU Networks in terms of Width and Depth](https://mlsanity.com/paper/2103.00502/) (2021)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces](https://mlsanity.com/paper/1804.10556/) (2018)
- [Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: II -- The Finite Horizon Case](https://mlsanity.com/paper/1908.01613/) (2019)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [Deep Learning Methods for Mean Field Control Problems with Delay](https://mlsanity.com/paper/1905.00358/) (2019)
- [Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis](https://mlsanity.com/paper/2210.13300/) (2022)
- [Numerical resolution of McKean-Vlasov FBSDEs using neural networks](https://mlsanity.com/paper/1909.12678/) (2019)
- [Integral Probability Metrics PAC-Bayes Bounds](https://mlsanity.com/paper/2207.00614/) (2022)
- [Deep learning architectures for nonlinear operator functions and nonlinear inverse problems](https://mlsanity.com/paper/1912.11090/) (2019)
- [Instance-Dependent Generalization Bounds via Optimal Transport](https://mlsanity.com/paper/2211.01258/) (2022)
- [Bridging the Gap Between Approximation and Learning via Optimal Approximation by ReLU MLPs of Maximal Regularity](https://mlsanity.com/paper/2409.12335/) (2024)
- [On the implementation of a primal-dual algorithm for second order time-dependent mean field games with local couplings](https://mlsanity.com/paper/1802.07902/) (2018)
- [Connecting GANs, MFGs, and OT](https://mlsanity.com/paper/2002.04112/) (2020)
- [Linear-Quadratic Mean Field Games](https://mlsanity.com/paper/1404.5741/) (2014)
- [Approximation theory for 1-Lipschitz ResNets](https://mlsanity.com/paper/2505.12003/) (2025)
- [The Mean Field Ensemble Kalman Filter: Near-Gaussian Setting](https://mlsanity.com/paper/2212.13239/) (2022)
- [Neural Operators Can Play Dynamic Stackelberg Games](https://mlsanity.com/paper/2411.09644/) (2024)
- [Learning algorithms for mean field optimal control](https://mlsanity.com/paper/2503.17869/) (2025)
- [Generating Rectifiable Measures through Neural Networks](https://mlsanity.com/paper/2412.05109/) (2024)
- [Individual vaccination as Nash equilibrium in a SIR model with application to the 2009-10 Influenza A(H1N1) epidemic in France](https://mlsanity.com/paper/2410.03567/) (2024)
- [Learning from one graph: transductive learning guarantees via the geometry of small random worlds](https://mlsanity.com/paper/2509.06894/) (2025)
- [A Mean-Field Game Approach to Equilibrium Pricing in Solar Renewable Energy Certificate Markets](https://mlsanity.com/paper/2003.04938/) (2020)
- [Hilbert Space-Valued LQ Mean Field Games: An Infinite-Dimensional Analysis](https://mlsanity.com/paper/2403.01012/) (2024)

## Built on by (most cited first, 3 of 3)

- [One model to solve them all: 2BSDE families via neural operators](https://mlsanity.com/paper/2511.01125/) (2025)
- [Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control](https://mlsanity.com/paper/2603.19463/) (2026)
- [Self-supervised In-context Operator Learning for Stochastic Mean-Field Control](https://mlsanity.com/paper/2608.18282/) (2026)

## Abstract (arXiv)

Traditional mean-field game (MFG) solvers operate on an instance-by-instance basis, which becomes infeasible when many related problems must be solved (e.g., for seeking a robust description of the solution under perturbations of the dynamics or utilities, or in settings involving continuum-parameterized agents.). We overcome this by training neural operators (NOs) to learn the rules-to-equilibrium map from the problem data (``rules'': dynamics and cost functionals) of LQ MFGs defined on separable Hilbert spaces to the corresponding equilibrium strategy. Our main result is a statistical guarantee: an NO trained on a small number of randomly sampled rules reliably solves unseen LQ MFG variants, even in infinite-dimensional settings. The number of NO parameters needed remains controlled under appropriate rule sampling during training. Our guarantee follows from three results: (i) local-Lipschitz estimates for the highly nonlinear rules-to-equilibrium map; (ii) a universal approximation theorem using NOs with a prespecified Lipschitz regularity (unlike traditional NO results where the NO's Lipschitz constant can diverge as the approximation error vanishes); and (iii) new sample-complexity bounds for L-Lipschitz learners in infinite dimensions, directly applicable as the Lipschitz constants of our approximating NOs are controlled in (ii).