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Unsupervised Solution Operator Learning for Mean-Field Games via Sampling-Invariant Parametrizations

Han Huang, Rongjie Lai

TL;DR

This work develops a novel framework to learn the MFG solution operator that is discretization-free, making it particularly suitable for learning operators of high-dimensional MFGs, and proves the notion of sampling invariance for the model, establishing its convergence to a continuous operator in the sampling limit.

Abstract

Recent advances in deep learning has witnessed many innovative frameworks that solve high dimensional mean-field games (MFG) accurately and efficiently. These methods, however, are restricted to solving single-instance MFG and demands extensive computational time per instance, limiting practicality. To overcome this, we develop a novel framework to learn the MFG solution operator. Our model takes a MFG instances as input and output their solutions with one forward pass. To ensure the proposed parametrization is well-suited for operator learning, we introduce and prove the notion of sampling invariance for our model, establishing its convergence to a continuous operator in the sampling limit. Our method features two key advantages. First, it is discretization-free, making it particularly suitable for learning operators of high-dimensional MFGs. Secondly, it can be trained without the need for access to supervised labels, significantly reducing the computational overhead associated with creating training datasets in existing operator learning methods. We test our framework on synthetic and realistic datasets with varying complexity and dimensionality to substantiate its robustness.

Unsupervised Solution Operator Learning for Mean-Field Games via Sampling-Invariant Parametrizations

TL;DR

This work develops a novel framework to learn the MFG solution operator that is discretization-free, making it particularly suitable for learning operators of high-dimensional MFGs, and proves the notion of sampling invariance for the model, establishing its convergence to a continuous operator in the sampling limit.

Abstract

Recent advances in deep learning has witnessed many innovative frameworks that solve high dimensional mean-field games (MFG) accurately and efficiently. These methods, however, are restricted to solving single-instance MFG and demands extensive computational time per instance, limiting practicality. To overcome this, we develop a novel framework to learn the MFG solution operator. Our model takes a MFG instances as input and output their solutions with one forward pass. To ensure the proposed parametrization is well-suited for operator learning, we introduce and prove the notion of sampling invariance for our model, establishing its convergence to a continuous operator in the sampling limit. Our method features two key advantages. First, it is discretization-free, making it particularly suitable for learning operators of high-dimensional MFGs. Secondly, it can be trained without the need for access to supervised labels, significantly reducing the computational overhead associated with creating training datasets in existing operator learning methods. We test our framework on synthetic and realistic datasets with varying complexity and dimensionality to substantiate its robustness.
Paper Structure (30 sections, 6 theorems, 46 equations, 10 figures, 3 tables)

This paper contains 30 sections, 6 theorems, 46 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Works
  4. MFG and OT
  5. Operator Learning
  6. Background
  7. Mean-Field Games
  8. Trajectory-Based MFG
  9. Interaction-free MFG and Optimal Transport
  10. MFG Solution Operator
  11. Methodology
  12. An Unsupervised Operator Learning Framework
  13. Input Representation
  14. Operator Parametrization
  15. Theoretical Results
  16. ...and 15 more sections

Key Result

Theorem 4.1

Let $X^n_0, X^n_1 \in \mathbb{R}^{n \times d}$ with rows sampled iid from $P_0, P_1$, respectively. The proposed parametrization $G_\theta (X^n_0, X^n_1)({\bm{x}},t)$ is permutation and sampling invariant.

Figures (10)

  • Figure 1: MNIST digits 0,6,5,4 represented by 1053 samples from their pixel value densities.
  • Figure 2: Illustration of the proposed parametrization $G_\theta(X_0, X_1)({\bm{x}})$ for MFG solution operators. MLP: Point-wise multi-layer perception. MHT: Multi-headed attention block.
  • Figure 3: Learned MFG trajectories for Gaussians in 10 (top) and 20 (bottom) dimensions projected to the first two components. Each row shows 3 MFG instances. The green, yellow, and blue dots represent samples from $P_0, P_1,$ and $G_\theta(P_0,P_1)_*P_0$, respectively. The black lines are learned trajectories for 8 selected landmarks.
  • Figure 4: Learned MFG trajectories for Gaussian mixture in 10(top) and 20(bottom) dimensions projected to the first two components. Each row shows 4 MFG instances. The green, yellow, and blue dots represent samples from $P_0, P_1,$ and $G_\theta(P_0,P_1)_*P_0$, respectively. The black lines represent learned trajectories for 16 selected landmarks.
  • Figure 5: Left: Normalized relative $L_1$ errors on the MFG value for Gaussians in $d=2,5,10,20$ with sample size 1024 compared to the statistically optimal MFG value. All errors are normalized by $\sqrt{\frac{d}{d_{max}}} = \sqrt{\frac{d}{20}}$ so that the optimal values are identical across dimensions. Middle: Learned MFG costs for Gaussian mixture in $d=2,5,10,20$. Right: Learned MFG costs for crowd motion in $d=2,5,10,20$.
  • ...and 5 more figures

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 3.1: Permutation Invariance
  • Definition 3.2: Sampling invariance
  • Theorem 4.1
  • Remark 4.2
  • Theorem 4.3
  • Remark 4.4
  • Remark 4.5
  • Proposition 4.6
  • Proposition 4.7
  • ...and 8 more