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Forecasting Public Sentiments via Mean Field Games

Michael V. Klibanov, Kevin McGoff, Trung Truong

TL;DR

This work develops a globally convergent numerical method for forecasting public sentiments within Mean Field Games by constructing a Carleman-weighted convexification functional J_{λ,α}. The authors prove global strong convexity on a ball for sufficiently large λ, establish Lipschitz continuity of the derivative, and derive error bounds linking initial-data inaccuracies to solution accuracy. They show global convergence of a gradient-descent scheme to the unique minimizer and validate the approach through 1D numerical experiments, including ideal, more realistic, and data-mimicking scenarios. The results demonstrate the method's potential to forecast MFG solutions from initial data with controlled error, offering a non-marching, stable alternative for inverse-type forecasting problems in social dynamics. The technique leverages Carleman estimates via a Carleman weight φ_λ and yields a robust framework for forecasting public sentiment modeled by a coupled parabolic MFG system.

Abstract

Motivated by the goal of forecasting public sentiments, we consider a forecasting problem in the context of the Mean Field Games theory. We develop a numerical method, which is a version of the so-called convexification method. We provide theoretical convergence analysis that establishes global convergence of the method with a convergence rate. We also conduct numerical experiments that demonstrate the accurate performance of the convexification technique and highlight some promising features of this approach.

Forecasting Public Sentiments via Mean Field Games

TL;DR

This work develops a globally convergent numerical method for forecasting public sentiments within Mean Field Games by constructing a Carleman-weighted convexification functional J_{λ,α}. The authors prove global strong convexity on a ball for sufficiently large λ, establish Lipschitz continuity of the derivative, and derive error bounds linking initial-data inaccuracies to solution accuracy. They show global convergence of a gradient-descent scheme to the unique minimizer and validate the approach through 1D numerical experiments, including ideal, more realistic, and data-mimicking scenarios. The results demonstrate the method's potential to forecast MFG solutions from initial data with controlled error, offering a non-marching, stable alternative for inverse-type forecasting problems in social dynamics. The technique leverages Carleman estimates via a Carleman weight φ_λ and yields a robust framework for forecasting public sentiment modeled by a coupled parabolic MFG system.

Abstract

Motivated by the goal of forecasting public sentiments, we consider a forecasting problem in the context of the Mean Field Games theory. We develop a numerical method, which is a version of the so-called convexification method. We provide theoretical convergence analysis that establishes global convergence of the method with a convergence rate. We also conduct numerical experiments that demonstrate the accurate performance of the convexification technique and highlight some promising features of this approach.

Paper Structure

This paper contains 14 sections, 5 theorems, 115 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Statement of the Problem
  3. Carleman Estimates
  4. The Convexification Functional
  5. The functional
  6. A high level intuitive sketch on why the functional $J_{\lambda ,\alpha }$ is strongly convex on $\overline{B\left( R\right) }$
  7. The Global Strong Convexity Property
  8. The Accuracy of the Minimizer
  9. Global Convergence of the Gradient Descent Method
  10. Numerical Studies
  11. The ideal case
  12. A more realistic case
  13. The Case Mimicking the Real-Life Data
  14. An Interesting Observation

Key Result

Theorem 3.1

Let $c$ be the number in (4.01), (4.02). There exists a sufficiently large number $\lambda _{0,1}=\lambda _{0,1}\left( c,Q_{T}\right) \geq 1$ depending only on listed parameters such that the following Carleman estimate holds: where the number $C_{1}=C_{1}\left( c,Q_{T}\right) >0$ depends only on listed parameters.

Figures (13)

  • Figure 1: Test 1.1 (an ideal case): Predicted and true $u(x,t)$ for the extended time interval $t \in [0,2]$.
  • Figure 2: Test 1.1 (an ideal case): Relative cost (\ref{['num:rel-cost']}) for the extended time interval $t \in [0,2]$.
  • Figure 3: Test 1.1 (an ideal case): True and predicted $u(x,t)$ over time for $u(x,t) = (x^2-1)^2(t^2+1)$ and $m(x,0)=\text{exp}\left( 1/(x^2-1) \right)+0.28$.
  • Figure 4: Test 1.1 (an ideal case): True and predicted $m(x,t)$ over time for $u(x,t) = (x^2-1)^2(t^2+1)$ and $m(x,0)=\text{exp}\left( 1/(x^2-1) \right)+0.28$.
  • Figure 5: Test 1.2 (an ideal case): True and predicted $u(x,t)$ over time for $u(x,t) = 0.1\cos{(2\pi x)}(t+1)$ and $m(x,0)=0.5$.
  • ...and 8 more figures

Theorems & Definitions (13)

  • Definition 1.1
  • Remark 2.1
  • Theorem 3.1: MFG2, section 2.3.1 of MFGbook
  • Theorem 3.2: a quasi-Carleman estimate, MFG2, section 2.3.2 of MFGbook
  • Theorem 5.1: the central result
  • Proof 1
  • Theorem 6.1
  • Proof 2: Proof of \ref{['Theorem 6.1']}
  • Theorem 7.1
  • Proof 3
  • ...and 3 more