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Solving McKean-Vlasov Equation by deep learning particle method

Jingyuan Li, Wei Liu

TL;DR

This work addresses efficient numerical solving of MV-SDEs beyond propagation-of-chaos limitations by introducing a meshless, PINN-based solver. The approach constructs a pseudo MV-SDE via Itô calculus and minimizes an $L^2$ loss to align drift and diffusion with the MV-SDE, with the key advantage that the loss is independent of the time step. The paper offers rigorous error analyses (in $L^p$ and $W_p$ metrics) and compares two learning regimes: per-particle networks for interacting systems and a single shared network for non-interacting cases, plus a self-interacting diffusion pathway for stationary distributions. Simulations demonstrate higher accuracy, meshless flexibility, and compatibility with diverse noise models, including fractional Brownian motion, with practical benefits for large particle counts and GPU acceleration.

Abstract

We introduce a novel meshless simulation method for the McKean-Vlasov Stochastic Differential Equation (MV-SDE) utilizing deep learning, applicable to both self-interaction and interaction scenarios. Traditionally, numerical methods for this equation rely on the interacting particle method combined with techniques based on the Itô-Taylor expansion. The convergence rate of this approach is determined by two parameters: the number of particles $N$ and the time step size $h$ for each Euler iteration. However, for extended time horizons or equations with larger Lipschitz coefficients, this method is often limited, as it requires a significant increase in Euler iterations to achieve the desired precision $ε$. To overcome the challenges posed by the difficulty of parallelizing the simulation of continuous interacting particle systems, which involve solving high-dimensional coupled SDEs, we propose a meshless MV-SDE solver grounded in Physics-Informed Neural Networks (PINNs) that does not rely on the propagation of chaos result. Our method constructs a pseudo MV-SDE using Itô calculus, then quantifies the discrepancy between this equation and the original MV-SDE, with the error minimized through a loss function. This loss is controlled via an optimization algorithm, independent of the time step size, and we provide an error estimate for the loss function. The advantages of our approach are demonstrated through corresponding simulations.

Solving McKean-Vlasov Equation by deep learning particle method

TL;DR

This work addresses efficient numerical solving of MV-SDEs beyond propagation-of-chaos limitations by introducing a meshless, PINN-based solver. The approach constructs a pseudo MV-SDE via Itô calculus and minimizes an loss to align drift and diffusion with the MV-SDE, with the key advantage that the loss is independent of the time step. The paper offers rigorous error analyses (in and metrics) and compares two learning regimes: per-particle networks for interacting systems and a single shared network for non-interacting cases, plus a self-interacting diffusion pathway for stationary distributions. Simulations demonstrate higher accuracy, meshless flexibility, and compatibility with diverse noise models, including fractional Brownian motion, with practical benefits for large particle counts and GPU acceleration.

Abstract

We introduce a novel meshless simulation method for the McKean-Vlasov Stochastic Differential Equation (MV-SDE) utilizing deep learning, applicable to both self-interaction and interaction scenarios. Traditionally, numerical methods for this equation rely on the interacting particle method combined with techniques based on the Itô-Taylor expansion. The convergence rate of this approach is determined by two parameters: the number of particles and the time step size for each Euler iteration. However, for extended time horizons or equations with larger Lipschitz coefficients, this method is often limited, as it requires a significant increase in Euler iterations to achieve the desired precision . To overcome the challenges posed by the difficulty of parallelizing the simulation of continuous interacting particle systems, which involve solving high-dimensional coupled SDEs, we propose a meshless MV-SDE solver grounded in Physics-Informed Neural Networks (PINNs) that does not rely on the propagation of chaos result. Our method constructs a pseudo MV-SDE using Itô calculus, then quantifies the discrepancy between this equation and the original MV-SDE, with the error minimized through a loss function. This loss is controlled via an optimization algorithm, independent of the time step size, and we provide an error estimate for the loss function. The advantages of our approach are demonstrated through corresponding simulations.
Paper Structure (22 sections, 7 theorems, 74 equations, 15 figures, 3 tables, 2 algorithms)

This paper contains 22 sections, 7 theorems, 74 equations, 15 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Introduction of the methodology
  3. Methods for approximating interacting particle system
  4. Methods for approximating independent particle system
  5. Approximating stationary distributions for self-interacting systems
  6. Reasons and Details
  7. Theoretical properties and error analysis
  8. Notations and Definition
  9. Preparation Lemmas
  10. Theoretical results
  11. Algorithms and simulations
  12. Algorithms
  13. Simulations
  14. Compare the order of convergence of the two methods
  15. Simulate particle systems in the case of meshless
  16. ...and 7 more sections

Key Result

Lemma 3.1

\newlabellemma10 For any process $X=\left(X_{t}\right)_{t \geq 0}$, for every $p \in[2, \infty)$ and for every $T \in[0,+\infty]$,

Figures (15)

  • Figure 1: Two different approximation methods for MV-SDE
  • Figure 1: loss functions vs $MSE_{dist}$ (Same $F$)
  • Figure 2: loss functions vs $MSE_{dist}$ (Different $F$)
  • Figure 3: Comparison of $MSE_{dist}$
  • Figure 4: Comparison of the solution trajectories under different methods.
  • ...and 10 more figures

Theorems & Definitions (13)

  • Lemma 3.1: The Generalized Minkowski Inequality
  • Lemma 3.2: Burk?lder-Davis-Gundy Inequality
  • Lemma 3.3: la Gronwall Inequality
  • Lemma 3.4
  • Theorem 3.6: Estimation of moments and Wasserstein distance bound in particle system
  • Proof 1
  • Theorem 3.7
  • Proof 2
  • Theorem 3.9: Estimation of moments and Wasserstein distance bound in self-interaction particle system
  • Proof 3
  • ...and 3 more