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Data-dependent density estimation for the Fokker-Planck equation in higher dimensions

Max Jensen, Fabian Merle, Andreas Prohl

TL;DR

This work tackles the challenge of obtaining the global density $p(T,\cdot)$ solving the high-dimensional Fokker-Planck equation by combining a simple Euler discretization of the associated SDE with data-driven, histogram-based density estimators on adaptive state-space partitions. It introduces two partition rules, Gessaman and Binary Tree Cuboid (BTC), to build data-dependent meshes and corresponding estimators $\widehat{\mathtt{p}}^{J,\mathtt{M}}_{\text{Ges}}$ and $\widehat{\mathtt{p}}^{J,\mathtt{M}}_{\text{BTC}}$, demonstrating their ability to adapt to local solution features. The paper proves almost-sure $L^{1}$-consistency for the Gessaman-based estimator (under suitable growth of $\mathtt{M}$ and $k_{\mathtt{M}}$) and extends Bally-type convergence results to random initial data, providing a rigorous foundation for the method. Computational experiments in low and high dimensions illustrate adaptive mesh refinement and show that BTC often outperforms Ges in efficiency and accuracy, particularly for non-constant operators, highlighting the method's potential for scalable, interpretable FP density estimation in complex, high-dimensional settings.

Abstract

We present a new strategy to approximate the global solution of the Fokker-Planck equation efficiently in higher dimensions and show its convergence. The main ingredients are the Euler scheme to solve the associated stochastic differential equation and a histogram method for tree-structured density estimation on a data-dependent partitioning of the state space R^d.

Data-dependent density estimation for the Fokker-Planck equation in higher dimensions

TL;DR

This work tackles the challenge of obtaining the global density solving the high-dimensional Fokker-Planck equation by combining a simple Euler discretization of the associated SDE with data-driven, histogram-based density estimators on adaptive state-space partitions. It introduces two partition rules, Gessaman and Binary Tree Cuboid (BTC), to build data-dependent meshes and corresponding estimators and , demonstrating their ability to adapt to local solution features. The paper proves almost-sure -consistency for the Gessaman-based estimator (under suitable growth of and ) and extends Bally-type convergence results to random initial data, providing a rigorous foundation for the method. Computational experiments in low and high dimensions illustrate adaptive mesh refinement and show that BTC often outperforms Ges in efficiency and accuracy, particularly for non-constant operators, highlighting the method's potential for scalable, interpretable FP density estimation in complex, high-dimensional settings.

Abstract

We present a new strategy to approximate the global solution of the Fokker-Planck equation efficiently in higher dimensions and show its convergence. The main ingredients are the Euler scheme to solve the associated stochastic differential equation and a histogram method for tree-structured density estimation on a data-dependent partitioning of the state space R^d.
Paper Structure (10 sections, 4 theorems, 43 equations, 34 figures, 2 tables, 1 algorithm)

This paper contains 10 sections, 4 theorems, 43 equations, 34 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Assumptions and the law of the Euler scheme
  3. Assumptions on data of the SDE \ref{['eq:SDE']}
  4. Convergence and properties of the density $p^{J}$ of $\mathbf{Y}^{J}$ from \ref{['eq:euler']}
  5. Data-dependent density estimators
  6. On data-dependent partitioning strategies
  7. Gessaman's splitting rule
  8. A Binary Tree Cuboid (BTC)-based splitting rule
  9. Strong consistency of the density estimator $\widehat{\mathtt{p}}_{\mathtt{Ges}}^{J,\mathtt{M}}$
  10. Computational experiments

Key Result

Theorem 2.1

Let $J\in \mathbb{N}$, and $\{t_{j}\}_{j=0}^{J}\subset [0,T]$ be a mesh with uniform step size $\tau=\frac{T}{J}$. For all $\mathbf{y},\mathbf{x}\in \mathbb{R}^{d}$ we have where $c>0$, and $C(T)>0$ is a constant that depends on $T$.

Figures (34)

  • Figure 1: $t=0$, $j=0$
  • Figure 2: $t=0.5$, $j=50$
  • Figure 3: $t=1$, $j=100$
  • Figure 5: $t=0$, $j=0$
  • Figure 6: $t=0.5$, $j=50$
  • ...and 29 more figures

Theorems & Definitions (12)

  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Theorem 2.1: bally
  • Theorem 2.2
  • proof
  • Example 3.2
  • Theorem 3.4: lugosi
  • Theorem 3.5
  • proof : Proof of Theorem \ref{['thmgessamanconvergence']}.
  • ...and 2 more