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A Spectral Approach to Optimal Control of the Fokker-Planck Equation

Dante Kalise, Lucas M. Moschen, Grigorios A. Pavliotis, Urbain Vaes

TL;DR

The paper develops a spectral optimal-control framework for the linear Fokker-Planck equation by applying the ground-state transformation to map the FP operator to a Schrödinger operator 𝒢 = -σ Δ + W(x). It then discretizes spectrally in the eigenbasis of 𝒢 and solves a reduced open-loop optimal-control problem via Pontryagin conditions and Barzilai-Borwein updates, with a designed shape-function α_i to amplify slow-decaying modes. A practical initialization from an infinite-horizon LQR Riccati solution enhances convergence, and numerical tests on ill-conditioned Gaussian and double-well potentials demonstrate substantial acceleration in approaching the steady state. The approach broadens the toolbox for diffusive control in unbounded domains, though it faces scalability challenges in high dimensions and does not guarantee exact controllability to arbitrary targets; future work includes model reduction and data-driven mode selections to mitigate this.

Abstract

In this paper, we present a spectral optimal control framework for Fokker-Planck equations based on the standard ground state transformation that maps the Fokker-Planck operator to a Schrodinger operator. Our primary objective is to accelerate convergence toward the (unique) steady state. To fulfill this objective, a gradient-based iterative algorithm with Pontryagin's maximum principle and the Barzilai-Borwein update is developed to compute time-dependent controls. Numerical experiments on two-dimensional ill-conditioned normal distributions and double-well potentials demonstrate that our approach effectively targets slow-decaying modes, thus increasing the spectral gap.

A Spectral Approach to Optimal Control of the Fokker-Planck Equation

TL;DR

The paper develops a spectral optimal-control framework for the linear Fokker-Planck equation by applying the ground-state transformation to map the FP operator to a Schrödinger operator 𝒢 = -σ Δ + W(x). It then discretizes spectrally in the eigenbasis of 𝒢 and solves a reduced open-loop optimal-control problem via Pontryagin conditions and Barzilai-Borwein updates, with a designed shape-function α_i to amplify slow-decaying modes. A practical initialization from an infinite-horizon LQR Riccati solution enhances convergence, and numerical tests on ill-conditioned Gaussian and double-well potentials demonstrate substantial acceleration in approaching the steady state. The approach broadens the toolbox for diffusive control in unbounded domains, though it faces scalability challenges in high dimensions and does not guarantee exact controllability to arbitrary targets; future work includes model reduction and data-driven mode selections to mitigate this.

Abstract

In this paper, we present a spectral optimal control framework for Fokker-Planck equations based on the standard ground state transformation that maps the Fokker-Planck operator to a Schrodinger operator. Our primary objective is to accelerate convergence toward the (unique) steady state. To fulfill this objective, a gradient-based iterative algorithm with Pontryagin's maximum principle and the Barzilai-Borwein update is developed to compute time-dependent controls. Numerical experiments on two-dimensional ill-conditioned normal distributions and double-well potentials demonstrate that our approach effectively targets slow-decaying modes, thus increasing the spectral gap.

Paper Structure

This paper contains 9 sections, 1 theorem, 39 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Fokker-Planck Equation and Preliminaries
  3. Optimal Control Problem Formulation
  4. Numerical Method: Spectral Discretization
  5. The Iterative Method
  6. Choice of the Shape Function
  7. Initialization of the Control Functions
  8. Numerical Experiments and Results
  9. Conclusions

Key Result

Proposition 1

Define $b_i \coloneqq \nabla\alpha_i \cdot \nabla\log\sqrt{\rho_\infty}$. Then, for all $\phi \in L^2(\mathbb{R}^d)$, Moreover, the formal adjoint of $\mathcal{N}_i$ in $L^2(\mathbb{R}^d)$ is

Figures (4)

  • Figure 1: Time evolution of the error norm for the quadratic potential with $a = 1$; line styles distinguish values of $b$ (solid for $b = 0.1$, dashed for $b = 0.05$), while colors indicate the control strategy (black: uncontrolled, blue: LQR feedback control, red: optimal control).
  • Figure 2: The four optimized control functions for the quadratic potential experiments with $a=1$ and $b=0.1$; control is shown only up to time $t = 0.5$ as it remains close to zero for $t > 0.5$.
  • Figure 3: Time evolution of the error norm $\|a(t)-a^\dagger\|_2$ for the double-well potential. The plot compares uncontrolled dynamics (dashed), LQR feedback control (dotted), and optimized open-loop control (solid).
  • Figure 4: Time evolution of the coefficients of the spectral decomposition of the double-well potential.

Theorems & Definitions (5)

  • Proposition 1
  • proof
  • remark 1
  • remark 2
  • proof