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

This repository accompanies the paper:

A Spectral Approach to Optimal Control of the Fokker–Planck Equation
by Dante Kalise, Lucas M. Moschen, Grigorios A. Pavliotis, and Urbain Vaes
Accepted in IEEE Control Systems Letters (L-CSS), 2025: https://ieeexplore.ieee.org/document/11015582/

The ArXiV version is located https://arxiv.org/abs/2503.15125.

To reproduce the examples, check the release with tag lcss2025-v1.

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Overview

This code implements a spectral method for the optimal control of the Fokker–Planck equation based on a Schrödinger operator formulation. The framework supports:

• Eigenfunction-based discretization in 1D and 2D
• Analytic, finite-difference, and PDE-based solvers
• Forward–backward optimal control via reduced-order modeling
• Visualization and benchmark scripts for controlled convergence

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Repository Structure

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├── schrodinger_operator.py        # Main class-based solver library
├── example_control.py             # Script: example run with 1–4 control fields
├── main_examples.ipynb            # Reproduces the main results from the paper
├── explore_schrodinger_operators.ipynb  # Experimental notebook for spectral operators
├── experimental_fp_solver.py      # Prototype for Fokker–Planck dynamics (unused)
├── experimental_LQ_solver.py      # Experimental LQR control script
├── requirements.txt               # Python dependencies
├── README.md                      # Project documentation

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Setup

Install the required dependencies with:

pip install -r requirements.txt

To run notebooks, you may also use:

jupyter notebook

Note: Some solvers require access to Wolfram Engine via the wolframclient Python interface.

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How to Run

To reproduce the key control results (Figure 1 in the paper):

python example_control.py

This generates a plot of the $L^2$ distance between the evolving state and target, under 0 to 4 control terms.

To explore the examples used in the paper:

jupyter notebook main_examples.ipynb

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Citation

If this code contributes to your research, please cite:

@article{kalise2025spectralfp,
  author={Kalise, Dante and Moschen, Lucas M. and Pavliotis, Grigorios A. and Vaes, Urbain},
  journal={IEEE Control Systems Letters}, 
  title={A Spectral Approach to Optimal Control of the Fokker–Planck Equation}, 
  year={2025},
  doi={10.1109/LCSYS.2025.3573604}}

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Notes

• experimental_fp_solver.py and experimental_LQ_solver.py are prototypes not used in the final results.
• explore_schrodinger_operators.ipynb is an exploratory notebook for testing spectral properties and interfaces.
• All computational routines are written in Python using NumPy, SciPy, and optionally Wolfram Language for PDE-based solvers.