PhysicInformedDeepLearningColloidalControl

This repository contains the simulation files for the paper:

I. Nodozi, J. O'Leary, A. Mesbah, and A. Halder, "A Physics-informed Deep Learning Approach for Minimum Effort Stochastic Control of Colloidal Self-Assembly," presented at ACC 2023.

In this work, we formulate the finite-horizon stochastic optimal control problem for colloidal self-assembly in the space of probability density functions (PDFs) of the underlying state variables (namely, order parameters). The control objective is to steer the state PDFs from a prescribed initial probability measure towards a prescribed terminal probability measure with minimum control effort.

Controlled self-assembly process is used in precision (e.g., sub nm scale) manufacturing of materials with advanced electrical, magnetic, or optical properties.

The prescribed initial PDF $\rho_0$ (solid line) at the initial time $t=0$, and the prescribed terminal PDF $\rho_{T}$ (dashed line) at the final time $t=T$. Both PDFs are supported over $[0,6]$, which is the range of values for the state variable $\langle C_6\rangle$ denoting a crystallinity order parameter. In particular, $\langle C_6\rangle \approx 0$ implies a disordered state and $\langle C_6\rangle \approx$ 5-6 implies a highly ordered state.

The architecture of the physics-informed neural network with the system order parameter and time as the input features $\mathbf{x}:=(\langle C_6\rangle, t)$. The output $\mathbf{y}$ comprises the value function, optimally controlled PDF, and optimal control policy.

Training data in the domain $\Omega =[0,6]\times[0,200]$. We choose 1000 training points at each of the initial ($t=0$ s) and terminal ($t=200$ s) times and another 5000 state-time points inside the domain $\Omega:=[0,6]\times[0,200]$.

After 15,000 training epochs, the residuals for all loss functions go below $10^{-3}$.

Snapshots of the optimally controlled joint PDFs steering the order parameters distribution from the given initial distribution to the given terminal distribution over the given time horizon $[0,T]\equiv [0,200]$ s subject to the controlled noisy nonlinear sample path dynamics. The solid black curves with grey-filled areas are obtained from the PINN. The stem plots are the KDE approximants of the optimally controlled PDF snapshots obtained from the closed-loop sample paths.

Repository Contents

• /code: Contains the source code for the simulations.
• /data: Includes datasets obtained from the simulations.
• /results: Directory for storing the figures.

Requirements

The Jupyter notebook provided in this repository includes the following Python libraries:

• deepxde
• numpy
• scipy
• math
• matplotlib

Usage

1. Clone the repository:

git clone https://github.com/inodozi/PhysicInformedDeepLearningColloidalControl.git
cd PhysicInformedDeepLearningColloidalControl

2. Navigate to the notebooks directory and open the Jupyter notebooks to explore the simulations:

jupyter notebook

3. Run the provided scripts in the directory to reproduce the results.

Citation

If you find this repository useful in your research, please consider citing our paper:

@inproceedings{nodozi2023physics,
  title={A physics-informed deep learning approach for minimum effort stochastic control of colloidal self-assembly},
  author={Nodozi, Iman and O’Leary, Jared and Mesbah, Ali and Halder, Abhishek},
  booktitle={2023 American Control Conference (ACC)},
  pages={609--615},
  year={2023},
  organization={IEEE}
}

Contact

For any questions or inquiries, please contact inodozi@ucsc.edu