(51f) From Brownian Dynamics Simulations and Experimental Observations of Colloidal Suspensions to Data-Driven Observables and Effective Sdes with Manifold Learning
AIChE Annual Meeting
2021
2021 Annual Meeting
Topical Conference: Applications of Data Science to Molecules and Materials
T3 Virtual Talks: Applications of Data Science to Molecules and Materials
Wednesday, November 17, 2021 - 9:00am to 9:12am
We construct a reduced data-driven surrogate SDE model for electric-field mediated colloidal crystallization using data obtained from Brownian dynamics [1].
These simulations were previously matched to optical microscopy experiments on quasi-2D colloidal assembly in multipolar AC electric fields (that generate reconfigurable energy landscapes for charged colloidal particles[2-3])). Ongoing extensions of these experiments and simulations, also to be analyzed using the approaches reported in this work, include systematically varying particle shape and assembly on curved surfaces, which both introduce topological defects and new dynamics features.
To achieve this, we first simulate a system of 210 particles in the canonical ensemble under fixed voltage. The obtained configurations include fluid, polycrystalline and crystalline states [1]. The configurations are aligned to a reference configuration using the Kabsch algorithm [4-5] and then we use the manifold learning method Diffusion maps [6], with an appropriate metric, to learn the intrinsic geometry of the collected data set. Diffusion maps eigenvectors give a set of reduced/effective/latent variables in which we can learn the SDE. Those latent variables can also be tested for their physical explainability by utilization of the Inverse Function theorem in a data-assisted way with the use of the extension scheme Geometric Harmonics [7].
In order to construct an S.D.E. on our reduced/latent variables an effective gradient needs to be obtained. The identified drift and the diffusion coefficients provide a way to study the evolution on those latent coordinates through an effective Langevin equation [8]. We illustrate the estimation of the drift and diffusivity in the latent space in two ways; (a) a data-driven approach that allows us to approximate the coefficients using their statistical definition [8-9] (b) through a Neural Network architecture in which the state dependent drift and diffusivity are computed from paired snapshots (xt,xt+h), scattered over the domain of interest. This latter approach works pointwise and the loss function is formulated for Gaussian noise. Our reduced/effective trajectories are being "lifted'' on the ambient space with the help of a Generative Adversarial Network (GANs). Lastly, we construct an effective potential based on the learned SDE.
[1] Yuguang Yang, Bevan, Michael A., Raghuram Thyagarajan, and David M.Ford. Dynamic colloidal assembly pathways via low dimensional models.Journal of Chemical Physics,144(20), 5 2016.
[2]. Zhang, Y. Zhang, M. A. Bevan, Spatially Varying Colloidal Phase Behavior on Multi-Dimensional Energy Landscapes. J. Chem. Phys. 152, 054905 (2020).
[3]. J. Zhang, J. Yang, Y. Zhang, M. A. Bevan, Controlling colloidal crystals via morphing energy landscapes and reinforcement learning. Science Advances 6, eabd6716 (2020).
[4] Kabsch W (1976) A solution for the best rotation to relate two sets of vectors. Acta Crystallogr A 32:922–933. 57.
[5] Kabsch W (1978) A discussion of the solution for the best rotation to relate two sets of vectors. Acta Crystallogr A 34:827–828.
[6] Ronald R Coifman and Stephane Lafon. Diffusion maps. Applied and Computational Harmonic Analysis, 21(1):5–30, 2006.
[7] Ronald R Coifman and Stephane Lafon. Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions. Applied and Computational Harmonic Analysis,21(1):31–52, 2006
[8] Ping Liu, CI Siettos, CW Gear, and IG Kevrekidis. Equation-free model reduction in agent-based computations: Coarse-grained bifurcation and variable-free rare event analysis. Mathematical Modelling of Natural Phenomena, 10(3):71–90, 2015
[9] H. Risken, T Frank ,The Fokker-Planck Equation: Methods of Solution and Applications Springer , 2012.
These simulations were previously matched to optical microscopy experiments on quasi-2D colloidal assembly in multipolar AC electric fields (that generate reconfigurable energy landscapes for charged colloidal particles[2-3])). Ongoing extensions of these experiments and simulations, also to be analyzed using the approaches reported in this work, include systematically varying particle shape and assembly on curved surfaces, which both introduce topological defects and new dynamics features.
To achieve this, we first simulate a system of 210 particles in the canonical ensemble under fixed voltage. The obtained configurations include fluid, polycrystalline and crystalline states [1]. The configurations are aligned to a reference configuration using the Kabsch algorithm [4-5] and then we use the manifold learning method Diffusion maps [6], with an appropriate metric, to learn the intrinsic geometry of the collected data set. Diffusion maps eigenvectors give a set of reduced/effective/latent variables in which we can learn the SDE. Those latent variables can also be tested for their physical explainability by utilization of the Inverse Function theorem in a data-assisted way with the use of the extension scheme Geometric Harmonics [7].
In order to construct an S.D.E. on our reduced/latent variables an effective gradient needs to be obtained. The identified drift and the diffusion coefficients provide a way to study the evolution on those latent coordinates through an effective Langevin equation [8]. We illustrate the estimation of the drift and diffusivity in the latent space in two ways; (a) a data-driven approach that allows us to approximate the coefficients using their statistical definition [8-9] (b) through a Neural Network architecture in which the state dependent drift and diffusivity are computed from paired snapshots (xt,xt+h), scattered over the domain of interest. This latter approach works pointwise and the loss function is formulated for Gaussian noise. Our reduced/effective trajectories are being "lifted'' on the ambient space with the help of a Generative Adversarial Network (GANs). Lastly, we construct an effective potential based on the learned SDE.
[1] Yuguang Yang, Bevan, Michael A., Raghuram Thyagarajan, and David M.Ford. Dynamic colloidal assembly pathways via low dimensional models.Journal of Chemical Physics,144(20), 5 2016.
[2]. Zhang, Y. Zhang, M. A. Bevan, Spatially Varying Colloidal Phase Behavior on Multi-Dimensional Energy Landscapes. J. Chem. Phys. 152, 054905 (2020).
[3]. J. Zhang, J. Yang, Y. Zhang, M. A. Bevan, Controlling colloidal crystals via morphing energy landscapes and reinforcement learning. Science Advances 6, eabd6716 (2020).
[4] Kabsch W (1976) A solution for the best rotation to relate two sets of vectors. Acta Crystallogr A 32:922–933. 57.
[5] Kabsch W (1978) A discussion of the solution for the best rotation to relate two sets of vectors. Acta Crystallogr A 34:827–828.
[6] Ronald R Coifman and Stephane Lafon. Diffusion maps. Applied and Computational Harmonic Analysis, 21(1):5–30, 2006.
[7] Ronald R Coifman and Stephane Lafon. Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions. Applied and Computational Harmonic Analysis,21(1):31–52, 2006
[8] Ping Liu, CI Siettos, CW Gear, and IG Kevrekidis. Equation-free model reduction in agent-based computations: Coarse-grained bifurcation and variable-free rare event analysis. Mathematical Modelling of Natural Phenomena, 10(3):71–90, 2015
[9] H. Risken, T Frank ,The Fokker-Planck Equation: Methods of Solution and Applications Springer , 2012.