# American Institute of Mathematical Sciences

December  2020, 7(2): 511-536. doi: 10.3934/jcd.2020021

## Manifold learning for accelerating coarse-grained optimization

 1 Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, USA 2 Department of Applied Mathematics and Statistics (AMS), Johns Hopkins University, Baltimore, MD 21218, USA 3 Department of Mathematics, Imperial College London, London SW7 2AZ, UK 4 Department of Chemical and Biomolecular Engineering & AMS, Johns Hopkins University, Baltimore, MD 21218, USA

* Corresponding author

Received  October 2019 Published  July 2020

Algorithms proposed for solving high-dimensional optimization problems with no derivative information frequently encounter the "curse of dimensionality, " becoming ineffective as the dimension of the parameter space grows. One feature of a subclass of such problems that are effectively low-dimensional is that only a few parameters (or combinations thereof) are important for the optimization and must be explored in detail. Knowing these parameters/combinations in advance would greatly simplify the problem and its solution. We propose the data-driven construction of an effective (coarse-grained, "trend") optimizer, based on data obtained from ensembles of brief simulation bursts with an "inner" optimization algorithm, that has the potential to accelerate the exploration of the parameter space. The trajectories of this "effective optimizer" quickly become attracted onto a slow manifold parameterized by the few relevant parameter combinations. We obtain the parameterization of this low-dimensional, effective optimization manifold on the fly using data mining/manifold learning techniques on the results of simulation (inner optimizer iteration) burst ensembles and exploit it locally to "jump" forward along this manifold. As a result, we can bias the exploration of the parameter space towards the few, important directions and, through this "wrapper algorithm, " speed up the convergence of traditional optimization algorithms.

Citation: Dmitry Pozharskiy, Noah J. Wichrowski, Andrew B. Duncan, Grigorios A. Pavliotis, Ioannis G. Kevrekidis. Manifold learning for accelerating coarse-grained optimization. Journal of Computational Dynamics, 2020, 7 (2) : 511-536. doi: 10.3934/jcd.2020021
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Evolution in time of the probability density of current points using either the Langevin equation or SA at constant $T$. The objective function is $f(x) = 0.5x^2$; $10^4$ realizations are used with starting point $x = 1$, $T = 0.5$; and the time step is $dt = 10^{-3}$
Applying Diffusion Maps to the Swiss roll data set
One complete run of the algorithm that approaches the global maximum. Six total "coarse iterations" (shown in red) were performed. In addition, a single run of SA using the same number of function evaluations is shown in green. Our algorithm visibly approaches the maximum much faster
A comparison of the evolving maximum objective value for both methods. SA combined with DMaps needs only a fraction of the total function evaluations compared to simple SA
Diffusion Maps applied to bursts on a rectangular strip]{Diffusion maps applied to (A) the original 2D data with Euclidean distances, and (B) the transformed data with Mahalanobis distances. In both cases, the inherent dimensionality is recovered in the first two eigenvectors, as the coloring of the data by the leading Diffusion Map coordinates corresponding to these two eigenvectors show
Estimation of the first drift coefficient $\theta_1$. "Theoretical" are obtained numerically from (5), "Euclidean" via DMaps on the original data, and "Mahalanobis" from DMaps on the transformed data. The last subplot shows results from Euclidean DMaps on transformed data, which, as expected, yields incorrect estimates
Estimation of the second drift coefficient $\theta_2$. The subplots are analogous to those in Figure 6
Coarse-grained (two-dimensional) optimization on a cylindrical surface in three dimensions. One complete run of the algorithm, illustrating the short bursts of trajectories and new points after being lifted by geometric harmonics. Observe that, although the lifting does not perfectly locate the cylinder, the burst trajectories are quickly attracted back to it. The second plot is a top-down view of the first
Simulation of a short trajectory initialized at perturbed initial conditions $(x_{ic}, \bf{y}_{ic})$
The first diffusion coordinate parameterizes $x$
Estimation of the drift coefficient in diffusion map space
Estimation of the diffusion coefficient in diffusion map space
Nonlinear transformation of the slow variable $x$ onto a semicircle in the plane
Embeddings of the original and transformed data set
Estimation of the drift coefficient in Diffusion Map space
Estimation of the diffusion coefficient in Diffusion Map space
Estimation of the drift coefficient in diffusion map space
Estimation of the diffusion coefficient in diffusion map space
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