Anastasia Georgiou

Juan M. Bello-Rivas, Anastasia Georgiou, Hannes Vandecasteele et al.

Finding saddle points of dynamical systems is an important problem in practical applications such as the study of rare events of molecular systems. Gentlest ascent dynamics (GAD) is one of a number of algorithms in existence that attempt to find saddle points in dynamical systems. It works by deriving a new dynamical system in which saddle points of the original system become stable equilibria. GAD has been recently generalized to the study of dynamical systems on manifolds (differential algebraic equations) described by equality constraints and given in an extrinsic formulation. In this paper, we present an extension of GAD to manifolds defined by point-clouds, formulated using the intrinsic viewpoint. These point-clouds are adaptively sampled during an iterative process that drives the system from the initial conformation (typically in the neighborhood of a stable equilibrium) to a saddle point. Our method requires the reactant (initial conformation), does not require the explicit constraint equations to be specified, and is purely data-driven.

Juan M. Bello-Rivas, Anastasia Georgiou, John Guckenheimer et al.

We introduce a method to successively locate equilibria (steady states) of dynamical systems on Riemannian manifolds. The manifolds need not be characterized by an a priori known atlas or by the zeros of a smooth map. Instead, they can be defined by point-clouds and sampled as needed through an iterative process. If the manifold is an Euclidean space, our method follows isoclines, curves along which the direction of the vector field $X$ is constant. For a generic vector field $X$, isoclines are smooth curves and every equilibrium lies on isoclines. We generalize the definition of isoclines to Riemannian manifolds through the use of parallel transport: generalized isoclines are curves along which the directions of $X$ are parallel transports of each other. As in the Euclidean case, generalized isoclines of generic vector fields $X$ are smooth curves that connect equilibria of $X$. Our algorithm can be regarded as an extension of the method of Newton trajectories to the manifold setting when the manifold is unknown. This work is motivated by computational statistical mechanics, specifically high dimensional (stochastic) differential equations that model the dynamics of molecular systems. Often, these dynamics concentrate near low-dimensional manifolds and have transitions (saddle points with a single unstable direction) between metastable equilibria. We employ iteratively sampled data and isoclines to locate these saddle points. Coupling a black-box sampling scheme (e.g., Markov chain Monte Carlo) with manifold learning techniques (diffusion maps in the case presented here), we show that our method reliably locates equilibria of $X$.

Anastasia Georgiou, Daniel Jungen, Luise Kaven et al.

Bayesian Optimization (BO) with Gaussian Processes relies on optimizing an acquisition function to determine sampling. We investigate the advantages and disadvantages of using a deterministic global solver (MAiNGO) compared to conventional local and stochastic global solvers (L-BFGS-B and multi-start, respectively) for the optimization of the acquisition function. For CPU efficiency, we set a time limit for MAiNGO, taking the best point as optimal. We perform repeated numerical experiments, initially using the Muller-Brown potential as a benchmark function, utilizing the lower confidence bound acquisition function; we further validate our findings with three alternative benchmark functions. Statistical analysis reveals that when the acquisition function is more exploitative (as opposed to exploratory), BO with MAiNGO converges in fewer iterations than with the local solvers. However, when the dataset lacks diversity, or when the acquisition function is overly exploitative, BO with MAiNGO, compared to the local solvers, is more likely to converge to a local rather than a global ly near-optimal solution of the black-box function. L-BFGS-B and multi-start mitigate this risk in BO by introducing stochasticity in the selection of the next sampling point, which enhances the exploration of uncharted regions in the search space and reduces dependence on acquisition function hyperparameters. Ultimately, suboptimal optimization of poorly chosen acquisition functions may be preferable to their optimal solution. When the acquisition function is more exploratory, BO with MAiNGO, multi-start, and L-BFGS-B achieve comparable probabilities of convergence to a globally near-optimal solution (although BO with MAiNGO may require more iterations to converge under these conditions).

Arjun Manoj, Anastasia S. Georgiou, Dimitris G. Giovanis et al.

Multi-fidelity optimization employs surrogate models that integrate information from varying levels of fidelity to guide efficient exploration of complex design spaces while minimizing the reliance on (expensive) high-fidelity objective function evaluations. To advance Gaussian Process (GP)-based multi-fidelity optimization, we implement a proximity-based acquisition strategy that simplifies fidelity selection by eliminating the need for separate acquisition functions at each fidelity level. We also enable multi-fidelity Upper Confidence Bound (UCB) strategies by combining them with multi-fidelity GPs rather than the standard GPs typically used. We benchmark these approaches alongside other multi-fidelity acquisition strategies (including fidelity-weighted approaches) comparing their performance, reliance on high-fidelity evaluations, and hyperparameter tunability in representative optimization tasks. The results highlight the capability of the proximity-based multi-fidelity acquisition function to deliver consistent control over high-fidelity usage while maintaining convergence efficiency. Our illustrative examples include multi-fidelity chemical kinetic models, both homogeneous and heterogeneous (dynamic catalysis for ammonia production).