Anastasia Georgiou

Sujit Roy, Udayshankar Nair, Yuling Wu et al.

The martian atmosphere hosts dynamical phenomena ranging from planet-encircling dust storms to mesoscale orographic clouds and nocturnal low-level jets. General circulation model show capability to simulate these phenomena, but is computationally expensive at resolution needed to resolve mesoscale features. While assimilation of satellite remote sensing observation enable forecasting capabilities using such models, observation record is often sparse, short and fragmented across instrument generators. These constraints motivate the development of a data-driven foundation model for the Martian atmosphere. Foundation models live in a complex design landscape. There is an interplay between the available data, the physics of the underlying processes and corresponding developments in AI. Even though the idea of a foundation model is to address multiple use cases in a data- and compute-efficient manner, it is important to have a clear picture what applications can sensibly addressed by a single model. The purpose of this paper is to elucidate this design landscape. We discuss available data ranging from atmospheric retrievals to reanalysis datasets as well as existing physical models. Moreover, we identify a wide range of candidate downstream applications. Finally, we consider relevant recent developments in artificial intelligence (AI) that can be leveraged in this context. Here, we put a particular emphasis on AI models for atmospheric physics, data-driven approaches to data assimilation as well as methods to work in a limited data setting.

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).