High-entropy nanoparticles have become a rapidly growing area of research in recent years. Because of their multielemental compositions and unique high-entropy mixing states (i.e., solid-solution) that can lead to tunable activity and enhanced stability, these nanoparticles have received notable attention for catalyst design and exploration. However, this strong potential is also accompanied by grand challenges originating from their vast compositional space and complex atomic structure, which hinder comprehensive exploration and fundamental understanding. Through a multidisciplinary view of synthesis, characterization, catalytic applications, high-throughput screening, and data-driven materials discovery, this review is dedicated to discussing the important progress of high-entropy nanoparticles and unveiling the critical needs for their future development for catalysis, energy, and sustainability applications.

Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD)⁵¹ and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables which spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the solution reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky partical differential equation as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy, we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems.

A data-driven, kernel-based method for approximating the leading Koopman eigenvalues, eigenfunctions, and modes in problems with high-dimensional state spaces is presented. This approach uses a set of scalar observables (functions that map a state to a scalar value) that are defined implicitly by the feature map associated with a user-defined kernel function. This circumvents the computational issues that arise due to the number of functions required to span a ``sufficiently rich'' subspace of all possible scalar observables in such applications. We illustrate this method on two examples: the first is the FitzHugh-Nagumo PDE, a prototypical one-dimensional reaction-diffusion system, and the second is a set of vorticity data computed from experimentally obtained velocity data from flow past a cylinder at Reynolds number 413. In both examples, we use the output of Dynamic Mode Decomposition, which has a similar computational cost, as the benchmark for our approach.

In traditional physicochemical modeling, one derives evolution equations at the (macroscopic, coarse) scale of interest; these are used to perform a variety of tasks (simulation, bifurcation analysis, optimization) using an arsenal of analytical and numerical techniques. For many complex systems, however, although one observes evolution at a macroscopic scale of interest, accurate models are only given at a more detailed (fine-scale, microscopic) level of description (e.g., lattice Boltzmann, kinetic Monte Carlo, molecular dynamics). Here, we review a framework for computer-aided multiscale analysis, which enables macroscopic computational tasks (over extended spatiotemporal scales) using only appropriately initialized microscopic simulation on short time and length scales. The methodology bypasses the derivation of macroscopic evolution equations when these equations conceptually exist but are not available in closed form—hence the term equation-free. We selectively discuss basic algorithms and underlying principles and illustrate the approach through representative applications. We also discuss potential difficulties and outline areas for future research.

Concise, accurate descriptions of physical systems through their conserved quantities abound in the natural sciences. In data science, however, current research often focuses on regression problems, without routinely incorporating additional assumptions about the system that generated the data. Here, we propose to explore a particular type of underlying structure in the data: Hamiltonian systems, where an “energy” is conserved. Given a collection of observations of such a Hamiltonian system over time, we extract phase space coordinates and a Hamiltonian function of them that acts as the generator of the system dynamics. The approach employs an autoencoder neural network component to estimate the transformation from observations to the phase space of a Hamiltonian system. An additional neural network component is used to approximate the Hamiltonian function on this constructed space, and the two components are trained jointly. As an alternative approach, we also demonstrate the use of Gaussian processes for the estimation of such a Hamiltonian. After two illustrative examples, we extract an underlying phase space as well as the generating Hamiltonian from a collection of movies of a pendulum. The approach is fully data-driven and does not assume a particular form of the Hamiltonian function.