Machine learning interatomic potentials have become an indispensable tool for materials science, enabling the study of larger systems and longer timescales. State-of-the-art models are generally graph neural networks that employ message passing to iteratively update atomic embeddings that are ultimately used for predicting properties. In this work we extend the message passing formalism with the inclusion of a continuous variable that accounts for fractional atomic existence. This allows us to calculate the gradient of the Gibbs free energy with respect to both the Cartesian coordinates of atoms and their existence. Using this we propose a gradient-based grand canonical optimization method and document its capabilities for a Cu(110) surface oxide.

Theories with a sign problem due to a complex action or Boltzmann weight can sometimes be numerically solved using a stochastic process in the complexified configuration space. However, the probability distribution effectively sampled by this complex Langevin process is not known a priori and notoriously hard to understand. In generative AI, diffusion models can learn distributions, or their log derivatives, from data. We explore the ability of diffusion models to learn the distributions sampled by a complex Langevin process, comparing score-based and energy-based diffusion models, and speculate about possible applications.

Understanding material surfaces and interfaces is vital in applications like catalysis or electronics. By combining energies from electronic structure with statistical mechanics, ab initio simulations can in principle predict the structure of material surfaces as a function of thermodynamic variables. However, accurate energy simulations are prohibitive when coupled to the vast phase space that must be statistically sampled. Here, we present a bi-faceted computational loop to predict surface phase diagrams of multi-component materials that accelerates both the energy scoring and statistical sampling methods. Fast, scalable, and data-efficient machine learning interatomic potentials are trained on high-throughput density-functional theory calculations through closed-loop active learning. Markov-chain Monte Carlo sampling in the semi-grand canonical ensemble is enabled by using virtual surface sites. The predicted surfaces for GaN(0001), Si(111), and SrTiO3(001) are in agreement with past work and suggest that the proposed strategy can model complex material surfaces and discover previously unreported surface terminations.

In recent years, the use of machine learning has become increasingly popular in the context of lattice field theories. An essential element of such theories is represented by symmetries, whose inclusion in the neural network properties can lead to high reward in terms of performance and generalizability. A fundamental symmetry that usually characterizes physical systems on a lattice with periodic boundary conditions is equivariance under spacetime translations. Here we investigate the advantages of adopting translationally equivariant neural networks in favor of non-equivariant ones. The system we consider is a complex scalar field with quartic interaction on a two-dimensional lattice in the flux representation, on which the networks carry out various regression and classification tasks. Promising equivariant and non-equivariant architectures are identified with a systematic search. We demonstrate that in most of these tasks our best equivariant architectures can perform and generalize significantly better than their non-equivariant counterparts, which applies not only to physical parameters beyond those represented in the training set, but also to different lattice sizes.

The free energy plays a fundamental role in descriptions of many systems in continuum physics. Notably, in multiphysics applications, it encodes thermodynamic coupling between different fields. It thereby gives rise to driving forces on the dynamics of interaction between the constituent phenomena. In mechano-chemically interacting materials systems, even consideration of only compositions, order parameters and strains can render the free energy to be reasonably high-dimensional. In proposing the free energy as a paradigm for scale bridging, we have previously exploited neural networks for their representation of such high-dimensional functions. Specifically, we have developed an integrable deep neural network (IDNN) that can be trained to free energy derivative data obtained from atomic scale models and statistical mechanics, then analytically integrated to recover a free energy density function. The motivation comes from the statistical mechanics formalism, in which certain free energy derivatives are accessible for control of the system, rather than the free energy itself. Our current work combines the IDNN with an active learning workflow to improve sampling of the free energy derivative data in a high-dimensional input space. Treated as input-output maps, machine learning accommodates role reversals between independent and dependent quantities as the mathematical descriptions change with scale bridging. As a prototypical system we focus on Ni-Al. Phase field simulations using the resulting IDNN representation for the free energy density of Ni-Al demonstrate that the appropriate physics of the material have been learned. To the best of our knowledge, this represents the most complete treatment of scale bridging, using the free energy for a practical materials system, that starts with electronic structure calculations and proceeds through statistical mechanics to continuum physics.

Focusing on the grand-canonical extension of the ordinary restricted Boltzmann machine, we suggest an energy-based model for feature extraction that uses a layer of hidden units with varying size. By an appropriate choice of the chemical potential and given a sufficiently large number of hidden resources the generative model is able to efficiently deduce the optimal number of hidden units required to learn the target data with exceedingly small generalization error. The formal simplicity of the grand-canonical ensemble combined with a rapidly converging ansatz in mean-field theory enable us to recycle well-established numerical algothhtims during training, like contrastive divergence, with only minor changes. As a proof of principle and to demonstrate the novel features of grand-canonical Boltzmann machines, we train our generative models on data from the Ising theory and MNIST.

Machine learning methods for solving the equations of dynamical mean-field theory are developed. The method is demonstrated on the three dimensional Hubbard model. The key technical issues are defining a mapping of an input function to an output function, and distinguishing metallic from insulating solutions. Both metallic and Mott insulator solutions can be predicted. The validity of the machine learning scheme is assessed by comparing predictions of full correlation functions, of quasi-particle weight and particle density to values directly computed. The results indicate that with modest further development, machine learning approach may be an attractive computational efficient option for real materials predictions for strongly correlated systems.

The problem of resource allocation in sparse graphs with real variables is studied using methods of statistical physics. An efficient distributed algorithm is devised on the basis of insight gained from the analysis and is examined using numerical simulations, showing excellent performance and full agreement with the theoretical results.

The problem of resource allocation in sparse graphs with real variables is studied using methods of statistical physics. An efficient distributed algorithm is devised on the basis of insight gained from the analysis and is examined using numerical simulations, showing excellent performance and full agreement with the theoretical results.

The problem of resource allocation in sparse graphs with real variables is studied using methods of statistical physics. An efficient distributed algorithm is devised on the basis of insight gained from the analysis and is examined using numerical simulations, showing excellent performance and full agreement with the theoretical results.