Advanced computational methods for material science, a multifaceted exploration

Background

Material science plays a crucial role in space exploration by enabling the development of structures that can withstand the harsh conditions of outer space [1]. Space craft components, in particular, require outstanding characteristics such as radiation resistance, adaptability to extreme temperatures, optimal energy efficiency, and resilience against degradation. To achieve these goals, simulations based on efficient computational methods play a crucial role in predicting and discovering the behavior of materials.

Two interesting models which lay at the basis of such simulators are the Fokker-Planck equation and the Density Functional Theory (DFT). The former is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion; the Fokker-Planck equation describes a variety of relevant phenomena such as diffusion processes[2], electron transport in semiconductors [3], nanomaterials and mesoscopic systems [4]. The latter, gives a quantum-mechanical representation of interactions among electrons and between electrons and atomic nuclei; it has gained growing significance in the fields of solid-state physics, chemistry, and materials science [6]. These models differ in their treatment of electron-electron interactions, the nature of the mathematical formalism, and their computational efficiency and their applications. Both methods have their strengths and limitations, and researchers often choose the method that best suits the specific characteristics of the system under investigation.

Despite notable advancements in computational techniques and integrated electronics, exemplified by Moore's law [7], the course of dimensionality [8] still prevents from the fine description of the mechanics and structure of materials at nanometric scale. Here, major challenges are related to the impressive number of degrees of freedom required due to the highly dimensional spaces in which those models are defined.

Project Overview

This project wants to investigate the feasibility of advanced computational methods to solve the Fokker-Plank equation and the Density Functional Theory, with the aim of improving the generalization and accuracy of the results, other than the speed and computational efficiency of the simulation itself. Within the Advanced Concept Team, we explore two innovative toolkits made available from the scientific community in relatively recent times.

• Machine Learning is gaining increasing popularity not only in fields such as computer vision, fault detection, and control theory, but also in scientific computing, to the extent that a new branch of this discipline was born, called "Scientific Machine Learning". The latter encompasses methodologies such as Physics-Informed Neural Networks [9] and Neural Operators called DeepONET [10], that we aim to use in a completely physics driven way as solvers for the Fokker-Planck equation, and by building upon these endeavors we can enhance our comprehension of the potential role that artificial intelligence may play in this field.
• Combinatorial optimization problems are typically computationally intractable, recent advancements in both algorithmic techniques and specialized hardware accelerators, such as Ising machines, offer promising avenues for scaling optimization algorithms to tackle industrial-scale instances more efficiently [11]. We aim todevelop and innovative encoding of the generalized eigenvalue problem, which is the bottleneck of DFT simulation, to turn it in a Quadratic Binary Optimization problem (QUBO) and solve it on an hising-machine [11].

References

[1] https://www.esa.int/ESA_Multimedia/Keywords/Description/Materials_science/(result_type)/videos

[2] Sorokin, M. V. and Dubinko, V. I. and Borodin, V. A., Applicability of the Fokker-Planck equation to the description of diffusion effects on nucleation ,Physical Review E, Volume 95, Issue 1, 2017, Publisher American Physical Society, https://link.aps.org/doi/10.1103/PhysRevE.95.012801.

[3] Vladimir I. Kolobov, Fokker–Planck modeling of electron kinetics in plasmas and semiconductors, Computational Materials Science, Volume 28, Issue 2, 2003, Pages 302-320, ISSN 0927-0256, https://doi.org/10.1016/S0927-0256(03)00115-0.

[4] D. Reguera, J. M. Rubí, and J. M. G. Vilar, J. Phys. Chem. B 2005, 109, 46, 21502–21515, Publication Date:October 29, 2005 https://doi.org/10.1021/jp052904i

[5] Hafner J. Ab-initio simulations of materials using VASP: Density-functional theory and beyond. J Comput Chem. 2008 Oct;29(13):2044-78. doi: 10.1002/jcc.21057. PMID: 18623101.

[6] J.M. Thijssen. Computational Physics. Cambridge University Press, 1999.

[7] E. Mollick, "Establishing Moore's Law," in IEEE Annals of the History of Computing, vol. 28, no. 3, pp. 62-75, July-Sept. 2006, doi: 10.1109/MAHC.2006.45.

[8] https://en.wikipedia.org/wiki/Curse_of_dimensionality

[9] M. Raissi, P. Perdikaris, G.E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics, Volume 378, 2019, Pages 686-707, ISSN 0021-9991, https://doi.org/10.1016/j.jcp.2018.10.045.

[10] Somdatta Goswami and Aniruddha Bora and Yue Yu and George Em Karniadakis, Physics-Informed Deep Neural Operator Networks, arXiv, 2022, eprint 2207.05748, https://doi.org/10.48550/arXiv.2207.05748.

[11] Hod Wirzberger et al. Lightsolver challenges a leading deep learning solver for Max-2-SAT problems. arXiv preprint arXiv:2302.06926, 2023.

[12] Negre, Christian Francisco Andres, Lopez-Bezanilla, Alejandro, Zhang, Yu, Akrobotu, Prosper D., Mniszewski, Susan M., Tretiak, Sergei, and Dub, Pavel A. Toward a QUBO-Based Density Matrix Electronic Structure Method. United States: N. p., 2022. Web. doi:10.1021/acs.jctc.2c00090.