Periodic Reporting for period 1 - OTmeetsDFT (Multi-marginal Optimal Transport and Density Functional Theory: a mathematical setting for physical ideas)
Okres sprawozdawczy: 2019-05-01 do 2021-04-30
Being able, by sole means of computer simulations, to make a preselection of pharmaceutical drugs that deserve to be tested experimentally, to predict the mechanisms of DNA damage by specific compounds, or to design materials with specific properties, is of unquestionable interest for the whole society, in terms of both technological progress, health and economic and energetic saving.
In some aspects, this is already a reality: computer simulations of many physical, chemical, biochemical and biomedical processes are successful, and of great help in understanding and guiding experiments. Despite all these advances, there are still basic unsolved problems that hamper a complete reliability of the results and conclusions of such calculations.
Density Functional Theory (DFT) is the standard approach to quantum chemistry in simulations with more than a dozen electrons. The classical way of breaking the curse of dimensionality in DFT is through the Kohn-Sham (KS) formalism, which has been extremely successful in predicting properties, for instance, in materials science and chemistry. Unfortunately, KS DFT relies on hand-crafted, highly problem-dependent (semi-empirical) approximations (LDA, B3LYP, PBE, etc), which can only be validated a posteriori through experiments or extremely costly calculations. In particular, KS DFT approximations fail in accurately predicting the physics of systems in which electronic correlation plays a prominent role, e.g. transition metals, which are the workhorse of catalysis.
In response to this, Gori-Giorgi, Friesecke, and others developed a modification of the KS setup, by considering also the the Coulomb interaction term and developing the so-called Strictly-Correlated Electron (SCE) formalism in Density Functional Theory. Such an approach has shown to be very promising specially to describe strong-correlation effects in atoms and molecules and in describing dissociation energies at long range.
The SCE approach has been developed mainly in physics and chemistry literature and still lacks a rigorous mathematical and computational grounds. The objective of the MSCA-IF "OTmeetsDFT" is to develop a mathematical formalism towards a rigorous SCE DFT theory, by developing rigorous analytical and computational algorithms of a new instance of optimal transport problem with finitely many marginals and Coulomb cost.
In some aspects, this is already a reality: computer simulations of many physical, chemical, biochemical and biomedical processes are successful, and of great help in understanding and guiding experiments. Despite all these advances, there are still basic unsolved problems that hamper a complete reliability of the results and conclusions of such calculations.
Density Functional Theory (DFT) is the standard approach to quantum chemistry in simulations with more than a dozen electrons. The classical way of breaking the curse of dimensionality in DFT is through the Kohn-Sham (KS) formalism, which has been extremely successful in predicting properties, for instance, in materials science and chemistry. Unfortunately, KS DFT relies on hand-crafted, highly problem-dependent (semi-empirical) approximations (LDA, B3LYP, PBE, etc), which can only be validated a posteriori through experiments or extremely costly calculations. In particular, KS DFT approximations fail in accurately predicting the physics of systems in which electronic correlation plays a prominent role, e.g. transition metals, which are the workhorse of catalysis.
In response to this, Gori-Giorgi, Friesecke, and others developed a modification of the KS setup, by considering also the the Coulomb interaction term and developing the so-called Strictly-Correlated Electron (SCE) formalism in Density Functional Theory. Such an approach has shown to be very promising specially to describe strong-correlation effects in atoms and molecules and in describing dissociation energies at long range.
The SCE approach has been developed mainly in physics and chemistry literature and still lacks a rigorous mathematical and computational grounds. The objective of the MSCA-IF "OTmeetsDFT" is to develop a mathematical formalism towards a rigorous SCE DFT theory, by developing rigorous analytical and computational algorithms of a new instance of optimal transport problem with finitely many marginals and Coulomb cost.
The first part of the project concentrate on mathematical aspects of the Strictly-Correlated Electrons (SCE) approach in Density Functional Theory. I have obtained
(i) A complete characterisation of solutions of Entropy-regularized Optimal Transport problems and regularity of Kantorovich potentials.
(ii) Obtained closed-form solutions for Entropic-regularized MOT problems with attractive/repulsive harmonic cost functions, when the marginals/one-body densities are multi-variate Gaussians.
(iii) Prove the (Gamma) convergence entropic-regularised MOT problems to the MOT / SCE functional when the regularisation parameter goes to zero (zero temperature/SCE limit) or plus infinity (fully bosonic uncorrelated problem).
From a chemical/physical viewpoint, we have discussed the implications of the Entropy-MOT formalism for Density Functional Theory and establishing a link with earlier works on quantum kinetic energy and classical entropy. I have carried out a very preliminary investigation (using simplified models for the harmonic oscillator) based on these closed-form solution to build approximations for the kinetic correlation functional at large coupling strengths. Moreover, I have analyzed the lower and upper bounds to the Hohenberg−Kohn functional by computing numerical realisations of the entropic regularized MOT.
The analysis in (i) and (iii) have been generalized for more general classes of Entropies (e.g. Tsallis, quadratic, etc). The second part focus on computational aspects as well as on the full quantum mechanical problem, i.e. without neglecting the Kinetic energy as in the SCE approach.
The results obtained has been disseminated: 1) through publications in peer-review journals with either gold or green open access (arXiv repository); 2) through talks at several relevant international conferences in mathematics, chemistry and machine learning (e.g. SIAM Material Sciences, ICML, IPAM, Simons Institute); 3) by organizing a workshop (Pisa and Eindhoven, 2022) and a school (Pisa, 2022). Moreover, I plan to make ideas of this project acessible for a wider audience, by adding multi-media material at the Signs of Mathematics website (https://jyu.fi/somath(odnośnik otworzy się w nowym oknie)) - an educational and outreach project I have developed in collaboration with researchers in Finland.
Finally, substantial effort has been dedicated to cross-fertilization and to making the results accessible for these communities. This started at pedagogical level: I am co-authoring a survey on OT and DFT to be published as a book chapter in the Springer series on Mathematics and Molecular Modeling; (iii) co-writing a survey for a wide computational chemist audience to be published in WIREs Computational Molecular Science.
(i) A complete characterisation of solutions of Entropy-regularized Optimal Transport problems and regularity of Kantorovich potentials.
(ii) Obtained closed-form solutions for Entropic-regularized MOT problems with attractive/repulsive harmonic cost functions, when the marginals/one-body densities are multi-variate Gaussians.
(iii) Prove the (Gamma) convergence entropic-regularised MOT problems to the MOT / SCE functional when the regularisation parameter goes to zero (zero temperature/SCE limit) or plus infinity (fully bosonic uncorrelated problem).
From a chemical/physical viewpoint, we have discussed the implications of the Entropy-MOT formalism for Density Functional Theory and establishing a link with earlier works on quantum kinetic energy and classical entropy. I have carried out a very preliminary investigation (using simplified models for the harmonic oscillator) based on these closed-form solution to build approximations for the kinetic correlation functional at large coupling strengths. Moreover, I have analyzed the lower and upper bounds to the Hohenberg−Kohn functional by computing numerical realisations of the entropic regularized MOT.
The analysis in (i) and (iii) have been generalized for more general classes of Entropies (e.g. Tsallis, quadratic, etc). The second part focus on computational aspects as well as on the full quantum mechanical problem, i.e. without neglecting the Kinetic energy as in the SCE approach.
The results obtained has been disseminated: 1) through publications in peer-review journals with either gold or green open access (arXiv repository); 2) through talks at several relevant international conferences in mathematics, chemistry and machine learning (e.g. SIAM Material Sciences, ICML, IPAM, Simons Institute); 3) by organizing a workshop (Pisa and Eindhoven, 2022) and a school (Pisa, 2022). Moreover, I plan to make ideas of this project acessible for a wider audience, by adding multi-media material at the Signs of Mathematics website (https://jyu.fi/somath(odnośnik otworzy się w nowym oknie)) - an educational and outreach project I have developed in collaboration with researchers in Finland.
Finally, substantial effort has been dedicated to cross-fertilization and to making the results accessible for these communities. This started at pedagogical level: I am co-authoring a survey on OT and DFT to be published as a book chapter in the Springer series on Mathematics and Molecular Modeling; (iii) co-writing a survey for a wide computational chemist audience to be published in WIREs Computational Molecular Science.
Being a fundamental research project, there is no guarantee of an immediate impact. However, the further development of the new understanding and surprising new connections that arose from this research could help other researchers to eliminate some guessing and empiricism. My methodology has potential to build a systematic scheme for DFT approximations as well as to give new computational and theoretical insights, for instance, on the study of enzymatic catalysis, dispersion forces and transition metals clusters. Thanks to the rigorous results established, deploying these functionals in software will allow nonexpert users to choose the right balance between computational cost and accuracy for their specific (bio)chemical and materials science applications.
Training and career perspective: despite the obstacles due to the restrictions cause by COVID-19, I have obtained an excellent training and being exposed to a wide variety of concepts, state-of-art techniques, methods and problems in the field of theoretical/computational chemistry. The guidance of Prof. Gori-Giorgi (host) was fundamental to archive the goals of the project as well as to open new research directions for my career. In fact, my training under the MSCA-IF framework has been pivotal to allow me to get a very interdisciplinary permanent position in Canada. I am currently an assistant professor at both, Department of Mathematics and Statistics, and Department of Chemistry and Biomedical Sciences, at the University of Ottawa. I have submitted an application for a Canadian Research Chair at Artificial Intelligence at the interface of Mathematics and Chemistry.
Training and career perspective: despite the obstacles due to the restrictions cause by COVID-19, I have obtained an excellent training and being exposed to a wide variety of concepts, state-of-art techniques, methods and problems in the field of theoretical/computational chemistry. The guidance of Prof. Gori-Giorgi (host) was fundamental to archive the goals of the project as well as to open new research directions for my career. In fact, my training under the MSCA-IF framework has been pivotal to allow me to get a very interdisciplinary permanent position in Canada. I am currently an assistant professor at both, Department of Mathematics and Statistics, and Department of Chemistry and Biomedical Sciences, at the University of Ottawa. I have submitted an application for a Canadian Research Chair at Artificial Intelligence at the interface of Mathematics and Chemistry.