High-dimensional mathematical methods for LargE Agent and Particle systems

Projektinformationen

HighLEAP

ID Finanzhilfevereinbarung: 101077204

DOI

10.3030/101077204

EK-Unterschriftsdatum 28 Juli 2023

Startdatum 1 Dezember 2023

Enddatum 30 November 2028

Finanziert unter

• European Research Council (ERC)

Gesamtkosten

€ 1 379 858,00

EU-Beitrag

€ 1 379 858,00

Koordiniert durch

ECOLE NATIONALE DES PONTS ET CHAUSSEES
France

Periodic Reporting for period 1 - HighLEAP (High-dimensional mathematical methods for LargE Agent and Particle systems)

Berichtszeitraum: 2023-12-01 bis 2026-05-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

Interacting particle or agent-based systems are ubiquitous in science. They arise in an extremely wide variety of applications including materials science, biology, economics and social sciences. Several mathematical models exist to account for the evolution of such systems at different scales, among which stand optimal transport problems, Fokker-Planck equations, mean-field games systems or stochastic differential equations. However, all of them suffer from severe limitations when it comes to the simulation of high-dimensional problems, the high-dimensionality character coming either from the large number of particles or agents in the system, the high amount of features of each agent or particle, or the huge quantity of parameters entering the model. The objective of this project is to provide a new mathematical framework for the development and analysis of efficient and accurate numerical methods for the simulation of high-dimensional particle or agent systems, stemming from applications in materials science and stochastic game theory.

The main challenges which will be addressed in this project are:
-sparse optimization problems for multi-marginal optimal transport problems, using moment constraints;
-numerical resolution of high-dimensional partial differential equations, with randomized iterative algorithms;
-efficient approximation of parametric stochastic differential equations, by means of reduced-order modeling approaches.

The potential impacts of the project are huge: making possible such extreme-scale simulations will enable to gain precious insights on the predictive power of agent- or particle-based models, with applications in various fields, such as quantum chemistry, molecular dynamics, crowd motion or urban traffic.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

1) A new approximation of the Lieb functional, termed the Moment Constrained Approximation of the Lieb (MCAL) functional has been introduced. The Lieb functional is a central object in Density Functional Theory for electronic structure calculations, representing the convexification of the Lévy-Lieb functional. It is defined as a multimarginal quantum optimal transport problem, i.e. a minimization problem over a set of density operators (quantum states) that satisfy a specific marginal constraint (matching a given electronic density). Beyond simple convergence, we establish convergence rates for the associated approximation of the ground state energy, providing a theoretical bound on the error introduced by using a finite number of moments.

2) Together with Mathias Dus, the PI performed a mathematical analysis of infinite-width neural network approach for the resolution of Schrödinger eigenvalue problems (see below).

3) We achieved to analyze mathematically variance reduction techniques in the context of reduced basis methods. We provide a rigorous study of how Monte-Carlo sampling affects the theoretical convergence properties of greedy algorithms used for computing parameter-dependent expectations. Using concentration inequalities for empirical measures in Wasserstein distance, the paper establishes sufficient conditions for the number of Monte-Carlo samples required at each iteration of the greedy procedure. These conditions ensure that the algorithm maintains its convergence properties despite the noise introduced by empirical variance estimates. Acknowledging that the purely theoretical sampling requirements may not be practical for all applications, we propose a heuristic procedure to adaptively choose the number of Monte-Carlo samples at each iteration. This approach is designed to balance computational efficiency with the accuracy needed for the greedy selection process.

4) A novel approach to solving complex physics problems, in particular electronic structure theory, using optimal transport theory. We summarize here the main theoretical contributions related to the study of the decay of Kolmogorov widths. Indeed, we investigate why traditional linear "Reduced Basis" (RB) methods often fail for certain quantum chemistry problems. We compare the decay rate of the Kolmogorov $n$-width (a measure of how well a set of solutions can be approximated by a linear space) in the standard $L^2$-norm versus distances based on optimal transport (Wasserstein distance). We prove that for parametric eigenvalue problems where the solution (wavefunction) shifts or changes with nucleus positions, the approximation width decays significantly faster in the Wasserstein space than in the linear $L^2$ space. This theoretical finding justifies the use of nonlinear methods over traditional linear ones. Actually, we propose a new nonlinear reduced-order modeling framework that approximates solutions using mixture Wasserstein barycenters, which proves to be computationally extremely efficient.

5)We have established the theoretical foundation for treating the time-dependent Schrödinger equation as a global space-time quadratic optimization problem rather than a standard initial-value problem. The analysis is shown to be valid for the electronic many-body case, accounting for an arbitrary number of electrons and handling the complex Coulomb singularities present in interaction potentials. A significant contribution is the introduction of a variational principle for dynamical low-complexity approximations that serves as an alternative to the classical Dirac-Frenkel principle, offering better theoretical guarantees for the existence of solutions, in particular with low-rank tensors and linear combinations of Gaussian wave packets. (see below)

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

We would like to highlight two theoretical breakthroughs obtained during the project:

1) The analysis of the convergence of inifinite-width neural networks for the Schrödinger eigenvalue problem:
Unlike the Poisson equation (which leads to a convex functional), eigenvalue problems involve the minimization of an energy functional subject to a unit norm constraint, making it a non-convex problem. This paper is the first to provide a rigorous analysis of gradient curve dynamics in the 2-Wasserstein space for such non-convex functionals. The work bridges the gap between deep learning theory and computational physics, offering a rigorous mathematical foundation for "neural network ansätze" (like the Fermionic Neural Network or PauliNet) used to solve many-body Schrödinger equations in quantum chemistry.

2) Global space-time formulation for low-complexity approximation of time-dependent Schrödinger equations:
This global space-time formulation enables to define low-complexity approximations of the time-dependent wavefunction of a system of electrons globally in time, whereas traditional Dirac-Frenkel variational formulations only yields approximations that are valid locally in time. In addition, these approximations can be proved to be stable over long time simulations, in particular when dealing with Polynomial Gaussian wavepackets. Lastly, this formulation holds for Schrödinger problems defined on the whole space R^3, with Coulombic interaction potentials and an arbitrary number of electrons. This is a major breakthrough which was not exepcted in the context of the project.

We then would like to highlight two outputs of the project to assess the success of the methodology developped in the project when it comes to computational performance:

1) First prize of the 2024 NeurIPS contest ML4CFD:
A PhD student won the first prize of the 2024 NeurIPS contest ML4CFD thanks to the methodology and algorithms that were developed in [2] and [14]. (see https://arxiv.org/abs/2407.01641) They ranked first among a poll of several hundreds of teams worlwide.

2) Computational speedups for neutronics applications:
The novel methodology developped in the project enabled to achieve computational speed-ups by a factor of several hundreds compared to the algorithms which were used up to now by CEA in criticity calculations for neutronics applications. This is of vital importance because criticity calculations enable to predict whether a nuclear reactor will remain stable during the observed time period.