Periodic Reporting for period 4 - COMANFLO (Computation and analysis of statistical solutions of fluid flow)
Période du rapport: 2023-02-01 au 2023-07-31
The project COMANFLO focuses on the analysis and computation of statistical solutions for the nonlinear partial differential equations (PDEs) which model the flow of fluids. The study of fluid flow is crucial importance in several areas of
science and engineering. Currently, there is only partial understanding of the complex dynamics of fluids. In particular, notions of existence, uniqueness, stability and computability of solutions of the governing equations of fluids such as
hyperbolic systems of conservation laws and the incompressible Euler and Navier-Stokes equations. This project investigates a recently proposed solution concept, the so-called statistical solutions, which are time-parameterized probability
measures on infinite-dimensional function spaces, as an appropriate framework for describing complex fluid motion. We study whether these solutions exist, are unique and can be computable. We also seek to design efficient algorithms to
compute these solutions. At the mid-term stage of the project, several of these objectives have already been met and there is a very high probability for a successful completion of most of the aims of the project. If successful, the mathematical
analysis in the project will further validate the use of the nonlinear PDEs governing fluid dynamics in several contexts and the algorithms developed in the project can be used for efficient computation of fluid flows.
science and engineering. Currently, there is only partial understanding of the complex dynamics of fluids. In particular, notions of existence, uniqueness, stability and computability of solutions of the governing equations of fluids such as
hyperbolic systems of conservation laws and the incompressible Euler and Navier-Stokes equations. This project investigates a recently proposed solution concept, the so-called statistical solutions, which are time-parameterized probability
measures on infinite-dimensional function spaces, as an appropriate framework for describing complex fluid motion. We study whether these solutions exist, are unique and can be computable. We also seek to design efficient algorithms to
compute these solutions. At the mid-term stage of the project, several of these objectives have already been met and there is a very high probability for a successful completion of most of the aims of the project. If successful, the mathematical
analysis in the project will further validate the use of the nonlinear PDEs governing fluid dynamics in several contexts and the algorithms developed in the project can be used for efficient computation of fluid flows.
At the end of the project, we have successfully completed most of the goals in project. In particular, we have designed Monte Carlo based algorithms (together with
Finite Volumes and spectral viscosity methods) for the computation of statistical solutions of systems of conservation laws and incompressible Euler equations. These algorithms have been proved to converge, under certain
natural and verifiable hypotheses, to a statistical solution. Moreover, a large number of numerical experiments have been performed to illustrate both the algorithms as well as different qualitative aspects of statistical
solutions in both two- and three-space dimensions. Similarly, partial results on the existence, uniqueness of statistical solutions and on their connection with turbulence models have been obtained. Finally, significant progress has been made in the design of novel algorithms that accelerate the base Monte Carlo algorithm for computing statistical solutions. In particular, the use of novel machine learning techniques has made great impact and gone way beyond the original remit of the project. The results of the project has been published as research articles in leading journals in the field as well conference proceedings of the top machine learning conferences. A large amount of open source code and data has also been generated as a part of the project and has been released to the research community for further use.
Finite Volumes and spectral viscosity methods) for the computation of statistical solutions of systems of conservation laws and incompressible Euler equations. These algorithms have been proved to converge, under certain
natural and verifiable hypotheses, to a statistical solution. Moreover, a large number of numerical experiments have been performed to illustrate both the algorithms as well as different qualitative aspects of statistical
solutions in both two- and three-space dimensions. Similarly, partial results on the existence, uniqueness of statistical solutions and on their connection with turbulence models have been obtained. Finally, significant progress has been made in the design of novel algorithms that accelerate the base Monte Carlo algorithm for computing statistical solutions. In particular, the use of novel machine learning techniques has made great impact and gone way beyond the original remit of the project. The results of the project has been published as research articles in leading journals in the field as well conference proceedings of the top machine learning conferences. A large amount of open source code and data has also been generated as a part of the project and has been released to the research community for further use.
The results achieved within the purview of the project go significantly beyond the state of the art and push the frontiers of our knowledge about
nonlinear PDEs governing fluid flows and their computation. In particular, we were able to obtain the first convergence results for numerical algorithms to the statistical
solutions of multi-dimensional hyperbolic systems of conservation laws and incompressible Euler equations. Moreover, several novel algorithms for computing these statistical
solutions efficiently, based on the use of deep neural networks, have been developed and tested. Moreover, significant progress has been made on solving the corresponding
Bayesian inverse problems.
nonlinear PDEs governing fluid flows and their computation. In particular, we were able to obtain the first convergence results for numerical algorithms to the statistical
solutions of multi-dimensional hyperbolic systems of conservation laws and incompressible Euler equations. Moreover, several novel algorithms for computing these statistical
solutions efficiently, based on the use of deep neural networks, have been developed and tested. Moreover, significant progress has been made on solving the corresponding
Bayesian inverse problems.