Objectives

The project is interdisciplinary and targets important goals related to the machine learning of physically informed neural networks, applied not only to solving complex differential equations arising in the field of gravitational-wave astrophysics, but also to tasks that are described with little data, or experimental observations with noise.

Gravitational wave modeling must be extremely precise, as it is possible to miss events and physical effects when comparing with the resulting observational data. Wave models must be estimated relatively quickly and efficiently because searches and parameter estimation require tens to hundreds of millions of gravitational wave shape data.

The main difficulties are in two main directions:

1. Huge databases of numerical and observational data;
2. Solving highly complex nonlinear systems of hyperbolic equations on warped spatio-temporal manifolds.

Motivated by what was written above, we propose the following scientific goals, formulated in four tasks:

Task 1. Modeling the gravitational-wave ringing signal of black holes in general relativity and modified gravity by physically informed neural networks..

A natural extension of the first task is to study compact objects without a horizon, such as neutron stars, boson-fermion stars, topological solitons, etc. Therefore, the second task that will be worked on in this project is:

Task 2. Modeling the gravitational-wave ringing signal of compact horizonless objects (neutron stars, boson-fermion stars, topological solitons) by physically informed neural networks.

The next two tasks are related to the development of PINN architectures for solving differential equations with unknown parameters that are determined from experimental data, i.e. inverse problems.

Task 3. Application of physically informed neural networks to solve differential equations with unknown parameters that can be determined from observational data or experimental observations with noise.

Task 4. Development of machine learning algorithms using PINN as universal function approximators for solving complex hyperbolic equations on distorted space-time manifolds.

Additionally, the following learning objectives are set:

• To present to all participants in the project the interdisciplinarity of the researched topic, requiring specialists in applied mathematics, informatics, mathematical and theoretical physics to overcome difficulties in modeling, calculations and simulations.

The proposed scientific tasks coincide with the main scientific interests of the groups. It is necessary to emphasize that the proposed objectives are extremely up-to-date, challenging and contain the potential for future in-depth research. After the completion of the work on the project, the researchers will have the capacity and opportunities to expand its work in the field of mathematical sciences and informatics.