Einstein's Field Equations as Continuous-Time Recurrent Neural Networks

Einstein field equations are notoriously challenging to solve due to their complex mathematical form, with few analytical solutions available in the absence of highly symmetric systems or ideal matter distribution. However, accurate solutions are crucial, particularly in systems with strong gravitational field such as black holes or neutron stars. In this work, we use neural networks and auto differentiation to solve the Einstein field equations numerically inspired by the id...

On the long-established classification problems in general relativity we take a novel perspective by adopting fruitful techniques from machine learning and modern data-science. In particular, we model Petrov's classification of spacetimes, and show that a feed-forward neural network can achieve high degree of success. We also show how data visualization techniques with dimensionality reduction can help analyze the underlying patterns in the structure of the different types of...

This work introduces advanced computational techniques for modeling the time evolution of compact binary systems using machine learning. The dynamics of compact binary systems, such as black holes and neutron stars, present significant nonlinear challenges due to the strong gravitational interactions and the requirement for precise numerical simulations. Traditional methods, like the post-Newtonian approximation, often require significant computational resources and face chal...

As a network-based functional approximator, we have proposed a "Lagrangian Density Space-Time Deep Neural Networks" (LDDNN) topology. It is qualified for unsupervised training and learning to predict the dynamics of underlying physical science governed phenomena. The prototypical network respects the fundamental conservation laws of nature through the succinctly described Lagrangian and Hamiltonian density of the system by a given data-set of generalized nonlinear partial dif...

Deep learning has been widely and actively used in various research areas. Recently, in the gauge/gravity duality, a new deep learning technique so-called the AdS/Deep-Learning (DL) has been proposed [1, 2]. The goal of this paper is to describe the essence of the AdS/DL in the simplest possible setups, for those who want to apply it to the subject of emergent spacetime as a neural network. For prototypical examples, we choose simple classical mechanics problems. This method ...

The parametrized post-Einsteinian (ppE) framework and its variants are widely used to probe gravity through gravitational-wave tests that apply to a large class of theories beyond general relativity. However, the ppE framework is not truly theory-agnostic as it only captures certain types of deviations from general relativity: those that admit a post-Newtonian series representation in the inspiral of coalescencing compact objects. Moreover, each type of deviation in the ppE f...

A new semi-supervised machine learning package is introduced which successfully solves the Euclidean vacuum Einstein equations with a cosmological constant, without any symmetry assumptions. The model architecture contains subnetworks for each patch in the manifold-defining atlas. Each subnetwork predicts the components of a metric in that patch, with the associated Einstein conditions, of the form $R_{\mu \nu} - \lambda g_{\mu \nu} = 0$, being used as independent loss compon...

This paper gives a self-contained, elementary, and largely pictorial statement of Einstein's equation.

Holography relates gravitational theories in five dimensions to four-dimensional quantum field theories in flat space. Under this map, the equation of state of the field theory is encoded in the black hole solutions of the gravitational theory. Solving the five-dimensional Einstein's equations to determine the equation of state is an algorithmic, direct problem. Determining the gravitational theory that gives rise to a prescribed equation of state is a much more challenging, ...

Neural network is a dynamical system described by two different types of degrees of freedom: fast-changing non-trainable variables (e.g. state of neurons) and slow-changing trainable variables (e.g. weights and biases). We show that the non-equilibrium dynamics of trainable variables can be described by the Madelung equations, if the number of neurons is fixed, and by the Schrodinger equation, if the learning system is capable of adjusting its own parameters such as the numbe...