In modern Astrophysics one of the areas of interest is to study the formation and the evolution of interstellar molecular clouds, the sites of current-day star and planet formation. This is crucial to gain better insights on how the forces of nature shape the celestial bodies the way we observe it today. Gravity is the essential force that drives the evolution of interstellar molecular clouds. Scientists model such system by solving a set a gravito-hydrodynamic (GHD) equations which describes the dynamics as well the conservation laws which governs the physical system.
Usually, such systems do not have analytic solutions. Traditionally, numerical methods like finite difference (FD), finite element (FE), and finite volume (FV) are used to solve GHD equations. Most of these methods are good at solving such set of PDEs but are often limited as they require substantial resources or some restrictive assumptions.
We introduce Gravity informed neural network (GRINN), a PINN based to code, to solve self-gravitating hydrodynamic system to model evolution of interstellar molecular clouds. We develop GRINN to solve the following set of coupled time-dependent partial differential equations (PDEs) in three dimensions (3D). The governing GHD equations are:
What are PINNs?
Physics Informed Neural Network (PINN) introduces a novel mesh-free approach in applying neural networks to solve both ordinary and partial differential equations (PDEs). PINNs are trained to approximate a solution to a PDE while ensuring that the learned solution adheres to the governing physics (hence the name “Physics Informed”). It simulate a dynamical system by incorporating the physics equations in the loss function. PINN leverages the universal approximation capabilities of neural networks to find a global function that is the solution of the PDE
Case Study 1: Growing gravitational Instability:
As a first experiment we consider an initial sinusoidal perturbation in density and velocity to an ambient uniform background of molecular gas. The system is defined using the above equations. Due to gravitational instability the initial perturbation becomes unstable and density starts growing forming the initial seed for star formation. The animation depicts the growth of the initial perturbation in density with time solved using GRINN
Case Study 2: Pressure Waves:
If the initial perturbation is less than the instability length scale (Jean's Length) it propagates in the medium as density waves. The animation depicts the evolution of density pressure waves with time solved using GRINN.
PINNs vs Finite Difference
In 3D, GRINN's computation time is an order of magnitude (~10x) less than the grid code with similar accuracy.
With the advent of deep learning and in particular Physics-Informed Deep Learning we are entering into a new era of Computational Astrophysics, where algorithms like PINNs and its variant will revolutionize our understanding of the cosmos.
The article is published in the Journal of : Machine Learning Science and Technology
"GRINN: a physics-informed neural network for solving hydrodynamic systems in the presence of self-gravity"-- Sayantan Auddy, Ramit Dey, Neal J Turner and Shantanu Basu.
Github link: https://github.com/sauddy/GRINN