Application of Physics-Informed Neural Networks for Galaxy Dynamics

Abstract: Developing efficient and accurate numerical methods to simulate dynamics of physical systems has been an everlasting challenge in computational physics. Physics-Informed Neural Networks (PINNs) are neural networks that encode laws of physics into their structure. Utilizing auto-differentiation, they can efficiently solve partial differential equations (PDEs) by minimizing the loss function at certain points within the domain of interest. The remarkable efficiency exhibited by these networks when solving PDEs positions them as ideal solvers for simulating complex systems.  In this pioneering work, we take a first step towards simulating galaxy dynamics using PINNs by solving the gravitational Poisson equation. We initially substantiate the capacity of PINNs to solve the gravitational Poisson equation for the simple Hernquist (Hernquist, 1990) radial density profile, and for the parametric Dehnen (Dehnen, 1993) radial density profile. Following this, we extended our study to encompass a more complex axisymmetric density profile describing a Thick Exponentiel Disk.  The capacity of PINNs to generate comparatively accurate results has been validated with an average error of 1.71% and 3.75% respectively for the spherically symmetric Hernquist and Dehnen models. While for the axisymmetric thick exponentiel disk model the PINN demonstrated an average relative error of 0.36% with a maximum error of just 0.99% after fine-tuning the PINN’s hyperparameters. Although this model typically relies on the two coordinates R and z along with the ratio η of the model’s scale lengths, the PINN is here trained using a fixed, predetermined value of η.  Drawing upon the outcomes of the grid search implemented for the thick exponen- tial disk model, we provide a succinct examination of how the hyperparameters of the PINN impact the relative error. Given the limited quantity of datapoints, we refrain from formulating definitive conclusions, yet we do exhibit certain discernible patterns. Specifically, we demonstrate that the hyperbolic tangent (tanh) activation function con- sistently outperforms other activation functions in the context of our model. Addition- ally, it appears that augmenting the depth of the network offers superior error reduction in comparison to increasing its width, reinforcing the importance of architectural con- siderations in the optimization of Physics-Informed Neural Networks  Our results show clear advantages of PINNs over regular solvers in terms of efficiency. Despite the success of the two-parameter PINN for the thick exponential disk, further work is required to confirm its extension to three dimensions. This pioneering research offers a promising foundation for further developments in the field, and demonstrates the genuine practical utility of PINNs for simulating complex systems such as galaxies.