Physics-Informed Neural Network for Modeling the Gravitational Potential of an Irregular Body

Overview

This project implements a Physics-Informed Neural Network (PINN) to model the gravitational potential field $ϕ(x, y)$ generated by an irregular mass distribution (e.g., asteroid).

Unlike traditional numerical solvers that rely on discretized grids or meshes, the PINN learns the potential function continuously by embedding Poisson’s equation directly into its loss function, blending physics laws and neural networks for accurate and generalizable predictions.

Why a New Gravity Model?

Newtonian gravity, while elegant, breaks down at certain levels of precision — for instance, in Mercury’s orbital precession — and conflicts with relativity’s light-speed limit.

Although this project remains within the realm of Newtonian mechanics, it highlights how AI can model gravitational systems more flexibly and serve as a bridge toward relativistic PINNs in future work.

Mathematical Formulation

We model the potential ϕ(x, y) using Poisson’s equation

$$∇2ϕ(x,y)=4πGρ(x,y)$$

Where:

• $𝜙(𝑥,𝑦)$ = gravitational potential
• $𝜌(𝑥,𝑦)$ = mass density
• $𝐺$ = gravitational constant

The PINN enforces this physical law through its loss function:

$$L=Lphysics​+Ldata​$$

with:

$$Lphysics​=∣∣∇2ϕ−4πGρ∣∣²$$ , $$Ldata​=∣∣ϕpred​−ϕtrue​∣∣² $$

Tech Stack:

• Python 3.10+

• PyTorch – Deep learning framework

• DeepXDE – Physics-Informed Neural Networks toolkit

• NumPy – Numerical computations

• Matplotlib – Visualization

Evaluation Metrics (PINN vs FD)

Metric	Value
RMSE	4.401050
MAE	3.534299
MAPE	1.13e11
R²	-1.1096
Mean Prediction	-0.1768
Std	5.7719
Min	-12.9625
Max	14.1818

How to run

# Clone this repository
git clone https://github.com/<your-username>/Physics-Informed-Neural-Network-Gravitational-Potential.git
cd Physics-Informed-Neural-Network-Gravitational-Potential

# Launch the notebook
jupyter notebook PINNs_notebook.ipynb

Results

• The PINN successfully learned the gravitational potential of an irregular body, producing a smooth and physically consistent potential field.

• Residuals of Poisson’s equation remained low across the domain, showing strong physical adherence.

• The model generalizes to new density configurations unseen during training.