Neural networks that incorporate physical laws directly into the loss function, enabling data-efficient learning of gravity field solutions while respecting fundamental equations of satellite dynamics.
\[\mathcal{L}_{total} = \mathcal{L}_{data} + \lambda_{physics}\mathcal{L}_{physics} + \lambda_{BC}\mathcal{L}_{BC}\]
Where physics loss enforces Laplace's equation: \(\nabla^2 V = 0\) in free space
\[V(r,\theta,\phi) = \frac{GM}{r}\sum_{n=0}^{N}\sum_{m=0}^{n}\left(\frac{R}{r}\right)^n P_{nm}(\cos\theta)[C_{nm}\cos(m\phi) + S_{nm}\sin(m\phi)]\]
• Physics constraints reduce required training data by 60-80%
• Automatic differentiation ensures exact derivative computation
• Handles irregular data distributions naturally
• Implicit regularization from physics loss
• Established error propagation theory
• Faster for simple linear problems
• Well-understood convergence properties
• Mature software implementations
• Use traditional methods for initial solution
• Apply PINN for gap-filling and refinement
• Combine uncertainties using ensemble methods
• Best of both worlds for operational use