PINN Theory

Physics-Informed Neural Networks for scientific computing in Earth observation.

Overview

PINNs incorporate physical laws as soft constraints via the loss function.

Reference: Raissi, M., et al. (2019). Physics-informed neural networks. Journal of Computational Physics, 378, 686-707. DOI: 10.1016/j.jcp.2018.10.045

Formulation

General PDE

\mathcal{N}[u](x, t) = 0, \quad x \in \Omega

Composite Loss

\mathcal{L}(\theta) = \lambda_d \mathcal{L}_{data} + \lambda_p \mathcal{L}_{pde} + \lambda_b \mathcal{L}_{bc}

where:

\mathcal{L}_{data} = \frac{1}{N_d}\sum_{i=1}^{N_d}\|u_\theta(x_i) - u_i^{obs}\|^2

\mathcal{L}_{pde} = \frac{1}{N_c}\sum_{j=1}^{N_c}\|\mathcal{N}[u_\theta](x_j)\|^2

Common PDEs

Heat/Diffusion Equation

\frac{\partial u}{\partial t} = D \nabla^2 u

Advection-Diffusion

\frac{\partial u}{\partial t} + \mathbf{v} \cdot \nabla u = D \nabla^2 u

Reference: Okubo, A. (1971). Oceanic diffusion diagrams. Deep Sea Research, 18(8), 789-802. DOI: 10.1016/0011-7471(71)90046-5

Network Architectures

Fourier Features

\gamma(x) = [\cos(2\pi B x), \sin(2\pi B x)]^T

Reference: Tancik, M., et al. (2020). Fourier Features Let Networks Learn High Frequency Functions. NeurIPS. arXiv:2006.10739

Collocation Strategies

Reference: Lu, L., et al. (2021). DeepXDE: A deep learning library for solving differential equations. SIAM Review, 63(1), 208-228. DOI: 10.1137/19M1274067

EO Applications

Application	PDE	Observable
SST Interpolation	Advection-Diffusion	Temperature
Pollution Dispersion	Transport	Concentration
Groundwater Flow	Darcy	Hydraulic head