deepFOSLS

This code contains numerical experiments for solving elliptic transmission partial differential equations with neural-networks, using a novel Robust First Order System Least Squares (FOSLS) formulation implemented with JAX and Keras.

For a Poisson-type equation with spatially dependent diffusion coefficient,

$$ -\mathrm{div}(\kappa \nabla u) = f \quad \text{in } \Omega, $$

with homogeneous Dirichlet boundary conditions, the second-order equation is rewritten as the first-order system

$$ \mathrm{div} q = f, \qquad q = -\kappa \nabla u. $$

The neural network is trained to approximate the pair $(u,q)$ by minimizing least-squares residuals associated with this first-order system. The code also compares this FOSLS approach with Deep Ritz training in one-dimensional tests.

The repository is organized into one-dimensional and two-dimensional experiments:

• 1D: scripts for FOSLS, Deep Ritz, robustness, interface, and quasi-Gibbs experiments.
• 1D/src: source classes for neural-network models, losses, integration rules and several utilities scripts.
• 2D: scripts for continuous and discontinuous source experiments.
• 2D/src: source classes for the two-dimensional models, losses, integration rules and several utilities scripts.

The scripts are written for experimentation in Spyder, with hyperparameters kept near the top of the files so different numerical setups can be changed directly.