LatentPINNs: Generative physics-informed neural networks via a latent
  representation learning
Mohammad H. Taufik Tariq Alkhalifah
Physics-informed neural networks (PINNs) are promising to replace
conventional partial differential equation (PDE) solvers by offering more
accurate and flexible PDE solutions. However, they are hampered by the
relatively slow convergence and the need to perform additional, potentially
expensive, training for different PDE parameters. To solve this limitation, we
introduce latentPINN, a framework that utilizes latent representations of the
PDE parameters as additional (to the coordinates) inputs into PINNs and allows
for training over the distribution of these parameters. Motivated by the recent
progress on generative models, we promote the use of latent diffusion models to
learn compressed latent representations of the PDE parameters distribution and
act as input parameters to NN functional solutions. We use a two-stage training
scheme in which the first stage, we learn the latent representations for the
distribution of PDE parameters. In the second stage, we train a
physics-informed neural network over inputs given by randomly drawn samples
from the coordinate space within the solution domain and samples from the
learned latent representation of the PDE parameters. We test the approach on a
class of level set equations given by the nonlinear Eikonal equation. We
specifically share results corresponding to three different sets of Eikonal
parameters (velocity models). The proposed method performs well on new phase
velocity models without the need for any additional training.