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.

本研究提出了一种名为 latentPINN 的框架，通过将偏微分方程（PDE）参数的潜在表示作为额外的输入进行 PINN 模型的训练，使用两个阶段的训练来学习 PDE 参数的潜在表示，并通过在解决区域内随机生成空间坐标和 PDE 参数值的样本进行物理感知神经网络的训练，试验结果表明该方法在不需要任何额外训练的情况下可以很好地适用于不同的 PDE 参数。