LatentPINNs: Generative physics-informed neural networks via a latent representation learning

TL;DR

LatentPINNs introduce a generative framework using latent representations of PDE parameters to improve the efficiency and flexibility of physics-informed neural networks in solving PDEs across parameter distributions.

Contribution

The paper presents a novel latentPINN framework that incorporates learned latent representations of PDE parameters into PINNs, enabling rapid adaptation to new parameter sets without retraining.

Findings

Effective on nonlinear Eikonal equations with different velocity models

Performs well on unseen PDE parameter sets without additional training

Utilizes latent diffusion models for parameter representation learning

Abstract

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.