VAE-DNN: Energy-Efficient Trainable-by-Parts Surrogate Model For Parametric Partial Differential Equations

TL;DR

VAE-DNN is a novel energy-efficient surrogate model for solving parametric PDEs that trains its components independently, leading to faster training and higher accuracy than existing models like FNO and DeepONet.

Contribution

The paper introduces VAE-DNN, a surrogate model with separable training of its encoder, neural network, and decoder, reducing training time and energy while improving accuracy.

Findings

VAE-DNN outperforms FNO and DeepONet in accuracy for PDE solutions.

VAE-DNN requires less training time and energy.

The model effectively solves forward and inverse PDE problems.

Abstract

We propose a trainable-by-parts surrogate model for solving forward and inverse parameterized nonlinear partial differential equations. Like several other surrogate and operator learning models, the proposed approach employs an encoder to reduce the high-dimensional input $y(\bm{x})$ to a lower-dimensional latent space, $\bm\mu_{\bm\phi_y}$. Then, a fully connected neural network is used to map $\bm\mu_{\bm\phi_y}$ to the latent space, $\bm\mu_{\bm\phi_h}$, of the PDE solution $h(\bm{x},t)$. Finally, a decoder is utilized to reconstruct $h(\bm{x},t)$. The innovative aspect of our model is its ability to train its three components independently. This approach leads to a substantial decrease in both the time and energy required for training when compared to leading operator learning models such as FNO and DeepONet. The separable training is achieved by training the encoder as part of the variational autoencoder (VAE) for $y(\bm{x})$ and the decoder as part of the $h(\bm{x},t)$ VAE. We refer to this model as the VAE-DNN model. VAE-DNN is compared to the FNO and DeepONet models for obtaining forward and inverse solutions to the nonlinear diffusion equation governing groundwater flow in an unconfined aquifer. Our findings indicate that VAE-DNN not only demonstrates greater efficiency but also delivers superior accuracy in both forward and inverse solutions compared to the FNO and DeepONet models.