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

arXiv: https://arxiv.org/abs/2508.03839 · HTML: https://mlsanity.com/paper/2508.03839/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 20 references that are in the map.
- First posted on arXiv 5 Aug 2025.

## Builds on (most cited first, 11 of 11)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Watch your Up-Convolution: CNN Based Generative Deep Neural Networks are Failing to Reproduce Spectral Distributions](https://mlsanity.com/paper/2003.01826/) (2020)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Focal Frequency Loss for Image Reconstruction and Synthesis](https://mlsanity.com/paper/2012.12821/) (2020)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Regularized linear autoencoders recover the principal components, eventually](https://mlsanity.com/paper/2007.06731/) (2020)
- [Inversion using a new low-dimensional representation of complex binary geological media based on a deep neural network](https://mlsanity.com/paper/1710.09196/) (2017)
- [Randomized Physics-Informed Neural Networks for Bayesian Data Assimilation](https://mlsanity.com/paper/2407.04617/) (2024)

## Built on by (most cited first, 1 of 1)

- [Latent Diffusion Posterior Sampling with Surrogate Likelihood Guidance for PDE Inverse Problems](https://mlsanity.com/paper/2606.26592/) (2026)

## Abstract (arXiv)

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(bmx) to a lower-dimensional latent space, bmμbmφy. Then, a fully connected neural network is used to map bmμbmφy to the latent space, bmμbmφh, of the PDE solution h(bmx,t). Finally, a decoder is utilized to reconstruct h(bmx,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(bmx) and the decoder as part of the h(bmx,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.