# Explicit and data-Efficient Encoding via Gradient Flow

arXiv: https://arxiv.org/abs/2412.00864 · HTML: https://mlsanity.com/paper/2412.00864/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 15 references that are in the map.
- First posted on arXiv 1 Dec 2024.

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

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [VAE with a VampPrior](https://mlsanity.com/paper/1705.07120/) (2017)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Recent Advances in Autoencoder-Based Representation Learning](https://mlsanity.com/paper/1812.05069/) (2018)
- [Learning Latent Space Energy-Based Prior Model](https://mlsanity.com/paper/2006.08205/) (2020)
- [Semi-Amortized Variational Autoencoders](https://mlsanity.com/paper/1802.02550/) (2018)
- [Canonical normalizing flows for manifold learning](https://mlsanity.com/paper/2310.12743/) (2023)

## Built on by (most cited first, 3 of 3)

- [Localized FNO for Spatiotemporal Hemodynamic Upsampling in Aneurysm MRI](https://mlsanity.com/paper/2507.13789/) (2025)
- [Learning Gradient Flow: Using Equation Discovery to Accelerate Engineering Optimization](https://mlsanity.com/paper/2602.13513/) (2026)
- [Parameter-Efficient Adaptation of Pretrained Language Models for Time-Series Forecasting](https://mlsanity.com/paper/2609.15344/) (2026)

## Abstract (arXiv)

The autoencoder model typically uses an encoder to map data to a lower dimensional latent space and a decoder to reconstruct it. However, relying on an encoder for inversion can lead to suboptimal representations, particularly limiting in physical sciences where precision is key. We introduce a decoder-only method using gradient flow to directly encode data into the latent space, defined by ordinary differential equations (ODEs). This approach eliminates the need for approximate encoder inversion. We train the decoder via the adjoint method and show that costly integrals can be avoided with minimal accuracy loss. Additionally, we propose a 2nd order ODE variant, approximating Nesterov's accelerated gradient descent for faster convergence. To handle stiff ODEs, we use an adaptive solver that prioritizes loss minimization, improving robustness. Compared to traditional autoencoders, our method demonstrates explicit encoding and superior data efficiency, which is crucial for data-scarce scenarios in the physical sciences. Furthermore, this work paves the way for integrating machine learning into scientific workflows, where precise and efficient encoding is critical. footnoteThe code for this work is available at urlhttps://github.com/k-flouris/gfe.