# OCTANE -- Optimal Control for Tensor-based Autoencoder Network Emergence: Explicit Case

arXiv: https://arxiv.org/abs/2509.08169 · HTML: https://mlsanity.com/paper/2509.08169/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 21 references that are in the map.
- First posted on arXiv 9 Sep 2025.

## Builds on (most cited first, 9 of 9)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [Layer-Parallel Training of Deep Residual Neural Networks](https://mlsanity.com/paper/1812.04352/) (2018)
- [Fractional Deep Neural Network via Constrained Optimization](https://mlsanity.com/paper/2004.00719/) (2020)
- [Novel Deep neural networks for solving Bayesian statistical inverse](https://mlsanity.com/paper/2102.03974/) (2021)
- [Tensor Decompositions in Deep Learning](https://mlsanity.com/paper/2002.11835/) (2020)
- [Bilevel Optimization, Deep Learning and Fractional Laplacian Regularization with Applications in Tomography](https://mlsanity.com/paper/1907.09605/) (2019)
- [Neural Architecture Search via Bregman Iterations](https://mlsanity.com/paper/2106.02479/) (2021)
- [Optimal Control, Numerics, and Applications of Fractional PDEs](https://mlsanity.com/paper/2106.13289/) (2021)

## Abstract (arXiv)

This paper presents a novel, mathematically rigorous framework for autoencoder-type deep neural networks that combines optimal control theory and low-rank tensor methods to yield memory-efficient training and automated architecture discovery. The learning task is formulated as an optimization problem constrained by differential equations representing the encoder and decoder components of the network and the corresponding optimality conditions are derived via a Lagrangian approach. Efficient memory compression is enabled by approximating differential equation solutions on low-rank tensor manifolds using an adaptive explicit integration scheme. These concepts are combined to form OCTANE (Optimal Control for Tensor-based Autoencoder Network Emergence) -- a unified training framework that yields compact autoencoder architectures, reduces memory usage, and enables effective learning, even with limited training data. The framework's utility is illustrated with application to image denoising and deblurring tasks and recommendations regarding governing hyperparameters are provided.