# Optimal Control Theoretic Neural Optimizer: From Backpropagation to Dynamic Programming

arXiv: https://arxiv.org/abs/2510.14168 · HTML: https://mlsanity.com/paper/2510.14168/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 61 references that are in the map.
- First posted on arXiv 15 Oct 2025.

## Builds on (most cited first, 30 of 32)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
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- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Mask R-CNN](https://mlsanity.com/paper/1703.06870/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [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)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
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- [The UEA multivariate time series classification archive, 2018](https://mlsanity.com/paper/1811.00075/) (2018)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [You Only Propagate Once: Accelerating Adversarial Training via Maximal Principle](https://mlsanity.com/paper/1905.00877/) (2019)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Fast Approximate Natural Gradient Descent in a Kronecker-factored Eigenbasis](https://mlsanity.com/paper/1806.03884/) (2018)
- [Stochastic modified equations and adaptive stochastic gradient algorithms](https://mlsanity.com/paper/1511.06251/) (2015)
- [Practical Gauss-Newton Optimisation for Deep Learning](https://mlsanity.com/paper/1706.03662/) (2017)
- [A Kronecker-factored approximate Fisher matrix for convolution layers](https://mlsanity.com/paper/1602.01407/) (2016)
- [Understanding and Improving Transformer From a Multi-Particle Dynamic System Point of View](https://mlsanity.com/paper/1906.02762/) (2019)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Reversible Architectures for Arbitrarily Deep Residual Neural Networks](https://mlsanity.com/paper/1709.03698/) (2017)
- [A Mean-Field Optimal Control Formulation of Deep Learning](https://mlsanity.com/paper/1807.01083/) (2018)
- [Pontryagin Differentiable Programming: An End-to-End Learning and Control Framework](https://mlsanity.com/paper/1912.12970/) (2019)
- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [OL'E: Orthogonal Low-rank Embedding, A Plug and Play Geometric Loss for Deep Learning](https://mlsanity.com/paper/1712.01727/) (2017)
- [Layer-Parallel Training of Deep Residual Neural Networks](https://mlsanity.com/paper/1812.04352/) (2018)
- [An Optimal Control Approach to Deep Learning and Applications to Discrete-Weight Neural Networks](https://mlsanity.com/paper/1803.01299/) (2018)

## Abstract (arXiv)

Optimization of deep neural networks (DNNs) has been a driving force in the advancement of modern machine learning and artificial intelligence. With DNNs characterized by a prolonged sequence of nonlinear propagation, determining their optimal parameters given an objective naturally fits within the framework of Optimal Control Programming. Such an interpretation of DNNs as dynamical systems has proven crucial in offering a theoretical foundation for principled analysis from numerical equations to physics. In parallel to these theoretical pursuits, this paper focuses on an algorithmic perspective. Our motivated observation is the striking algorithmic resemblance between the Backpropagation algorithm for computing gradients in DNNs and the optimality conditions for dynamical systems, expressed through another backward process known as dynamic programming. Consolidating this connection, where Backpropagation admits a variational structure, solving an approximate dynamic programming up to the first-order expansion leads to a new class of optimization methods exploring higher-order expansions of the Bellman equation. The resulting optimizer, termed Optimal Control Theoretic Neural Optimizer (OCNOpt), enables rich algorithmic opportunities, including layer-wise feedback policies, game-theoretic applications, and higher-order training of continuous-time models such as Neural ODEs. Extensive experiments demonstrate that OCNOpt improves upon existing methods in robustness and efficiency while maintaining manageable computational complexity, paving new avenues for principled algorithmic design grounded in dynamical systems and optimal control theory.