# A Unified Framework for Neural Computation and Learning Over Time

arXiv: https://arxiv.org/abs/2409.12038 · HTML: https://mlsanity.com/paper/2409.12038/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 51 references that are in the map.
- First posted on arXiv 18 Sep 2024.

## Builds on (most cited first, 30 of 34)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [A Comprehensive Survey of Continual Learning: Theory, Method and Application](https://mlsanity.com/paper/2302.00487/) (2023)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Towards Continual Reinforcement Learning: A Review and Perspectives](https://mlsanity.com/paper/2012.13490/) (2020)
- [The Forward-Forward Algorithm: Some Preliminary Investigations](https://mlsanity.com/paper/2212.13345/) (2022)
- [Online Continual Learning in Image Classification: An Empirical Survey](https://mlsanity.com/paper/2101.10423/) (2021)
- [Griffin: Mixing Gated Linear Recurrences with Local Attention for Efficient Language Models](https://mlsanity.com/paper/2402.19427/) (2024)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Difference Target Propagation](https://mlsanity.com/paper/1412.7525/) (2014)
- [An Exponential Learning Rate Schedule for Deep Learning](https://mlsanity.com/paper/1910.07454/) (2019)
- [Dendritic cortical microcircuits approximate the backpropagation algorithm](https://mlsanity.com/paper/1810.11393/) (2018)
- [Predictive Coding Approximates Backprop along Arbitrary Computation Graphs](https://mlsanity.com/paper/2006.04182/) (2020)
- [ANODE: Unconditionally Accurate Memory-Efficient Gradients for Neural ODEs](https://mlsanity.com/paper/1902.10298/) (2019)
- [A Unified Algebraic Perspective on Lipschitz Neural Networks](https://mlsanity.com/paper/2303.03169/) (2023)
- [Pontryagin Differentiable Programming: An End-to-End Learning and Control Framework](https://mlsanity.com/paper/1912.12970/) (2019)
- [Biologically Motivated Algorithms for Propagating Local Target Representations](https://mlsanity.com/paper/1805.11703/) (2018)
- [A Unified Framework of Online Learning Algorithms for Training Recurrent Neural Networks](https://mlsanity.com/paper/1907.02649/) (2019)
- [The Predictive Forward-Forward Algorithm](https://mlsanity.com/paper/2301.01452/) (2023)
- [Brain-inspired Computational Intelligence via Predictive Coding](https://mlsanity.com/paper/2308.07870/) (2023)
- [Credit Assignment in Neural Networks through Deep Feedback Control](https://mlsanity.com/paper/2106.07887/) (2021)
- [Online learning of long-range dependencies](https://mlsanity.com/paper/2305.15947/) (2023)
- [On Numerical Integration in Neural Ordinary Differential Equations](https://mlsanity.com/paper/2206.07335/) (2022)
- [Exploring the Promise and Limits of Real-Time Recurrent Learning](https://mlsanity.com/paper/2305.19044/) (2023)
- [State Space Models as Foundation Models: A Control Theoretic Overview](https://mlsanity.com/paper/2403.16899/) (2024)
- [On the Resurgence of Recurrent Models for Long Sequences -- Survey and Research Opportunities in the Transformer Era](https://mlsanity.com/paper/2402.08132/) (2024)
- [Error Forward-Propagation: Reusing Feedforward Connections to Propagate Errors in Deep Learning](https://mlsanity.com/paper/1808.03357/) (2018)
- [Cognitive Action Laws: The Case of Visual Features](https://mlsanity.com/paper/1808.09162/) (2018)
- [Local Propagation in Constraint-based Neural Network](https://mlsanity.com/paper/2002.07720/) (2020)

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

- [LTLZinc: a Benchmarking Framework for Continual Learning and Neuro-Symbolic Temporal Reasoning](https://mlsanity.com/paper/2507.17482/) (2025)

## Abstract (arXiv)

This paper proposes Hamiltonian Learning, a novel unified framework for learning with neural networks "over time", i.e., from a possibly infinite stream of data, in an online manner, without having access to future information. Existing works focus on the simplified setting in which the stream has a known finite length or is segmented into smaller sequences, leveraging well-established learning strategies from statistical machine learning. In this paper, the problem of learning over time is rethought from scratch, leveraging tools from optimal control theory, which yield a unifying view of the temporal dynamics of neural computations and learning. Hamiltonian Learning is based on differential equations that: (i) can be integrated without the need of external software solvers; (ii) generalize the well-established notion of gradient-based learning in feed-forward and recurrent networks; (iii) open to novel perspectives. The proposed framework is showcased by experimentally proving how it can recover gradient-based learning, comparing it to out-of-the box optimizers, and describing how it is flexible enough to switch from fully-local to partially/non-local computational schemes, possibly distributed over multiple devices, and BackPropagation without storing activations. Hamiltonian Learning is easy to implement and can help researches approach in a principled and innovative manner the problem of learning over time.