# Energy-Based Dynamical Models for Neurocomputation, Learning, and Optimization

arXiv: https://arxiv.org/abs/2604.05042 · HTML: https://mlsanity.com/paper/2604.05042/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 53 of its 142 references that are in the map.
- First posted on arXiv 6 Apr 2026.

## Builds on (most cited first, 30 of 53)

- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Concrete Problems in AI Safety](https://mlsanity.com/paper/1606.06565/) (2016)
- [The Mythos of Model Interpretability](https://mlsanity.com/paper/1606.03490/) (2016)
- [Your Classifier is Secretly an Energy Based Model and You Should Treat it Like One](https://mlsanity.com/paper/1912.03263/) (2019)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Large Associative Memory Problem in Neurobiology and Machine Learning](https://mlsanity.com/paper/2008.06996/) (2020)
- [Generalized Energy Based Models](https://mlsanity.com/paper/2003.05033/) (2020)
- [Energy Transformer](https://mlsanity.com/paper/2302.07253/) (2023)
- [Universal Hopfield Networks: A General Framework for Single-Shot Associative Memory Models](https://mlsanity.com/paper/2202.04557/) (2022)
- [Modern Hopfield Networks and Attention for Immune Repertoire Classification](https://mlsanity.com/paper/2007.13505/) (2020)
- [Dense Associative Memory is Robust to Adversarial Inputs](https://mlsanity.com/paper/1701.00939/) (2017)
- [Unsupervised Learning by Competing Hidden Units](https://mlsanity.com/paper/1806.10181/) (2018)
- [Memorization to Generalization: Emergence of Diffusion Models from Associative Memory](https://mlsanity.com/paper/2505.21777/) (2025)
- [Hierarchical Associative Memory](https://mlsanity.com/paper/2107.06446/) (2021)
- [Equivalence of Equilibrium Propagation and Recurrent Backpropagation](https://mlsanity.com/paper/1711.08416/) (2017)
- [A Hebbian/Anti-Hebbian Neural Network for Linear Subspace Learning: A Derivation from Multidimensional Scaling of Streaming Data](https://mlsanity.com/paper/1503.00669/) (2015)
- [Local Unsupervised Learning for Image Analysis](https://mlsanity.com/paper/1908.08993/) (2019)
- [Simplicial Hopfield networks](https://mlsanity.com/paper/2305.05179/) (2023)
- [Long Sequence Hopfield Memory](https://mlsanity.com/paper/2306.04532/) (2023)
- [Training Coupled Phase Oscillators as a Neuromorphic Platform using Equilibrium Propagation](https://mlsanity.com/paper/2402.08579/) (2024)
- [End-to-end Differentiable Clustering with Associative Memories](https://mlsanity.com/paper/2306.03209/) (2023)
- [AI Safety for Everyone](https://mlsanity.com/paper/2502.09288/) (2025)
- [Operator Learning for Reconstructing Flow Fields from Sparse Measurements: an Energy Transformer Approach](https://mlsanity.com/paper/2501.08339/) (2025)
- [Dense Associative Memory with Epanechnikov Energy](https://mlsanity.com/paper/2506.10801/) (2025)
- [Neuron-Astrocyte Associative Memory](https://mlsanity.com/paper/2311.08135/) (2023)
- [Dynamics over Signed Networks](https://mlsanity.com/paper/1706.03362/) (2017)
- [Control Barrier Function Based Design of Gradient Flows for Constrained Nonlinear Programming](https://mlsanity.com/paper/2204.01930/) (2022)

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

- [Learning infinite context windows in recurrent architectures via spatial neural computing](https://mlsanity.com/paper/2610.10690/) (2026)

## Abstract (arXiv)

Recent advances at the intersection of control theory, neuroscience, and machine learning have revealed novel mechanisms by which dynamical systems perform computation. These advances encompass a wide range of conceptual, mathematical, and computational ideas, with applications for model learning and training, memory retrieval, data-driven control, and optimization. This tutorial focuses on neuro-inspired approaches to computation that aim to improve scalability, robustness, and energy efficiency across such tasks, bridging the gap between artificial and biological systems. Particular emphasis is placed on energy-based dynamical models that encode information through gradient flows and energy landscapes. We begin by reviewing classical formulations, such as continuous-time Hopfield networks and Boltzmann machines, and then extend the framework to modern developments. These include dense associative memory models for high-capacity storage, oscillator-based networks for large-scale optimization, and proximal-descent dynamics for composite and constrained reconstruction. The tutorial demonstrates how control-theoretic principles can guide the design of next-generation neurocomputing systems, steering the discussion beyond conventional feedforward and backpropagation-based approaches to artificial intelligence.