# Transformers as Intrinsic Optimizers: Forward Inference through the Energy Principle

arXiv: https://arxiv.org/abs/2511.00907 · HTML: https://mlsanity.com/paper/2511.00907/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 68 of its 83 references that are in the map.
- First posted on arXiv 2 Nov 2025.

## Builds on (most cited first, 30 of 68)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [RWKV: Reinventing RNNs for the Transformer Era](https://mlsanity.com/paper/2305.13048/) (2023)
- [Efficient Transformers: A Survey](https://mlsanity.com/paper/2009.06732/) (2020)
- [Scaling Vision Transformers to 22 Billion Parameters](https://mlsanity.com/paper/2302.05442/) (2023)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Retentive Network: A Successor to Transformer for Large Language Models](https://mlsanity.com/paper/2307.08621/) (2023)
- [Gated Delta Networks: Improving Mamba2 with Delta Rule](https://mlsanity.com/paper/2412.06464/) (2024)
- [Gated Linear Attention Transformers with Hardware-Efficient Training](https://mlsanity.com/paper/2312.06635/) (2023)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Scaling up Test-Time Compute with Latent Reasoning: A Recurrent Depth Approach](https://mlsanity.com/paper/2502.05171/) (2025)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Learning to (Learn at Test Time): RNNs with Expressive Hidden States](https://mlsanity.com/paper/2407.04620/) (2024)
- [From Softmax to Sparsemax: A Sparse Model of Attention and Multi-Label Classification](https://mlsanity.com/paper/1602.02068/) (2016)
- [Titans: Learning to Memorize at Test Time](https://mlsanity.com/paper/2501.00663/) (2024)
- [xLSTM: Extended Long Short-Term Memory](https://mlsanity.com/paper/2405.04517/) (2024)
- [The Forward-Forward Algorithm: Some Preliminary Investigations](https://mlsanity.com/paper/2212.13345/) (2022)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)

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

- [Krause Synchronization Transformers](https://mlsanity.com/paper/2602.11534/) (2026)
- [YuriiFormer: A Suite of Nesterov-Accelerated Transformers](https://mlsanity.com/paper/2601.23236/) (2026)
- [Black-Mamba: Biologically-Inspired Leaky Accumulation for Conceptual Knowledge under Distribution Drift](https://mlsanity.com/paper/2607.18899/) (2026)

## Abstract (arXiv)

Attention-based Transformers have demonstrated strong adaptability across a wide range of tasks and have become the backbone of modern Large Language Models (LLMs). However, their underlying mechanisms remain open for further exploration. The energy-based perspective has long provided a valuable principle for understanding neural computation. In this paper, we revisit the principle of energy as a lens to understand attention-based Transformer models. We present a unified energy-based framework which is composed of three key components: the local energy Ei, the global energy F, and the employed optimization algorithms. We show that different attention forms including unnormalized linear attention, gated linear attention and standard softmax attention can be induced by choosing their corresponding recipes within this framework. Building on this framework, we propose energy-based modifications of attention structures. Inspired by classical gradient descent (GD) algorithms, we extend the original attention formulation based on standard GD to the momentum-based GD, Nesterov Accelerated Gradient (NAG), and Newton's method, each inducing a corresponding new attention structure. Our experiments provide preliminary support for the potential of the energy-based framework for designing attention mechanisms.