# Can Stationary Distributions of Scale-Invariant Neural Networks Be Described by the Thermodynamics of an Ideal Gas?

arXiv: https://arxiv.org/abs/2511.07308 · HTML: https://mlsanity.com/paper/2511.07308/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 50 of its 80 references that are in the map.
- First posted on arXiv 10 Nov 2025.

## Builds on (most cited first, 30 of 50)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Deep Learning and the Information Bottleneck Principle](https://mlsanity.com/paper/1503.02406/) (2015)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [Entropy-SGD: Biasing Gradient Descent Into Wide Valleys](https://mlsanity.com/paper/1611.01838/) (2016)
- [How Does Batch Normalization Help Optimization?](https://mlsanity.com/paper/1805.11604/) (2018)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [Three Factors Influencing Minima in SGD](https://mlsanity.com/paper/1711.04623/) (2017)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [Understanding Batch Normalization](https://mlsanity.com/paper/1806.02375/) (2018)
- [The Power of Interpolation: Understanding the Effectiveness of SGD in Modern Over-parametrized Learning](https://mlsanity.com/paper/1712.06559/) (2017)
- [Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU Networks](https://mlsanity.com/paper/1811.08888/) (2018)
- [A Bayesian Perspective on Generalization and Stochastic Gradient Descent](https://mlsanity.com/paper/1710.06451/) (2017)
- [L2 Regularization versus Batch and Weight Normalization](https://mlsanity.com/paper/1706.05350/) (2017)
- [An Exponential Learning Rate Schedule for Deep Learning](https://mlsanity.com/paper/1910.07454/) (2019)
- [Three Mechanisms of Weight Decay Regularization](https://mlsanity.com/paper/1810.12281/) (2018)
- [Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks](https://mlsanity.com/paper/1710.11029/) (2017)
- [Statistical Mechanics of Deep Linear Neural Networks: The Back-Propagating Kernel Renormalization](https://mlsanity.com/paper/2012.04030/) (2020)
- [nGPT: Normalized Transformer with Representation Learning on the Hypersphere](https://mlsanity.com/paper/2410.01131/) (2024)
- [Theoretical Analysis of Auto Rate-Tuning by Batch Normalization](https://mlsanity.com/paper/1812.03981/) (2018)
- [Stochastic Gradient Descent on Separable Data: Exact Convergence with a Fixed Learning Rate](https://mlsanity.com/paper/1806.01796/) (2018)
- [Norm matters: efficient and accurate normalization schemes in deep networks](https://mlsanity.com/paper/1803.01814/) (2018)
- [Reconciling Modern Deep Learning with Traditional Optimization Analyses: The Intrinsic Learning Rate](https://mlsanity.com/paper/2010.02916/) (2020)
- [Normalization and effective learning rates in reinforcement learning](https://mlsanity.com/paper/2407.01800/) (2024)
- [Separation of Scales and a Thermodynamic Description of Feature Learning in Some CNNs](https://mlsanity.com/paper/2112.15383/) (2021)

## Abstract (arXiv)

Understanding the training dynamics of deep neural networks remains a major open problem, with physics-inspired approaches offering promising insights. Building on this perspective, we develop a thermodynamic framework to describe the stationary distributions of stochastic gradient descent (SGD) with weight decay for scale-invariant neural networks, a setting that both reflects practical architectures with normalization layers and permits theoretical analysis. We establish analogies between training hyperparameters (e.g., learning rate, weight decay) and thermodynamic variables such as temperature, pressure, and volume. Starting with a simplified isotropic noise model, we uncover a close correspondence between SGD dynamics and ideal gas behavior, validated through theory and simulation. Extending to training of neural networks, we show that key predictions of the framework, including the behavior of stationary entropy, align closely with experimental observations. This framework provides a principled foundation for interpreting training dynamics and may guide future work on hyperparameter tuning and the design of learning rate schedulers.