# Understanding Recurrent Neural Networks Using Nonequilibrium Response Theory

arXiv: https://arxiv.org/abs/2006.11052 · HTML: https://mlsanity.com/paper/2006.11052/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jun 2020.

## Built on by (most cited first, 10 of 10)

- [Noisy Recurrent Neural Networks](https://mlsanity.com/paper/2102.04877/) (2021)
- [Framing RNN as a kernel method: A neural ODE approach](https://mlsanity.com/paper/2106.01202/) (2021)
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- [Stateful ODE-Nets using Basis Function Expansions](https://mlsanity.com/paper/2106.10820/) (2021)
- [Feed-Forward Neural Networks as a Mixed-Integer Program](https://mlsanity.com/paper/2402.06697/) (2024)
- [p-Adic Statistical Field Theory and Deep Belief Networks](https://mlsanity.com/paper/2207.13877/) (2022)
- [Path classification by stochastic linear recurrent neural networks](https://mlsanity.com/paper/2108.03090/) (2021)
- [Neural Generalized Ordinary Differential Equations with Layer-varying Parameters](https://mlsanity.com/paper/2209.10633/) (2022)
- [Approximated Orthogonal Projection Unit: Stabilizing Regression Network Training Using Natural Gradient](https://mlsanity.com/paper/2409.15393/) (2024)
- [Mechanistic Interpretability of RNNs emulating Hidden Markov Models](https://mlsanity.com/paper/2510.25674/) (2025)

## Abstract (arXiv)

Recurrent neural networks (RNNs) are brain-inspired models widely used in machine learning for analyzing sequential data. The present work is a contribution towards a deeper understanding of how RNNs process input signals using the response theory from nonequilibrium statistical mechanics. For a class of continuous-time stochastic RNNs (SRNNs) driven by an input signal, we derive a Volterra type series representation for their output. This representation is interpretable and disentangles the input signal from the SRNN architecture. The kernels of the series are certain recursively defined correlation functions with respect to the unperturbed dynamics that completely determine the output. Exploiting connections of this representation and its implications to rough paths theory, we identify a universal feature -- the response feature, which turns out to be the signature of tensor product of the input signal and a natural support basis. In particular, we show that SRNNs, with only the weights in the readout layer optimized and the weights in the hidden layer kept fixed and not optimized, can be viewed as kernel machines operating on a reproducing kernel Hilbert space associated with the response feature.