# Drift-Diffusion Matching: Embedding dynamics in latent manifolds of asymmetric neural networks

arXiv: https://arxiv.org/abs/2602.14885 · HTML: https://mlsanity.com/paper/2602.14885/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 82 references that are in the map.
- First posted on arXiv 16 Feb 2026.

## Builds on (most cited first, 15 of 15)

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- [Linking connectivity, dynamics and computations in low-rank recurrent neural networks](https://mlsanity.com/paper/1711.09672/) (2017)
- [Recurrent Network Models Of Sequence Generation And Memory](https://mlsanity.com/paper/1603.04687/) (2016)
- [Recurrent neural network models for working memory of continuous variables: activity manifolds, connectivity patterns, and dynamic codes](https://mlsanity.com/paper/2111.01275/) (2021)
- [A unifying framework for mean-field theories of asymmetric kinetic Ising systems](https://mlsanity.com/paper/2002.04309/) (2020)
- [Neural manifold analysis of brain circuit dynamics in health and disease](https://mlsanity.com/paper/2203.11874/) (2022)
- [Nonequilibrium physics of brain dynamics](https://mlsanity.com/paper/2504.12188/) (2025)
- [Decomposing the local arrow of time in interacting systems](https://mlsanity.com/paper/2112.14721/) (2021)
- [Broken detailed balance and entropy production in directed networks](https://mlsanity.com/paper/2402.19157/) (2024)
- [Self-orthogonalizing attractor neural networks emerging from the free energy principle](https://mlsanity.com/paper/2505.22749/) (2025)
- [Curl Descent: Non-Gradient Learning Dynamics with Sign-Diverse Plasticity](https://mlsanity.com/paper/2510.02765/) (2025)
- [Inferring entropy production in many-body systems using nonequilibrium maximum entropy](https://mlsanity.com/paper/2505.10444/) (2025)
- [Shaping manifolds in equivariant recurrent neural networks](https://mlsanity.com/paper/2511.04802/) (2025)

## Abstract (arXiv)

Recurrent neural networks (RNNs) provide a theoretical framework for understanding computation in biological neural circuits, yet classical results, such as Hopfield's model of associative memory, rely on symmetric connectivity that restricts network dynamics to gradient-like flows. In contrast, biological networks support rich time-dependent behaviour facilitated by their asymmetry. Here we introduce a general framework, which we term drift-diffusion matching, for training continuous-time RNNs to represent arbitrary, nonlinear stochastic differential equations (SDEs), with given drift and diffusion coefficients, within a low-dimensional latent subspace. Allowing asymmetric connectivity, we show that RNNs can faithfully embed the drift and diffusion of a given SDE, including nonlinear and nonequilibrium dynamics such as chaotic attractors. As an application, we construct RNN realisations of stochastic systems that transiently explore various attractors through both input-driven switching and autonomous transitions driven by nonequilibrium currents, which we interpret as models of associative and sequential (episodic) memory. To elucidate how these dynamics are encoded in the network, we introduce decompositions of the RNN based on its asymmetric connectivity and its time-irreversibility. Our results extend attractor neural network theory beyond equilibrium, showing that asymmetric neural populations can implement a broad class of dynamical computations within low-dimensional manifolds, unifying ideas from associative memory, nonequilibrium statistical mechanics, and neural computation.