# DFORM: Diffeomorphic vector field alignment for assessing dynamics across learned models

arXiv: https://arxiv.org/abs/2402.09735 · HTML: https://mlsanity.com/paper/2402.09735/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 30 references that are in the map.
- First posted on arXiv 15 Feb 2024.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Universality and individuality in neural dynamics across large populations of recurrent networks](https://mlsanity.com/paper/1907.08549/) (2019)
- [Attractor and integrator networks in the brain](https://mlsanity.com/paper/2112.03978/) (2021)
- [Linking connectivity, dynamics and computations in low-rank recurrent neural networks](https://mlsanity.com/paper/1711.09672/) (2017)
- [Beyond Geometry: Comparing the Temporal Structure of Computation in Neural Circuits with Dynamical Similarity Analysis](https://mlsanity.com/paper/2306.10168/) (2023)
- [Implicit regularization of deep residual networks towards neural ODEs](https://mlsanity.com/paper/2309.01213/) (2023)
- [Reverse engineering recurrent neural networks with Jacobian switching linear dynamical systems](https://mlsanity.com/paper/2111.01256/) (2021)
- [ResNet-LDDMM: Advancing the LDDMM Framework using Deep Residual Networks](https://mlsanity.com/paper/2102.07951/) (2021)
- [Interpretable statistical representations of neural population dynamics and geometry](https://mlsanity.com/paper/2304.03376/) (2023)
- [Dynamics of random recurrent networks with correlated low-rank structure](https://mlsanity.com/paper/1909.04358/) (2019)
- [Residual Networks as Flows of Velocity Fields for Diffeomorphic Time Series Alignment](https://mlsanity.com/paper/2106.11911/) (2021)
- [Batch Normalization in Cytometry Data by kNN-Graph Preservation](https://mlsanity.com/paper/2304.00050/) (2023)

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

- [InputDSA: Demixing then Comparing Recurrent and Externally Driven Dynamics](https://mlsanity.com/paper/2510.25943/) (2025)
- [Geometry-aware similarity metrics for neural representations on Riemannian and statistical manifolds](https://mlsanity.com/paper/2603.28764/) (2026)
- [Characterizing Nonlinear Dynamics via Smooth Prototype Equivalences](https://mlsanity.com/paper/2503.10336/) (2025)

## Abstract (arXiv)

Dynamical system models such as Recurrent Neural Networks (RNNs) have become increasingly popular as hypothesis-generating tools in scientific research. Evaluating the dynamics in such networks is key to understanding their learned generative mechanisms. However, comparison of learned dynamics across models is challenging due to their inherent nonlinearity and because a priori there is no enforced equivalence of their coordinate systems. Here, we propose the DFORM (Diffeomorphic vector field alignment for comparing dynamics across learned models) framework. DFORM learns a nonlinear coordinate transformation which provides a continuous, maximally one-to-one mapping between the trajectories of learned models, thus approximating a diffeomorphism between them. The mismatch between DFORM-transformed vector fields defines the orbital similarity between two models, thus providing a generalization of the concepts of smooth orbital and topological equivalence. As an example, we apply DFORM to models trained on a canonical neuroscience task, showing that learned dynamics may be functionally similar, despite overt differences in attractor landscapes.