# Context-dependent manifold learning: A neuromodulated constrained autoencoder approach

arXiv: https://arxiv.org/abs/2603.11673 · HTML: https://mlsanity.com/paper/2603.11673/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 26 references that are in the map.
- First posted on arXiv 12 Mar 2026.

## Builds on (most cited first, 13 of 13)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Arbitrary Style Transfer in Real-time with Adaptive Instance Normalization](https://mlsanity.com/paper/1703.06868/) (2017)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [Idempotent Generative Network](https://mlsanity.com/paper/2311.01462/) (2023)
- [Introducing Neuromodulation in Deep Neural Networks to Learn Adaptive Behaviours](https://mlsanity.com/paper/1812.09113/) (2018)
- [Learning Nonlinear Projections for Reduced-Order Modeling of Dynamical Systems using Constrained Autoencoders](https://mlsanity.com/paper/2307.15288/) (2023)
- [A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics](https://mlsanity.com/paper/2410.18868/) (2024)

## Abstract (arXiv)

Many physical systems exhibit a low-dimensional structure that varies with external parameters: link lengths in a robot, forcing constants in a fluid, or Reynolds numbers in a flow shift the underlying manifold while preserving its intrinsic dimension. Constrained AutoEncoders (cAEs) learn such manifolds through an idempotent encoder-decoder projection, a property that unconstrained autoencoders cannot match and that is essential whenever the model is applied iteratively. However, the standard strategies for making a cAE context-dependent, namely concatenating the context to the input or affinely modulating hidden activations, break the encoder-decoder idempotency, sacrificing the projection guarantee precisely in the setting where it would be most valuable. To restore this guarantee under context variation, we developed the Neuromodulated Constrained Autoencoder (NcAE), which modulates the activation slope and bias of a cAE through a context-driven hyper-network. This paper presents the NcAE, its theoretical foundation, and its empirical validation. We prove that for every context, including contexts unseen at training time, the reconstruction map remains an idempotent projection, the topology of the learned manifold is invariant, and context perturbations induce smooth changes in the manifold. We evaluated our approach on a 16-DoF pendulum with context-dependent coupling and the Lorenz96 system across a bifurcation. The NcAE matched or exceeded the best of six baselines on reconstruction, idempotency, and latent-geometry metrics, while being the only architecture that preserves geometric consistency by construction. The NcAE thereby provides a stable, geometry-preserving coordinate system across families of physical regimes.