Context-dependent manifold learning: A neuromodulated constrained autoencoder approach

Abstract

Constrained autoencoders (cAE) provide a successful path towards interpretable dimensionality reduction by enforcing geometric structure on latent spaces. However, standard cAEs cannot adapt to varying physical parameters or environmental conditions without conflating these contextual shifts with the primary input. To address this, we integrated a neuromodulatory mechanism into the cAE framework to allow for context-dependent manifold learning. This paper introduces the Neuromodulated Constrained Autoencoder (NcAE), which adaptively parameterizes geometric constraints via gain and bias tuning conditioned on static contextual information. Experimental results on dynamical systems show that the NcAE accurately captures how manifold geometry varies across different regimes while maintaining rigorous projection properties. These results demonstrate that neuromodulation effectively decouples global contextual parameters from local manifold representations. This architecture provides a foundation for developing more flexible, physics-informed representations in systems subject to (non-stationary) environmental constraints.

Figures (10)

• Figure 1: Neuromodulation dynamically reconfigures the autoencoder manifold by parameterizing activation functions and biases based on external context. This schematic illustrates the integration of the context vector $\boldsymbol{c}$, which is processed by a fully connected MLP to generate the modulation signal $\boldsymbol{s}$. This signal is then multiplied by a layer-specific weight matrix $\boldsymbol{W}_{l, \alpha}$ to compute the activation parameters $\boldsymbol{\alpha}^{(l)}$. Simultaneously, $\boldsymbol{s}$ passes through a linear layer defined by parameters $\boldsymbol{W}_{l, b}$ and $\boldsymbol{b}_l$ to determine the layer-wise bias $\boldsymbol{b}^{(l)}$. Together, these components allow the model to adapt its internal transformations to the specific requirements of the provided context.
• Figure 2: The NcAE significantly outperforms standard and conditional autoencoders, especially in reconstructing the underlying system dynamics under context-dependent coupling. This boxplot compares the root mean square error (RMSE) distribution across 256 test trajectories for three architectures. The shaded grey region indicates the baseline performance range observed in the Standard pendulum experiment where context does not influence coupling. The left panel shows the reconstruction error for the pendulum DoFs, while the right panel displays the error for their first-order derivatives. The results demonstrate that the NcAE ability to modulate its internal representation leads to a more accurate capture of both the state and its temporal evolution compared to simple context concatenation.
• Figure 3: The NcAE exhibits a highly structured and continuous adaptation of its latent geometry in response to varying physical parameters. This figure visualizes latent trajectories for dimensions $z_1$ and $z_2$ across a range of configurations where all link lengths are held equal. The transparency gradient represents the transition from 0.35m (most transparent) to 0.65m (least transparent). While standard architectures often collapse or struggle to organize context-dependent data, the NcAE successfully partitions the latent space, demonstrating that the neuromodulation mechanism internalizes the relationship between the physical context (link lengths) and the resulting system manifold.
• Figure 4: The NcAE maintains high-fidelity reconstruction across critical bifurcations, where baseline architectures fail to adapt to changing system dynamics. These boxplots illustrate the distribution of the RMSE for both state reconstruction and first-order derivatives across test trajectories spanning the full range of forcing parameters $F$. The shaded grey region represents the performance benchmarks from the Standard Lorenz 96a and Standard Lorenz 96b experiments; in these stable regimes, all architectures perform similarly well, with error magnitudes that are negligible relative to the scale of the illustrated plots. However, in the context-dependent case, the NcAE neuromodulation allows it to resolve the shifting manifold structure that causes significant error spikes in the non-modulated models.
• Figure 5: The NcAE provides a context-specific manifold alignment that minimizes systematic spatio-temporal reconstruction errors. This figure displays Hovmöller diagrams of the absolute error between the ground truth and the reconstruction for a trajectory at $F=3.133$. By utilizing a neuromodulation signal tailored to this specific forcing parameter, the NcAE avoids the structured error patterns seen in architectures that lack the flexibility to adapt their internal transformations to the specific regime.