SOM-VQ: Topology-Aware Tokenization for Interactive Generative Models

Vector-quantized representations enable powerful discrete generative models but lack semantic structure in token space, limiting interpretable human control. We introduce SOM-VQ, a tokenization method that combines vector quantization with Self-Organizing Maps to learn discrete codebooks with explicit low-dimensional topology. Unlike standard VQ-VAE, SOM-VQ uses topology-aware updates that preserve neighborhood structure: nearby tokens on a learned grid correspond to semantically similar states, enabling direct geometric manipulation of the latent space. We demonstrate that SOM-VQ produces more learnable token sequences in the evaluated domains while providing an explicit navigable geometry in code space. Critically, the topological organization enables intuitive human-in-the-loop control: users can steer generation by manipulating distances in token space, achieving semantic alignment without frame-level constraints. We focus on human motion generation - a domain where kinematic structure, smooth temporal continuity, and interactive use cases (choreography, rehabilitation, HCI) make topology-aware control especially natural - demonstrating controlled divergence and convergence from reference sequences through simple grid-based sampling. SOM-VQ provides a general framework for interpretable discrete representations applicable to music, gesture, and other interactive generative domains.

Inverting Self-Organizing Maps: A Unified Activation-Based Framework

Self-Organizing Maps provide topology-preserving projections of high-dimensional data and have been widely used for visualization, clustering, and vector quantization. In this work, we show that the activation pattern of a SOM - the squared distances to its prototypes - can be inverted to recover the exact input under mild geometric conditions. This follows from a classical fact in Euclidean distance geometry: a point in $D$ dimensions is uniquely determined by its distances to $D{+}1$ affinely independent references. We derive the corresponding linear system and characterize the conditions under which the inversion is well-posed. Building upon this mechanism, we introduce the Manifold-Aware Unified SOM Inversion and Control (MUSIC) update rule, which enables controlled, semantically meaningful trajectories in latent space. MUSIC modifies squared distances to selected prototypes while preserving others, resulting in a deterministic geometric flow aligned with the SOM's piecewise-linear structure. Tikhonov regularization stabilizes the update rule and ensures smooth motion on high-dimensional datasets. Unlike variational or probabilistic generative models, MUSIC does not rely on sampling, latent priors, or encoder-decoder architectures. If no perturbation is applied, inversion recovers the exact input; when a target cluster or prototype is specified, MUSIC produces coherent semantic variations while remaining on the data manifold. This leads to a new perspective on data augmentation and controllable latent exploration based solely on prototype geometry. We validate the approach using synthetic Gaussian mixtures, the MNIST and the Faces in the Wild dataset. Across all settings, MUSIC produces smooth, interpretable trajectories that reveal the underlying geometry of the learned manifold, illustrating the advantages of SOM-based inversion over unsupervised clustering.

Lyapunov Learning at the Onset of Chaos

Handling regime shifts and non-stationary time series in deep learning systems presents a significant challenge. In the case of online learning, when new information is introduced, it can disrupt previously stored data and alter the model's overall paradigm, especially with non-stationary data sources. Therefore, it is crucial for neural systems to quickly adapt to new paradigms while preserving essential past knowledge relevant to the overall problem. In this paper, we propose a novel training algorithm for neural networks called \textit{Lyapunov Learning}. This approach leverages the properties of nonlinear chaotic dynamical systems to prepare the model for potential regime shifts. Drawing inspiration from Stuart Kauffman's Adjacent Possible theory, we leverage local unexplored regions of the solution space to enable flexible adaptation. The neural network is designed to operate at the edge of chaos, where the maximum Lyapunov exponent, indicative of a system's sensitivity to small perturbations, evolves around zero over time. Our approach demonstrates effective and significant improvements in experiments involving regime shifts in non-stationary systems. In particular, we train a neural network to deal with an abrupt change in Lorenz's chaotic system parameters. The neural network equipped with Lyapunov learning significantly outperforms the regular training, increasing the loss ratio by about $96\%$.

Dreaming Learning

Incorporating novelties into deep learning systems remains a challenging problem. Introducing new information to a machine learning system can interfere with previously stored data and potentially alter the global model paradigm, especially when dealing with non-stationary sources. In such cases, traditional approaches based on validation error minimization offer limited advantages. To address this, we propose a training algorithm inspired by Stuart Kauffman's notion of the Adjacent Possible. This novel training methodology explores new data spaces during the learning phase. It predisposes the neural network to smoothly accept and integrate data sequences with different statistical characteristics than expected. The maximum distance compatible with such inclusion depends on a specific parameter: the sampling temperature used in the explorative phase of the present method. This algorithm, called Dreaming Learning, anticipates potential regime shifts over time, enhancing the neural network's responsiveness to non-stationary events that alter statistical properties. To assess the advantages of this approach, we apply this methodology to unexpected statistical changes in Markov chains and non-stationary dynamics in textual sequences. We demonstrated its ability to improve the auto-correlation of generated textual sequences by $\sim 29\%$ and enhance the velocity of loss convergence by $\sim 100\%$ in the case of a paradigm shift in Markov chains.

Hybrid Physical-Neural ODEs for Fast N-body Simulations

We present a new scheme to compensate for the small-scales approximations resulting from Particle-Mesh (PM) schemes for cosmological N-body simulations. This kind of simulations are fast and low computational cost realizations of the large scale structures, but lack resolution on small scales. To improve their accuracy, we introduce an additional effective force within the differential equations of the simulation, parameterized by a Fourier-space Neural Network acting on the PM-estimated gravitational potential. We compare the results for the matter power spectrum obtained to the ones obtained by the PGD scheme (Potential gradient descent scheme). We notice a similar improvement in term of power spectrum, but we find that our approach outperforms PGD for the cross-correlation coefficients, and is more robust to changes in simulation settings (different resolutions, different cosmologies).