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).