Symplectic Reservoir Representation of Legendre Dynamics

Modern learning systems act on internal representations of data, yet how these representations encode underlying physical or statistical structure is often left implicit. In physics, conservation laws of Hamiltonian systems such as symplecticity guarantee long-term stability, and recent work has begun to hard-wire such constraints into learning models at the loss or output level. Here we ask a different question: what would it mean for the representation itself to obey a symplectic conservation law in the sense of Hamiltonian mechanics? We express this symplectic constraint through Legendre duality: the pairing between primal and dual parameters, which becomes the structure that the representation must preserve. We formalize Legendre dynamics as stochastic processes whose trajectories remain on Legendre graphs, so that the evolving primal-dual parameters stay Legendre dual. We show that this class includes linear time-invariant Gaussian process regression and Ornstein-Uhlenbeck dynamics. Geometrically, we prove that the maps that preserve all Legendre graphs are exactly symplectomorphisms of cotangent bundles of the form "cotangent lift of a base diffeomorphism followed by an exact fibre translation". Dynamically, this characterization leads to the design of a Symplectic Reservoir (SR), a reservoir-computing architecture that is a special case of recurrent neural network and whose recurrent core is generated by Hamiltonian systems that are at most linear in the momentum. Our main theorem shows that every SR update has this normal form and therefore transports Legendre graphs to Legendre graphs, preserving Legendre duality at each time step. Overall, SR implements a geometrically constrained, Legendre-preserving representation map, injecting symplectic geometry and Hamiltonian mechanics directly at the representational level.

A Minimal Genetic Circuit for Cellular Anticipation

Living systems have evolved cognitive complexity to reduce environmental uncertainty, enabling them to predict and prepare for future conditions. Anticipation, distinct from simple prediction, involves active adaptation before an event occurs and is a key feature of both neural and non-neural biological agents. Recent work by Steven Frank proposed a minimal anticipatory mechanism based on the moving average convergence-divergence principle from financial markets. Here, we implement this principle using synthetic biology to design and evaluate minimal genetic circuits capable of anticipating environmental trends. Through deterministic and stochastic analyses, we demonstrate that these motifs achieve robust anticipatory responses under a wide range of conditions. Our findings suggest that simple genetic circuits could be naturally exploited by cells to prepare for future events, providing a foundation for engineering predictive biological systems. ### Competing Interest Statement The authors have declared no competing interest.

Auszeichnungen auf dem Gebiet der Physik der kondensierten Materie für drei JGU-Physiker | FB 08 - Physik, Mathematik und Informatik

effconnpy/effconnpy/nonlinearGC.py at main · alecrimi/effconnpy

A package for causal inference and statistical modeling in brain time series - alecrimi/effconnpy

Higher-order Granger reservoir computing: simultaneously achieving scalable complex structures inference and accurate dynamics prediction - Nature Communications

For reservoir computing, improving prediction accuracy while maintaining low computing complexity remains a challenge. Inspired by the Granger causality, Li et al. design a data-driven and model-free framework by integrating the inference process and the inferred results on high-order structures.

Energy-saving computing with magnetic whirls