Broken-symmetry shape discrimination on a driven Duffing ring
Authors: Kaspar Anton Schindler
Organizations: Sleep-Wake-Epilepsy-Center, Department of Neurology, Inselspital, Bern University Hospital, University of Bern, Bern, Switzerland
Abstract
Distributed computational substrates rely on two elementary operations: bundling, the act of populating a shared physical medium with independently retrievable components, and binding, the act of composing components into outputs whose identity depends on their relations. We study these two primitives on the simplest closed substrate carrying a continuous symmetry, a cycle graph of N nodes, in two parameter regimes of a single master equation of motion. The linear regime sorts a temporal input across the substrate's U(1)-organised eigenmodes, providing a feature representation that matches a windowed-FFT baseline at high signal-to-noise ratio and modestly outperforms it for transient signals at low SNR. The Duffing regime activates a cubic mode-mixing operation constrained by the substrate's symmetry into a sparse selection rule on integer wavenumbers, generating shape-dependent harmonic content that the linear regime cannot produce. We identify a single-number observable, φ0, that summarises the bound representation's response to input shape, and we analyse its symmetry structure: a π-periodicity in the shape parameter is exact, while a time-reversal symmetry that would render φ0 degenerate is broken by the substrate's dissipation. The asymmetric status of these two symmetries is what licenses φ0 as a meaningful single-number observable; its trajectory across the quotient domain encodes the joint response of binding and dissipation to the input shape. Numerical experiments confirm that φ0 retains its information content under additive band-limited noise, with seed-averaged means staying clearly above the symmetric-attractor value down to 0 dB input SNR. The framework is developed on synthetic signals only; extensions to richer substrates, more elaborate drives, and real biological signals are open questions for the work that follows.
Physical reservoir computing harnesses nonlinear mechanical dynamics but, by convention, freezes the substrate and trains only a linear readout, presuming the substrate is not usefully trainable. We revisit that premise for networks of nonlinear oscillators whose mass, damping, and stiffness are learned end-to-end through a symplectic integrator. Our central result is a trilemma: memory horizon, gradient stability, and dynamical expressivity cannot be simultaneously maximized, because all three are governed by the damping. The backward gradient decays at a rate set by the damping, capping how far back credit can propagate, while forward sensitivities grow exponentially in the largest Lyapunov exponent, so usable gradients require damping above a stability floor. Since the Lyapunov exponent falls as damping rises while the memory ceiling falls as the horizon grows, stable training is confined to a band that contracts with horizon and closes at a critical point. We test every step on a twenty-oscillator network. A damping sweep finds the largest Lyapunov exponent monotone and crossing zero at a well-defined stability floor, confirming the theorem's key assumption. A compute-matched comparison of learned versus frozen substrate on delayed recall across nine horizons shows the learned substrate dominating at short horizons and the advantage closing and reversing near a horizon of eleven steps, the predicted signature of band closure; trained models settle near the stability floor, seeking the edge of chaos unprompted. The analytic ceiling overestimates the empirical crossover roughly fivefold, a gap between detectable and learnable gradient that we report rather than tune away. The contribution is a confirmed account of when training a physical substrate beats freezing it.
Observable Matrix Dynamics (OMD) is a diagnostic framework that probes the dynamics of high-dimensional internal representations of inputs by a neural network via a fixed-size N×N distance matrix M(t) on a held set of N inputs. OMD uses methods of random matrix theory and particle dynamics to explore spectral reorganisations that are missed by scalar loss functions, but are informative of the training process. We read M(t) against a perturbative ambient-versus-latent decomposition extending the Bogomolny--Bohigas--Schmit (BBS) theory of random distance matrices, with per-snapshot diagnostics for the top-of-spectrum band structure and ambient noise, trajectory-level observables linking snapshots, and a 3D MDS embedding (bottom-three eigenvectors) rendering training as a moving particle cloud. Across seven experiments, diffusive regimes lack stable top-of-spectrum band structure, while sharp endogenous or externally driven reorganisations produce stable fingerprints: consistent with smooth or product latent geometries in BBS-adjacent cases, and with finite-cluster or Fourier-soliton structures otherwise. OMD thus reads the geometric regime of a representation rather than reporting a single intrinsic dimension.
We show that an array of scatterers which has been designed to have latent ("hidden") symmetries can be used as a sensor. We use the capacitance matrix as a canonical model for three-dimensional hybridisation and study how the introduction of an "intruder'' scatterer breaks the latent symmetries. By analysing the degree to which each symmetry is broken, we identify the radius of the intruder and localize its position. This can be achieved using a dictionary-based approach, however Bayesian inference or an artificial neural network (multi-layer perceptron) perform better in the presence of measurement noise. To our knowledge, this is the first time latent symmetries have been exploited successfully for sensing problems. It is also the first time latent symmetries have been observed in a three-dimensional open system that cannot be approximated by a sparse graph.