Optimal Memory Encoding Through Fluctuation-Response Structure

Abstract

Physical reservoir computing exploits the intrinsic dynamics of physical systems for information processing, while keeping the internal dynamics fixed and training only linear readouts; yet the role of input encoding remains poorly understood. We show that optimal input encoding is a geometric problem governed by the system's fluctuation-response structure. By measuring steady-state fluctuations and linear response, we derive an analytical criterion for the input direction that maximizes task-specific linear memory under a fixed power constraint, termed Response-based Optimal Memory Encoding (ROME). Backpropagation-based encoder optimization is shown to be equivalent to ROME, revealing a trade-off between task-dependent feature mixing and intrinsic noise. We apply ROME to various reservoir platforms, including spin-wave waveguides and spiking neural networks, demonstrating effective encoder design across physical and neuromorphic reservoirs, even in non-differentiable systems.

Figures (2)

• Figure 1: Numerical demonstrations of ROME. (a) Polar plot of the MF for a linear reservoir, evaluated in the plane spanned by the task-only direction (blue) and the ROME-optimal direction (red); gray curve: MF landscape; black line: noise-minimizing direction projected onto this plane. (b) MF versus delay $k$ for a linear reservoir with different linear memory tasks. Blue curve and shaded band: mean$\pm$s.d. over random encoders; colored curves: short-term ($k=2,3,4$; orange) and long-term ($k=20,22,24$; green) ROME encoders, MC-optimal encoder (black, $\mathrm{MC}=\sum_k \mathrm{MF}(k)$), and single-delay–optimal encoder (red). (c) NARMA10 performance $R^2$ versus input power $P$ for a nonlinear ESN. Blue circles and shaded band: mean$\pm$s.d. over random encoders; Red squares: ROME encoder. (d) Training evolution for the BP encoder in a linear reservoir (single-delay task). Blue circles: alignment $|\cos(G_{\rm BP},G^*)|$ between learned input direction and ROME direction; red squares: BP performance $R^2$; dashed line: $R^2$ of the ROME direction $G^*$.
• Figure 2: PRC demonstrations of ROME. (a) Schematic illustration of the spin-wave physical reservoir and the performance $R^2$ versus delay $k$ on a single-delay memory task. Blue curve and shaded band: mean $\pm$ s.d. over random encoders; red and black: ROME encoders optimized for target delays $k_0$. (b) Performance of the heterogeneous E/I SNN neuromorphic PRC systems on a single-delay memory task. Red dashed line: ROME; blue line: mean over random encoders; gray dotted lines: standard deviation of random encoders.