Adaptive Sensing of Continuous Physical Systems for Machine Learning

Physical dynamical systems can be viewed as natural information processors: their systems preserve, transform, and disperse input information. This perspective motivates learning not only from data generated by such systems, but also how to measure them in a way that extracts the most useful information for a given task. We propose a general computing framework for adaptive information extraction from dynamical systems, in which a trainable attention module learns both where to probe the system state and how to combine these measurements to optimize prediction performance. As a concrete instantiation, we implement this idea using a spatiotemporal field governed by a partial differential equation as the underlying dynamics, though the framework applies equally to any system whose state can be sampled. Our results show that adaptive spatial sensing significantly improves prediction accuracy on canonical chaotic benchmarks. This work provides a perspective on attention-enhanced reservoir computing as a special case of a broader paradigm: neural networks as trainable measurement devices for extracting information from physical dynamical systems.

Attention-Enhanced Reservoir Computing

Photonic reservoir computing has been recently utilized in time series forecasting as the need for hardware implementations to accelerate these predictions has increased. Forecasting chaotic time series remains a significant challenge, an area where the conventional reservoir computing framework encounters limitations of prediction accuracy. We introduce an attention mechanism to the reservoir computing model in the output stage. This attention layer is designed to prioritize distinct features and temporal sequences, thereby substantially enhancing the forecasting accuracy. Our results show that a photonic reservoir computer enhanced with the attention mechanism exhibits improved forecasting capabilities for smaller reservoirs. These advancements highlight the transformative possibilities of reservoir computing for practical applications where accurate forecasting of chaotic time series is crucial.

Master memory function for delay-based reservoir computers with single-variable dynamics

We show that many delay-based reservoir computers considered in the literature can be characterized by a universal master memory function (MMF). Once computed for two independent parameters, this function provides linear memory capacity for any delay-based single-variable reservoir with small inputs. Moreover, we propose an analytical description of the MMF that enables its efficient and fast computation. Our approach can be applied not only to reservoirs governed by known dynamical rules such as Mackey-Glass or Ikeda-like systems but also to reservoirs whose dynamical model is not available. We also present results comparing the performance of the reservoir computer and the memory capacity given by the MMF.

Improving Delay Based Reservoir Computing via Eigenvalue Analysis

We analyze the reservoir computation capability of the Lang-Kobayashi system by comparing the numerically computed recall capabilities and the eigenvalue spectrum. We show that these two quantities are deeply connected, and thus the reservoir computing performance is predictable by analyzing the eigenvalue spectrum. Our results suggest that any dynamical system used as a reservoir can be analyzed in this way as long as the reservoir perturbations are sufficiently small. Optimal performance is found for a system with the eigenvalues having real parts close to zero and off-resonant imaginary parts.

Deep Time-Delay Reservoir Computing: Dynamics and Memory Capacity

The Deep Time-Delay Reservoir Computing concept utilizes unidirectionally connected systems with time-delays for supervised learning. We present how the dynamical properties of a deep Ikeda-based reservoir are related to its memory capacity (MC) and how that can be used for optimization. In particular, we analyze bifurcations of the corresponding autonomous system and compute conditional Lyapunov exponents, which measure the generalized synchronization between the input and the layer dynamics. We show how the MC is related to the systems distance to bifurcations or magnitude of the conditional Lyapunov exponent. The interplay of different dynamical regimes leads to a adjustable distribution between linear and nonlinear MC. Furthermore, numerical simulations show resonances between clock cycle and delays of the layers in all degrees of the MC. Contrary to MC losses in a single-layer reservoirs, these resonances can boost separate degrees of the MC and can be used, e.g., to design a system with maximum linear MC. Accordingly, we present two configurations that empower either high nonlinear MC or long time linear MC.