Reservoir Computing

Biological systems are promising substrates for computation because they naturally process environmental information through complex internal dynamics. In this study, we investigate whether bacterial metabolic models can act as physical reservoirs and whether their computational performance can be predicted from dynamical properties linked to separability and similarity. We simulated the growth dynamics of five bacterial species, one yeast species, and 29 Escherichia coli single-gene deletion mutants using dynamic flux balance analysis (dFBA), with glucose and xylose concentrations as inputs and growth curves as reservoir states. Computational performance was assessed on random nonlinear classification tasks using a linear readout, while reservoir properties linked to separability and similarity were characterised through kernel and generalisation ranks computed from growth-curve state matrices. Several microbial models achieved high classification accuracy, showing that bacterial metabolic dynamics can support nonlinear computation. Clear differences were observed between species, with some models converging more rapidly and others reaching higher maximum accuracy, revealing a trade-off between convergence speed and peak performance. In contrast, all E. coli mutants were dominated by the wild-type model, suggesting that gene deletions reduce the dynamical richness required for efficient computation. The difference between kernel and generalisation ranks was generally associated with improved accuracy, but deviations across models and sensitivity at low rank values limited its predictive power in practice. Overall, these results show that bacterial metabolic models constitute promising substrates for reservoir computing and provide a first step towards identifying microbial strains with favourable computational properties for future experimental implementations.

Understanding how training shapes the geometry of recurrent network dynamics is a central problem in time-series modeling. We study the emergence of low-dimensional dominant manifolds in the training of Reservoir Computing (RC) networks for temporal forecasting tasks. For a general linear continuous-time reservoir in the infinite-data limit, we show that the training data generate an invariant subspace of the trained reservoir, whose dimension equals the number of dominant modes. We then specialize to a simplified diagonal linear reservoir, where we link the dominant eigenvalues and eigenvectors to the spectrum of a backward Dynamic Mode Decomposition matrix, which yields a finite-dimensional approximation of the backward-time Koopman operator of the system generating the training data. We illustrate the emergence of these dominant modes during training in simulation, and discuss how the analysis may be extended to nonlinear RC via tangent dynamics and differential p-dominance.

We present PhasorFlow, an open-source Python library for computing on the $S^1$ unit circle. Inputs are encoded as complex phasors $z=e^{iφ}$ on the $N$-torus ($\mathbb{T}^N$); as computation proceeds through unitary wave-interference gates, global norm is preserved while components drift into $\mathbb{C}^N$, letting algorithms leverage continuous geometric gradients. PhasorFlow makes three contributions. First, we formalize the Phasor Circuit model ($N$ threads, $M$ gates) with a 22-gate library spanning standard-unitary, non-linear, neuromorphic, and encoding operations under full matrix-algebra simulation. Second, we introduce the Variational Phasor Circuit (VPC), a trainable phase-native classifier analogous to variational quantum circuits. Third, we introduce the Phasor Transformer block and Large Phasor Model (LPM), replacing $QK^TV$ attention with a parameter-free DFT token-mixing layer. We validate the framework on financial volatility detection, neuromorphic associative memory, neural binding, period finding, and algorithmic logic applications that are unique to the library. This positions unit-circle computing as a deterministic, lightweight paradigm on classical hardware. Available at https://github.com/mindverse-computing/phasorflow.

Most robotic systems rely on fixed-frequency discrete-time controllers, creating a trade-off between the efficiency of low-frequency control and the responsiveness of high-frequency feedback. As a result, systems typically default to high control rates for robustness, at the cost of wasted inference and unnecessary actuation. Addressing this, we introduce Time-Adaptive Robotic Control (TARC), a reinforcement learning framework in which the policy jointly predicts a control action and its duration of application. TARC learns temporally extended actions by optimizing task performance under soft or hard constraints on the number of control switches, enabling adaptive modulation of control rates. We evaluate TARC on two robotic hardware platforms: a high-speed RC car and the Unitree Go1 quadruped, and on a vision-language action model in simulation, where each query incurs a costly transformer forward pass. Across all settings, TARC matches the performance of high-frequency discrete-time controllers while operating at less than half their control frequency. Unlike fixed-rate controllers, TARC adapts its control frequency online, allocating high-frequency feedback only when required.

Scrambling quantum systems have attracted attention as effective substrates for temporal information processing. Here we consider a quantum reservoir processing framework that captures a broad range of physical computing models with quantum systems. We examine the scalability and memory retention of the model with scrambling reservoirs modelled by high-order unitary designs in both noiseless and noisy settings. In the former regime, we show that measurement readouts become exponentially concentrated with increasing reservoir size, yet strikingly do not worsen with the reservoir iterations. Thus, while repeatedly reusing a small scrambling reservoir with quantum data might be viable, scaling up the problem size deteriorates generalization unless one can afford an exponential shot overhead. In contrast, the memory of early inputs and initial states decays exponentially in both reservoir size and reservoir iterations. In the noisy regime, we also prove that memory decays exponentially in time for local noisy channels. These results required us to introduce new proof techniques for bounding concentration in temporal quantum models. Beyond this extreme scrambling regime, we numerically demonstrate that exponential concentration can still exist even with a physical reservoir such as an Ising model whenever the reservoir operates in a quantum-chaotic phase. In contrast, physical reservoirs in a many-body localized phase and at the edge of chaos appear to not suffer from such phenomena

We study how the degree of nonlinearity in the input data affects the optimal design of reservoir computers, focusing on how closely the model's nonlinearity should align with that of the data. By reducing minimal RCs to a single tunable nonlinearity parameter, we explore how the predictive performance varies with the degree of nonlinearity in the reservoir. To provide controlled testbeds, we generalize to the fractional Halvorsen system, a novel chaotic system with fractional exponents. Our experiments reveal that the prediction performance is maximized when the reservoir's nonlinearity matches the nonlinearity present in the data. In cases where multiple nonlinearities are present in the data, we find that the correlation dimension of the predicted signal is reconstructed correctly when the smallest nonlinearity is matched. We use this observation to propose a method for estimating the minimal nonlinearity in unknown time series by sweeping the reservoir exponent and identifying the transition to a successful reconstruction. Applying this method to both synthetic and real-world datasets, including financial time series, we demonstrate its practical viability. Finally, we transfer these insights to classical RC by augmenting traditional architectures with fractional, generalized reservoir states. This yields performance gains, particularly in resource-constrained scenarios such as physical reservoirs, where increasing reservoir size is impractical or economically unviable. Our work provides a principled route toward tailoring RCs to the intrinsic complexity of the systems they aim to model.