Robert Simon Fong

Robert Simon Fong, Gouhei Tanaka, Kazuyuki Aihara

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.

Robert Simon Fong, Boyu Li, Peter Tiňo

Reservoir Computing (RC) models, a subclass of recurrent neural networks, are distinguished by their fixed, non-trainable input layer and dynamically coupled reservoir, with only the static readout layer being trained. This design circumvents the issues associated with backpropagating error signals through time, thereby enhancing both stability and training efficiency. RC models have been successfully applied across a broad range of application domains. Crucially, they have been demonstrated to be universal approximators of time-invariant dynamic filters with fading memory, under various settings of approximation norms and input driving sources. Simple Cycle Reservoirs (SCR) represent a specialized class of RC models with a highly constrained reservoir architecture, characterized by uniform ring connectivity and binary input-to-reservoir weights with an aperiodic sign pattern. For linear reservoirs, given the reservoir size, the reservoir construction has only one degree of freedom -- the reservoir cycle weight. Such architectures are particularly amenable to hardware implementations without significant performance degradation in many practical tasks. In this study we endow these observations with solid theoretical foundations by proving that SCRs operating in real domain are universal approximators of time-invariant dynamic filters with fading memory. Our results supplement recent research showing that SCRs in the complex domain can approximate, to arbitrary precision, any unrestricted linear reservoir with a non-linear readout. We furthermore introduce a novel method to drastically reduce the number of SCR units, making such highly constrained architectures natural candidates for low-complexity hardware implementations. Our findings are supported by empirical studies on real-world time series datasets.

Peter Tino, Robert Simon Fong, Roberto Fabio Leonarduzzi

This work proposes a time series prediction method based on the kernel view of linear reservoirs. In particular, the time series motifs of the reservoir kernel are used as representational basis on which general readouts are constructed. We provide a geometric interpretation of our approach shedding light on how our approach is related to the core reservoir models and in what way the two approaches differ. Empirical experiments then compare predictive performances of our suggested model with those of recent state-of-art transformer based models, as well as the established recurrent network model - LSTM. The experiments are performed on both univariate and multivariate time series and with a variety of prediction horizons. Rather surprisingly we show that even when linear readout is employed, our method has the capacity to outperform transformer models on univariate time series and attain competitive results on multivariate benchmark datasets. We conclude that simple models with easily controllable capacity but capturing enough memory and subsequence structure can outperform potentially over-complicated deep learning models. This does not mean that reservoir motif based models are preferable to other more complex alternatives - rather, when introducing a new complex time series model one should employ as a sanity check simple, but potentially powerful alternatives/baselines such as reservoir models or the models introduced here.

Ziqiang Li, Robert Simon Fong, Kantaro Fujiwara et al.

Reservoir Computing (RC) is a time-efficient computational paradigm derived from Recurrent Neural Networks (RNNs). The Simple Cycle Reservoir (SCR) is an RC model that stands out for its minimalistic design, offering extremely low construction complexity and proven capability of universally approximating time-invariant causal fading memory filters, even in the linear dynamics regime. This paper introduces Multiple Simple Cycle Reservoirs (MSCRs), a multi-reservoir framework that extends Echo State Networks (ESNs) by replacing a single large reservoir with multiple interconnected SCRs. We demonstrate that optimizing MSCR using Particle Swarm Optimization (PSO) outperforms existing multi-reservoir models, achieving competitive predictive performance with a lower-dimensional state space. By modeling interconnections as a weighted Directed Acyclic Graph (DAG), our approach enables flexible, task-specific network topology adaptation. Numerical simulations on three benchmark time-series prediction tasks confirm these advantages over rival algorithms. These findings highlight the potential of MSCR-PSO as a promising framework for optimizing multi-reservoir systems, providing a foundation for further advancements and applications of interconnected SCRs for developing efficient AI devices.