The goal of this paper is to develop an optimization framework for tuning the topology of a linear reservoir computer, making it competitive with its nonlinear counterpart while enhancing the interpretability of reservoir computing systems.
The governing equation for a conventional decoupled reservoir computer is: (See the paper for details on how it is computed based on a conventional coupled reservoir computer)
where q is the decoupled reservoir state, Λ is a diagonal matrix of eigenvalues, c is the input weight, and γ is a positive constant.
With the input and output frequencies of:
where a and b are the amplitudes of the input and output signals, respectively. ω is the frequency of the input signal. ϕ is the phase shift in the output signal.
By solving Eq. 9, the reservoir's state q can be expressed as in Eq. 14.
M and θ are according to Eq. 15
The reservoir state matrix in the frequency domain (Eq. 17) will be computed after taking the Fourier transformation of Eq. 14 and equating it to the Fourier transformation output signal. (To see details, check the paper)
The following optimization formulation, based on Eq. 17, is developed to determine the optimal eigenvalues and output weights.
Results of the reservoir computer before and after optimization for a reservoir with 10 nodes and three distinct input frequencies:
Before running the optimization code, please ensure that Julia is installed and the following packages are added: DelimitedFiles, JuMP, Ipopt, LinearAlgebra, MathOptInterface, and Distributions.
You can add these packages by running the following commands in the Julia REPL:
using Pkg
Pkg.add("JuMP")
Pkg.add("Ipopt")
Pkg.add("LinearAlgebra")
Pkg.add("Distributions")
Pkg.add("MathOptInterface")
Pkg.add("DelimitedFiles")
(The code is compatible with MATLAB version 2022a and later.)
1- To run the code, open "runapp.mlapp" in MATLAB.
2- Run the code from the Designer tab.
3- Replace the parameters with your own values, then click the "Initialize" button.
4- Fill the new empty fields with the parameters of your signals.
5- Click the "Run" button to start the optimization.
If you find our repo or paper useful, please cite us as follows:
Tangerami, S., Mecholsky, N. A., & Sorrentino, F. (2025). Optimizing the Network Topology of a Linear Reservoir Computer. arXiv:2509.23391. https://arxiv.org/abs/2509.23391