Optimising complexity and learning for photonic reservoir computing with gain-controlled multimode fibres

Nonlinear photonics is a promising platform for neuromorphic hardware, offering high-speed processing, broad bandwidth, and scalable integration. Within this framework, Reservoir Computing (RC) and Extreme Learning Machines (ELM) are powerful approaches that leverage the dynamics of a complex nonlinear system to process information. In photonics, a key open challenge is controlling the nonlinear response required by photonic RC systems to tailor the photonic substrate (i.e., the physical implementation of the reservoir) to the specific task requirements. In this theoretical work, we propose a nonlinear photonic reservoir based on Erbium-Doped Multi-Mode Fibres (ED-MMF). In our approach, RC is implemented by structuring the pump and probe beams using phase-only spatial light modulators. Thanks to the nonlinear interactions between signal and pump modes within the gain medium, we show how the ED-MMF implements a tunable nonlinear transformation of the input field, where the degree of nonlinear coupling between different fibre modes can be controlled through easily accessible global parameters, such as pump and signal power. The ability to dynamically tune the degree of nonlinearity in our system enables us to identify the best operating conditions for our reservoir system across regression, classification, and time-series prediction tasks. We discuss the physical origin of the optimal regions by analysing the information theory and linear algebra properties of the readout matrix, unveiling a deep connection between the computational performance of the system and the Kolmogorov algorithmic complexity of the nonlinear features generated by the reservoir. Our results pave the way to developing optimised nonlinear photonic reservoirs leveraging structured complexity and controllable nonlinearity as fundamental design principles.