Importance of reservoir architectures in comparing physical reservoir systems

G. Venkat¹, I. Vidamour¹, C. Swindells¹, T. J. Hayward¹, P. W. Fry², M. Foerster³, M. A. Niño³, M. C. Rosamond⁴, D. A. Allwood¹

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¹Department of Materials Science and Engineering, University of Sheffield, Sheffield, S1 3JD, UK

²Nanoscience and Technology Centre, University of Sheffield, Sheffield, S3 7HQ, UK

³ALBA Synchrotron Light Facility, Carrer de la Llum 2-26, Cerdanyola del Vallés, 08290 Barcelona, Spain

⁴School of Electronic and Electrical Engineering, University of Leeds, Leeds, LS2 9JT, UK

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Take home: Choice of reservoir architecture is important to exploit physical reservoir dynamics!

Talk Outline

• Reservoir computing with physics
• Magnetic states in rings and arrays of rings
• Measurements of magnetic nanostructures
• Lattice geometry variation in ring arrays
• Dynamic responses of different lattices
• Timescales of dynamics
• What happens in the arrays microscopically?
• Measuring reservoir metrics in arrays
• Task independent metrics
• Different reservoir architectures
• How do we measure these metrics?
• Computational quality and memory capacity of different lattices
• Conclusions

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Reservoir Computing with Physics

• Dynamic systems functionally analogous to reservoir layer
• Can substitute reservoir layer for dynamic system
• Paradigm requires method for injecting and extracting data
• Memory and Computation occur in tandem as inherent property of material

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Magnetic states in single rings and junctions

• Magnetisation likes to lie around the ring
• Domain wall – change in magnetic configuration
• Onion state – Two domain walls at opposite ends
• Vortex state – No domain walls

• Junction allows domain wall interactions
• Low field regime – Walls are pinned
• High field regime – Walls are mobile
• Intermediate regime
• Walls show stochastic pinning and depinning behaviour
• Leads to nucleation and de-nucleation of domain walls

Interconnected ring arrays

• Ring arrays fabricated using nanofabrication techniques
• System perturbed using rotating magnetic fields

• Highly non-linear response to rotating field amplitude
• Intermediate regime where a combination of magnetic states stochastically exist

• Magnetic states show markedly different evolutions at successive field amplitudes
• Washing out of evolution indicates fading memory

Dawidek et al, Adv. Funct. Mater. 2021 2008389.

Measurements of magnetic nanostructures

• Magneto-optical Kerr effect (MOKE) magnetometry [measurement of local dynamic magnetisation]

• X-ray photoelectron emission microscopy (X-PEEM)[imaging local magnetic configuration]

• Anisotropic magnetoresistance (AMR)[measurement of domain wall transitions – used for computation]

How can we vary the behaviours of ring arrays?

Lattice arrangement of arrays

• Periodic arrangement of rings in the array
• Number of nearest neighbours change
• Square – Four neighbours
• Trigonal – Six neighbours
• Kagome – Three neighbours

• Significant differences in stimuli response expected as stochastic dynamics affected junction configurations

J. Phys.: Condens. Matter 19 (2007) 140301

Dynamic responses of different lattices

• How do we find the dynamic response?
• Apply a pulsed magnetic field and set the ground state
• Apply rotating magnetic fields and measure the magnetic response
• Monitor the amplitude of signal for different field amplitudes

• Low and high field responses of the lattices are similar in behaviour

• Intermediate field regime shows significant differences in behaviour

What happens to timescales in dynamics?

• Probe the temporal variation at various fields
• Starts off with a low amplitude and constant envelope
• Slower settling envelope at intermediate fields
• High amplitude waveform and fast settling envelope

• Increase in settling times at intermediate fields indicative of system ability to process time-varying inputs

• Different decay in settling time for trigonal compared to square indicates different abilities for temporal signal processing

8 cycles/Oe

13 cycles/Oe

What happens in the arrays microscopically?

• Start: Onion states after saturation
• Some domain walls start moving and three-quarter states start forming
• Then vortex pairs start forming (low energy states) – Less number of vortex states

• Finally it is propagating domain walls which look like onion states in an image

• Pinned -> Stochastic -> Propagating states in the array lead to macroscopic response

• PEEM images obtained after setting ground state and 30 rotations of field

Square

What happens in the arrays microscopically?

• Start: Onion states after saturation
• Some domain walls start moving and three-quarter states start forming
• Much larger number of vortex pairs form – leads to lower magnetisation
• Extended field regime for vortex pairs

• Significantly varied responses from the different lattice arrangements

Kagome

Message 1: Varying the lattice arrangements of rings varies behaviour significantly!

How do we compare their ”computing” performance?

Signal Sub-sample Reservoir

• Single rotation of magnetic field per input, scaled linearly
• Sample evolving time signal at fixed intervals
• Interplay of different frequency signals leads to nonlinearly varying nodes
• Simple method providing dimensional expansion with minimal pre/post processing

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Single Dynamical Node Reservoir

• Single dynamical node approach
• Inputs combined with fixed mask
• Masked input fed into reservoir
• Reservoir state evaluated over time
• Simple reservoir construction
• Commonly used for dynamic system implementation

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Appeltant, L., Soriano, M., Van der Sande, G. et al. Information processing using a single dynamical node as complex system. Nat Commun 2, 468 (2011). https://doi.org/10.1038/ncomms1476

Rotating Neurons Reservoir

• Rotating Neurons Reservoir introduced by Liang et al. (2022)
• Distinct dynamical nodes with revolving input mask
• Leverages non-volatility of system state
• Negative mask input -> no change of magnetic state
• Leads to shift-register like behaviour

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Task independent metrics

• Kernel rank (KR) – Test ability of reservoir to map distinct inputs to separate states [1]
• Input matrix – 100 sequences of 10 randomly sampled data points from a uniform distribution between ±1 – Uncorrelated sequences with linear dependence
• Generalisation rank (GR) – Test ability of reservoir to map noisy version of same inputs to similar reservoir states [1]
• Input same as for KR but last 3 floating points in each sequence same as 1^st three points in 1^st sequence
• Computational Quality (CQ=KR-GR) monitored as higher KR and lower GR is desirable [2]
• Fixed mask consisting of 50 random floating points sampled from a uniform distribution between ±1; rank of output matrix provides metric

[1] Dale, M. et al. Proc. Math. Phys. Eng 475, (2019)

[2] Jensen, J. H., et al. in ALIFE 2018 - 2018 Conference on Artificial Life: Beyond AI 15–22 (MIT Press - Journals, 2020).

How do we measure these metrics?

• Modulate applied field strength according to input

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• Measure evolving AMR response

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• Extract features from different frequency components of signals

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• Repeat for different input encodings and generate metric maps

Computational Quality

• Sub-sampling: Similar regions of operation and CQ for the three lattices
• Single dynamical node: Different regions of operation and CQ for three lattices
• Might indicate differences in computational performance

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Sub-Sampling

Single dynamical node

Memory capacity

• Sub-sampling: Similar regions of operation and MC for the three lattices
• Revolving neurons: Different regions of operation and similar MC values for the three lattices
• Revolving neurons architecture sensitive to 1^st transition of dynamical response
• Similar MC for the different lattices indicates architecture has prominent effect

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Sub-Sampling

Revolving neurons

Conclusion: Reservoir architecture might be dominating physical reservoir dynamics in measurements of metrics

Message 2: Choice of reservoir architecture is important to exploit physical reservoir dynamics!

Backup slides

Importance of reservoir architectures in comparing physical reservoir systems

Guru Venkat, Ian T Vidamour, Charles Swindells, Paul W Fry, M.C. Rosamond, M. Foerster, M. A. Niño, Dan A Allwood, Tom J Hayward.

Lattice geometry variation of arrays

• Periodic arrangement of rings in the array
• Number of nearest neighbours change
• Square – Four neighbours
• Trigonal – Six neighbours
• Kagome – Three neighbours

• Ground state – Magnetic saturation in some direction and then allow it relax

• Kagome array ground state– domain walls closer at some junctions

• Trigonal array ground state– a domain wall chain forms at the junctions

Device/System Setup

• Layer 1 - Array of 25x25 permalloy nanorings
• Layer 2 - Electrical contacts for transport measurements
• Electrical contacts wire-bonded to chip carrier
• 4 Electromagnets provide rotating magnetic field

Measuring Array Response

• Magnetic response measured via AMR
• 2 mechanisms leading to resistance change
• DWs propagating around rings → 2 x clock frequency
• Stretching of pinned DWs → 1 x clock frequency
• Sub-signals have different nonlinear relationships to field

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Functional Behaviours

• Interplay of 1f and 2f signals leads to complex time signals
• Non-volatile system state under low driving field
• Range of settling times to reach dynamic equilibrium

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How do we extract these behaviours?

• Changing driving field ranges leads to different dynamic responses
• Method for encoding/extracting data drastically changes response
• Tailor the reservoir configuration to leverage specific device properties for different computational demands

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Single Dynamical Node Reservoir

• Modulate applied field strength according to input

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• Measure evolving AMR response

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• Split AMR response into 1f and 2f components

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• Extract features from envelope of signals

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Demonstration 3- Linear and Nonlinear Memory

• Linear memory capacity (MC) and NARMA-10
• Tasked with reproducing linear and nonlinear representations of past inputs from current reservoir states
• MC curve shows correlation between reconstruction and true past inputs
• NARMA-10 expressed as reconstruction error -> Shift register NMSE ≈ 0.4

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