Hysteresis-Based Reservoirs: Memory Mechanisms

• Hysteresis-based reservoirs are systems where history-dependent, nonlinear hysteretic elements encode memory through path-dependent state evolution.
• They are implemented via methods like independent Preisach banks, programmable hysteron networks, and nanoscale devices, supporting both fading and persistent memory regimes.
• Key insights include optimal input range design, noise and sampling challenges, and tuning operating regimes to balance computational efficiency and memory retention.

Searching arXiv for papers on hysteresis-based reservoirs and related hysteretic physical computing. Hysteresis-based reservoirs are systems in which path-dependent internal variables provide the operative memory of the reservoir. In physical reservoir computing, the reservoir state is generated by hysteretic elements, hysterons, or hysteretic device networks driven by external signals, so that the present state cannot be determined from the instantaneous input alone. In porous-media reservoir science, the same phrase also applies to reservoirs whose constitutive behavior is governed by drainage/imbibition hysteresis, capillary entry contrasts, and path-dependent accumulation laws. Across these usages, the common structure is a nonlinear input-state map with memory encoded in branch occupancy, threshold crossings, or internal constitutive variables rather than in an explicitly trained recurrent core (Yamada, 8 Jul 2025, Gershenzon et al., 2015).

1. Defining characteristics

In reservoir computing, the generic state-update and readout structure is written as

$\bm{r}(t) = \bm{F}(\bm{r}(t-1), u(t)),$

with linear readout

$\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$

Hysteresis is attractive in this setting because it naturally provides both history dependence and nonlinearity. In the practical Preisach-bank architecture studied in 2025, hysteresis is explicitly framed as a source of fading memory, with the reservoir state depending on prior extrema of the input through thresholded branch retention rather than through arbitrary delayed interactions (Yamada, 8 Jul 2025).

A distinct but closely related memory regime appears in programmable hysteron networks. In an electronic network of nonlinear resistive elements with tunable negative differential resistivity, the memory is persistent under constant input, history dependent, and distributed across the network as a binary pattern of branch voltages. Two distinct states, 010111010 and 101010101, are both stable at $V_0 = 12~\text{V}$ with current $0.19~\text{mA}$, differing solely because of the prior voltage-ramping history. This establishes that hysteresis-based reservoirs need not be restricted to fading-memory operators; they may also realize persistent multistable memory (Altman et al., 8 Feb 2025).

A recurrent misconception is that hysteresis-based reservoirs must be recurrently coupled dynamical systems. The literature includes both coupled and uncoupled constructions. One simple reservoir is composed of independent hysteretic systems driven in parallel by the same scalar input, whereas other platforms rely on Kirchhoff-law coupling, Coulomb blockade, or electrothermal feedback to produce distributed path dependence. This diversity indicates that hysteresis-based reservoirs are better understood as a class of memory mechanisms than as a single architecture.

2. Canonical reservoir constructions

A deliberately minimal physical reservoir architecture is composed of $M=10$ independent hysteretic systems. Each system is a Preisach-model hysteresis operator built by averaging many elementary hysterons, and the full reservoir is driven by a single scalar input applied in parallel to all elements. For the benchmark tasks, the input is transformed as

$u(t) = -5 + 20u'(t),$

so that the reservoir input lies in $[-5,5]$. The reservoir coordinates are the macroscopic outputs $Y_j(t)$, with design matrix entries

$\Phi_{tj} = Y_j(t),$

and width diversity is introduced through threshold ranges

$$\left[-\frac{d}{2}j,\frac{d}{2}j\right], \qquad j=1,2,\dots,10.$$

This construction is intentionally simple: the elements are independent, only the readout is trained, and the design objective is practical implementation rather than maximal benchmark performance (Yamada, 8 Jul 2025).

A more strongly collective construction is a series network of $\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$0 nonlinear differential resistance elements plus a measurement resistor $\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$1, driven only by a global source voltage $\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$2. Here each edge acts as a hysteron with a low-voltage stable branch, a negative differential resistance region, and a high-voltage stable branch. The observable memory state is the vector of branch voltages $\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$3, equivalently a binary on/off string. Under appropriate threshold ordering,

$\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$4

all binary strings are reachable from saturated states by monotonic switch-on and switch-off ordering. This yields a programmable collective memory substrate written and erased by a global drive alone (Altman et al., 8 Feb 2025).

At the nanoscale, a silicon MOS double quantum dot coupled to a single electron reservoir provides a different hysteretic node. Dot $\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$5 is tunnel-coupled to the reservoir, $\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$6 is not directly coupled to any reservoir, and $\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$7 can exchange electrons with the outside world only indirectly through $\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$8. In the few-electron regime, Coulomb blockade in $\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$9 and strong mutual electrostatic coupling produce history-dependent charge transitions, with mutual coupling $V_0 = 12~\text{V}$0. The device operates as a single-electron Set/Reset memory latch, showing how coupling topology alone can create a hysteretic memory-bearing element (Yang et al., 2014).

3. Mathematical formalisms

The simplest explicit hysteresis model used as a reservoir element in the recent literature is the Preisach hysteron. Each elementary hysteron $V_0 = 12~\text{V}$1 has thresholds $V_0 = 12~\text{V}$2 and $V_0 = 12~\text{V}$3, and output

$V_0 = 12~\text{V}$4

where $V_0 = 12~\text{V}$5 takes the last value when $V_0 = 12~\text{V}$6 is outside the interval $V_0 = 12~\text{V}$7. A macroscopic hysteretic system is then the average

$V_0 = 12~\text{V}$8

In the numerical examples, $V_0 = 12~\text{V}$9, and the reservoir state is the collection $0.19~\text{mA}$0. Readout training is ordinary least squares,

$0.19~\text{mA}$1

with initialization $0.19~\text{mA}$2 because the operator is path dependent (Yamada, 8 Jul 2025).

Programmable hysteron networks are formulated at the constitutive-law level by

$0.19~\text{mA}$3

where $0.19~\text{mA}$4 is the voltage drop across an element. For one nonlinear element in series with a resistor,

$0.19~\text{mA}$5

and the operating point satisfies

$0.19~\text{mA}$6

With parallel capacitors included, node charges satisfy

$0.19~\text{mA}$7

and current mismatch gives

$0.19~\text{mA}$8

These equations make explicit that collective hysteresis can arise from local bistability combined with graph constraints rather than from conventional recurrent neural dynamics (Altman et al., 8 Feb 2025).

Magnetic hysteresis-based reservoirs are often analyzed through continuous-time dynamical models. In the Landau–Lifshitz–Gilbert study of dynamic hysteresis,

$0.19~\text{mA}$9

and the loop area is treated as a function

$M=10$0

The reported empirical behavior is that the area increases with frequency until a peak and then decreases, while remaining proportional to field amplitude. This gives a compact description of how rate dependence tunes hysteretic richness in magnetic substrates (Liu et al., 2016).

4. Task performance and operating regimes

The independent Preisach-bank reservoir has been evaluated on imitation tasks for NARMA-type benchmarks. Training uses $M=10$1 to $M=10$2, and imitation/testing uses $M=10$3 to $M=10$4. For the second-order NARMA system,

$M=10$5

with

$M=10$6

the paper reports a representative result at $M=10$7: linear regression gives $M=10$8, whereas the hysteretic reservoir gives $M=10$9. The main empirical design rule is that the input range must be covered by at least one hysteresis range: imitation is not successful when $u(t) = -5 + 20u'(t),$0, becomes successful for larger $u(t) = -5 + 20u'(t),$1, and once $u(t) = -5 + 20u'(t),$2 is sufficiently large, NMSE varies little with further increases in $u(t) = -5 + 20u'(t),$3 (Yamada, 8 Jul 2025).

The same study also clarifies the computational profile of this simple architecture. Standard NARMA-$u(t) = -5 + 20u'(t),$4 performance is much worse than second-order NARMA, and the paper attributes the degradation to the delayed multiplicative input term

$u(t) = -5 + 20u'(t),$5

When that term is removed in the modified NARMA-$u(t) = -5 + 20u'(t),$6 system and replaced by $u(t) = -5 + 20u'(t),$7, the NMSE becomes significantly smaller and remains approximately below $u(t) = -5 + 20u'(t),$8, even though it increases with $u(t) = -5 + 20u'(t),$9. The authors therefore conclude that independent hysteretic systems provide useful memory, but not the specific delayed-input interaction required by standard NARMA-$[-5,5]$0 (Yamada, 8 Jul 2025).

Dynamic magnetic hysteresis suggests a complementary operating-regime interpretation. The reported two-regime behavior,

$[-5,5]$1

together with the observation that the hysteresis area peaks at an intermediate frequency, suggests that hysteresis-based reservoirs can be tuned between quasi-static, maximally lagged, and overdriven regimes by matching the input timescale to the internal hysteretic timescale (Liu et al., 2016).

5. Related hysteretic substrates and adjacent architectures

Several adjacent device classes are not reservoir computers in the strict sense, but they illuminate the design space of hysteresis-based reservoirs. A prominent example is volatile hysteresis in PrMnO$[-5,5]$2-based RRAM. Under triangular voltage ramps with $[-5,5]$3 mV and dwell times $[-5,5]$4 ns, $[-5,5]$5 ns, $[-5,5]$6 ns, and $[-5,5]$7 ns, the device exhibits volatile hysteresis generated by electrothermal feedback. The paper reports that the voltage window initially increases and later decreases as the ramp rate is increased, while the current window reduces monotonically, and models the behavior with Ohmic plus space-charge-limited current coupled to a Fourier heat equation. This suggests a compact reservoir node with finite-memory dynamics determined by self-heating and thermal relaxation rather than by nonvolatile structural change (Sakhuja et al., 2020).

A different direction is provided by HystRNN, an ordinary-differential-equation-based recurrent neural network for magnetic hysteresis modeling. Its hidden dynamics are written as a second-order system with hysteresis-inspired absolute-value terms,

$[-5,5]$8

$[-5,5]$9

with readout $Y_j(t)$0. The model is trained, not used as a fixed reservoir, but it generalizes from a major loop to previously unseen first-order reversal curves and minor loops better than standard RNN, LSTM, and GRU baselines. This suggests that hysteresis-structured recurrence can be more appropriate than generic recurrent dynamics when the target process is branch-dependent and path-dependent (Chandra et al., 2023).

Hybrid control offers a further contrast. In HyRL, hysteresis appears as a mode-retention mechanism with overlap, implemented by a discrete logic state $Y_j(t)$1. The controller flows with one local policy inside an extended region and switches only after exiting that region. This is not a reservoir architecture, but it is an explicit example of how even one bit of hysteretic internal state can suppress noise-induced chatter and preserve branch consistency in systems with ambiguous decision boundaries (Priester et al., 2022).

6. Hysteresis in porous-media reservoirs

In subsurface and porous-media applications, hysteresis-based reservoirs refer to reservoirs whose multiphase behavior depends on drainage/imbibition history, facies contrasts, and internal constitutive state. A representative example is a deep saline aquifer with multi-scale fluvial stratigraphic heterogeneity, represented as $Y_j(t)$2 sandstone with geometric mean permeability $Y_j(t)$3 mD and $Y_j(t)$4 open framework conglomerate with geometric mean permeability $Y_j(t)$5 mD. Hysteresis is implemented through distinct drainage and imbibition characteristic curves, with $Y_j(t)$6 characteristic curves total, and the Brooks–Corey versus van Genuchten comparison changes only the drainage capillary pressure branch. In the heterogeneous reservoir, total capillary trapping is about $Y_j(t)$7 larger in the van Genuchten case; trapped CO$Y_j(t)$8 in sandstone is more than $Y_j(t)$9 larger; trapped CO$\Phi_{tj} = Y_j(t),$0 in open framework conglomerate is about $\Phi_{tj} = Y_j(t),$1 smaller; dissolved CO$\Phi_{tj} = Y_j(t),$2 differs by $\Phi_{tj} = Y_j(t),$3–$\Phi_{tj} = Y_j(t),$4 depending on time; and the end-of-simulation ratio of mobile CO$\Phi_{tj} = Y_j(t),$5 in Brooks–Corey to van Genuchten is more than $\Phi_{tj} = Y_j(t),$6, with mobile gas negligible for the van Genuchten case near the end. The key point is that hysteresis cannot be separated from small-scale facies heterogeneity and capillary entry-pressure contrasts (Gershenzon et al., 2015).

A more abstract constitutive framework is developed for first-order conservation laws with hysteresis in the accumulation term: $\Phi_{tj} = Y_j(t),$7 Here $\Phi_{tj} = Y_j(t),$8 is represented by generalized play, $\Phi_{tj} = Y_j(t),$9-nonlinear play, or regularized $$\left[-\frac{d}{2}j,\frac{d}{2}j\right], \qquad j=1,2,\dots,10.$$0-Preisach operators, and the local update is reduced to scalar nonlinear solves based on clipping-type resolvents. The paper proves well-posedness and numerical stability for these hysteresis operators, and for first-order conservation laws uses the scheme

$$\left[-\frac{d}{2}j,\frac{d}{2}j\right], \qquad j=1,2,\dots,10.$$1

In this setting, the reservoir state is not a readout feature vector but a field of local internal hysteresis variables that alter shocks, rarefactions, and breakthrough behavior through the conserved quantity $$\left[-\frac{d}{2}j,\frac{d}{2}j\right], \qquad j=1,2,\dots,10.$$2 (Peszynska et al., 2020).

At the grain scale, water-retention hysteresis in partially saturated granular media can be reproduced by a particle-water DEM model in which local contact angles evolve according to bridge-volume changes,

$$\left[-\frac{d}{2}j,\frac{d}{2}j\right], \qquad j=1,2,\dots,10.$$3

For the tested relatively mono-disperse materials, the paper concludes that hysteresis during wetting and drainage arises from the dynamics of solid-liquid contact angles as a function of local liquid volume changes, with correlation coefficients of at least $$\left[-\frac{d}{2}j,\frac{d}{2}j\right], \qquad j=1,2,\dots,10.$$4, and mostly above $$\left[-\frac{d}{2}j,\frac{d}{2}j\right], \qquad j=1,2,\dots,10.$$5. This is a constitutive, pore-scale meaning of hysteresis-based reservoir behavior rather than a computational one (Gan et al., 2013).

7. Limitations, distinctions, and open directions

The recent literature is unusually clear that simple hysteresis reservoirs are not universally expressive. The independent Preisach-bank architecture is intentionally minimal, and its authors explicitly state that “the imitation performance of this reservoir is not high.” They compare it with a more advanced hysteretic reservoir using hysteresis plus multiplexing, for which standard NARMA-10 performance is reported as $$\left[-\frac{d}{2}j,\frac{d}{2}j\right], \qquad j=1,2,\dots,10.$$6 $$\left[-\frac{d}{2}j,\frac{d}{2}j\right], \qquad j=1,2,\dots,10.$$7. They also note that it is “not obvious whether the results in this work can be applied to all actual systems.” Hysteresis therefore supplies useful memory, but width diversity alone does not reproduce the richer internal interactions of recurrent or multiplexed reservoirs (Yamada, 8 Jul 2025).

A second distinction concerns memory type. The programmable hysteron network emphasizes persistent multistable memory, whereas the independent Preisach bank is framed as fading memory. These are not interchangeable regimes. Persistent branch occupancy is advantageous for stateful computation, protocol recognition, and distributed memory patterns, but can hinder continuously driven tasks that require smooth washout of old inputs. Conversely, fading-memory hysteresis is better suited to low-order temporal processing but can struggle when delayed interactions must be recombined explicitly (Altman et al., 8 Feb 2025).

Finally, identification and modeling remain delicate. SINDyHybrid augments sparse system identification with relay hysterons and a proximity hysteron, and in the tank example correctly identifies state transitions of a hysteresis-controlled pump system. Yet the same study reports strong noise sensitivity: SNR $$\left[-\frac{d}{2}j,\frac{d}{2}j\right], \qquad j=1,2,\dots,10.$$8 works properly, SNR $$\left[-\frac{d}{2}j,\frac{d}{2}j\right], \qquad j=1,2,\dots,10.$$9 gives inaccurate slopes, SNR $\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$00 yields faulty transitions and dynamics, and SNR $\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$01 collapses toward an average response. Standard hysterons also fail to detect state transitions for sampling slower than $\hat{y}(t) = \sum_{j=1}^M w_j r_j(t).$02 s, while the proximity hysteron is introduced to improve robustness under coarse sampling. This indicates that hysteresis-based reservoirs are not only a hardware problem but also a state-estimation and identification problem, especially when switching events must be inferred from sampled and noisy measurements (Thiele et al., 2020).