Beyond density snapshots: Dynamical memory in Chern-Simons reservoirs

Jyotiranjan Beuria; Venkatesh H. Chembrolu

cs.ETSep 10, 2026

Beyond density snapshots: Dynamical memory in Chern-Simons reservoirs

Authors: Jyotiranjan Beuria, Venkatesh H. Chembrolu

Organizations: Center for Philosophical and Cognitive Sciences, ISS Delhi, India · Indian Institute of Technology Mandi, Mandi, India

Abstract

How a reservoir encodes input history jointly in density and geometric degrees of freedom remains insufficiently understood. We address this mechanistic gap with a continuous density on an internal manifold coupled to an independently evolving Chern-Simons (CS) connection through a conserved current. On the same internal manifold, we compare reciprocal transport, instantaneous transverse reconstruction, local nonlinear feedback, and coupled CS evolution. These controls isolate how transport and independent connection evolution contribute to memory. We identify a current-generated exact connection component that the tested instantaneous reconstruction and scalar local closure miss. Resetting this component at fixed density and Gauss-law curl changes the future density response within the chosen gauge and drift law and the small effect persists under numerical refinement. We then distinguish dynamical memory from its readout accessibility. Although the four models have similar scalar-memory scores and density traces of pulse order, their circulating and longitudinal traces differ. With all measured features, coupled CS improves pulse-order decoding by $6.5\pm1.9$ percentage points over the instantaneous-transverse control in paired seedwise comparisons, while remaining comparable to local feedback. Coupled CS shows less lag than the instantaneous and local-feedback controls on the first cycle, but more on subsequent cycles, revealing a dependence on preparation. We thus connect a field-theoretic mechanism of history retention to controlled tests of its dynamical role and computational use. The four-model framework provides a way to determine what an evolving transport field contributes when density snapshots and aggregate performance alone leave the memory mechanism unresolved.

Figures & tables

Panel of Figure 1 — Figure 1: Study design and controlled model hierarchy. We use the same spatial domain, input protocol, numerical scheme, initial conditions, and seeds in all four models, and vary how density couples to the velocity or gauge connection. We test scalar memory, pulse-order dependence, and cyclic-drive lag with the reservoir-computing protocols. With the gauge-dynamics protocols, we test Chern-Simons consistency, snapshot reconstruction, connection reset, and feedback-dependent cross-response. We then separate properties shared by the field reservoir family from effects that require an independently evolving gauge connection.

Figure 2: Reconstruction and closure at

t_{f}\simeq 4

. (a) Mean

E_{a}(t)

with SEM band at

\alpha=0.70

. (b) Mean

E_{v}

for instantaneous transverse and fitted-local reconstruction, with SEM. (c) Their paired percentage difference (mean

\pm

SEM).

Figure 3: CS-equation residuals on the

16^{2}

grid with

\Delta t=0.015

and

t_{f}\simeq 4

. (a) Mean Gauss residual

R_{G}

. (b) Mean seedwise time-median spatial residual

R_{S}

Diagnostic	Result
Maximum $R_{G}$ , main run	$4.42\times 10^{-14}$
Coupled spatial residual, $\Delta t$	$1.07\times 10^{-2}$
Coupled spatial residual, $\Delta t/2$	$5.30\times 10^{-3}$
$\left\lVert\rho\right\rVert_{2}$ , $N=12/16/24$	$0.261882/0.261881/0.261881$
$\left\lVert\rho\right\rVert_{2}$ , $\Delta t/2$	$0.263887$
$\left\lVert\rho\right\rVert_{2}$ , three noise seeds	$0.261844$ to $0.261888$

Table 1: Numerical validation of coupled CS. Unless indicated otherwise, values use the

16^{2}

grid and

\Delta t=0.015

. The two spatial-residual convergence runs have nominal duration 2.

Observable	model	$A_{\rm p}=0.50$	$A_{\rm p}=0.75$	$A_{\rm p}=1.00$	$A_{\rm p}=1.25$
density contrast $\Delta_{\rho}$ (%)	reciprocal	$0.319\pm 0.210$	$0.369\pm 0.206$	$0.437\pm 0.201$	$0.523\pm 0.197$
	inst. transverse	$-0.010\pm 0.001$	$-0.014\pm 0.002$	$-0.018\pm 0.002$	$-0.024\pm 0.002$
	local feedback	$-0.011\pm 0.002$	$-0.014\pm 0.002$	$-0.018\pm 0.002$	$-0.024\pm 0.003$
	coupled CS	$0$ (reference)	$0$ (reference)	$0$ (reference)	$0$ (reference)
circulation contrast $\Delta_{\omega}$ (%)	reciprocal	$-100.000$ ( $D_{\omega}\simeq 0$ )	$-100.000$ ( $D_{\omega}\simeq 0$ )	$-100.000$ ( $D_{\omega}\simeq 0$ )	$-100.000$ ( $D_{\omega}\simeq 0$ )
	inst. transverse	$-0.009\pm 0.002$	$-0.011\pm 0.002$	$-0.014\pm 0.002$	$-0.017\pm 0.002$

Table 2: Endpoint pulse-order contrasts versus amplitude, after

R=12

zero-input steps. For each seed and

A_{\rm p}

, we average raw distances over

W\in\{8,12,16\}

before forming contrasts relative to coupled CS. Entries are ten-seed means

\pm

SEM; coupled CS is the exact reference.

Panel of Figure 4 — Figure 4: Held-out readout diagnostics. (a) Scalar-memory coefficient $MC(d)$ for continuous delayed-input regression; curves and bands show the ten-seed mean and SEM, and legend values give $C=\sum_{d}MC(d)$ . (b) Binary pulse-order classification accuracy for the five feature families. Cells give the mean $\pm$ SEM over 50 outer folds, with ridge penalties selected by nested seed-block validation; chance accuracy is $0.5$ .

Observable	model	$t_{\rm rel}=0$	$t_{\rm rel}=0.24$	$t_{\rm rel}=0.48$	$t_{\rm rel}=0.72$
density contrast $\Delta_{\rho}$ (%)	reciprocal	$0.552\pm 0.208$	$0.404\pm 0.200$	$0.299\pm 0.194$	$0.222\pm 0.189$
	inst. transverse	$-0.010\pm 9.89\times 10^{-4}$	$-0.021\pm 0.002$	$-0.030\pm 0.003$	$-0.039\pm 0.004$
	local feedback	$-0.010\pm 0.002$	$-0.021\pm 0.003$	$-0.030\pm 0.003$	$-0.039\pm 0.004$
	coupled CS	$0$ (reference)	$0$ (reference)	$0$ (reference)	$0$ (reference)
circulation contrast $\Delta_{\omega}$ (%)	reciprocal	$-100.000$ ( $D_{\omega}\simeq 0$ )	$-100.000$ ( $D_{\omega}\simeq 0$ )	$-100.000$ ( $D_{\omega}\simeq 0$ )	$-100.000$ ( $D_{\omega}\simeq 0$ )
	inst. transverse	$-0.008\pm 0.001$	$-0.016\pm 0.002$	$-0.023\pm 0.003$	$-0.029\pm 0.004$

Table 3: Post-drive pulse-order contrasts. Here

t_{\rm rel}=0

is immediately after the second pulse; the endpoint in Table 2 is at

t_{\rm rel}=0.18

. For each seed and time, we average raw distances over

W

and

A_{\rm p}

before forming contrasts and ratios relative to coupled CS. Entries are ten-seed means

\pm

SEM; coupled CS is the exact reference.

Figure 5: First-cycle density lag without pre-equilibration. (a) Mean forward (solid) and backward (dashed) branches at

r_{s}=0.2083

; bands show SEM. We show only the reciprocal and coupled-CS branches because the instantaneous transverse controls overlap visually. (b) Loop area

A_{\rm cyc}

versus sweep rate. (c) Mean loop area with SEM error bars at the representative rate.

Figure 6: Incremental feedback response. (a) Background-subtracted excess

\Delta R_{\chi}

; the dashed line gives the uncorrected

\alpha=0

scale. (b) Raw asymmetry

R_{\chi}

for coupled CS and the instantaneous baseline.

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Organization of computation in reservoir computing

Mohab Abdalla, Damien Rontani

CentraleSupélec, Université de Lorraine, LMOPS EA-4423 Laboratory, 57070 Metz, France

Reservoir Computing

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Reservoir computing exploits nonlinear dynamical systems to encode temporal inputs into high-dimensional state space representations. Although reservoir performance is often characterized through memory, nonlinearity, and their tradeoff, such aggregate measures do not reveal how task-relevant information is organized within the state space. Here, we introduce an eigen-spectral decomposition framework linking the degree-wise information processing capacity to the corresponding state space modes. As a result, we are able to quantify the degree-wise representation energy, and show that in some cases, substantial amounts of information processing capacity may reside in low-energy modes that are vulnerable to experimental noise. These results suggest that useful reservoir computation depends not only on dimensionality expansion, but also on the geometric organization of task-relevant information, with direct implications on physical reservoir computers.

Abstract

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