Asymmetric Coupling Anisotropy for Causal Information Filtering in Physical Reservoirs

Takashi Hikihara; Yuma Aoki

nlin.AOAug 27, 2026

Asymmetric Coupling Anisotropy for Causal Information Filtering in Physical Reservoirs

Authors: Takashi Hikihara, Yuma Aoki

Organizations: Kyoto University, Yoshida-honmachi, Sakyo, Kyoto 606-8501, Japan · Department of Electrical Engineering, Kyoto University, Kyotodaigaku-katsura, Nishikyo, Kyoto 615-8510, Japan

Abstract

We demonstrate a physical mechanism for causal information filtering in a physical reservoir computing (PRC) by exploiting asymmetric coupling anisotropy. Using a network of coupled Duffing oscillators, we show that the directionality of internal coupling induces a spatial gradient in the effective potential, establishing a deterministic upstream-to-downstream information flow. This anisotropy allows for the selective amplification of semantic drifts, triggering a macroscopic saddle-node bifurcation as a physical interlock before global computational failure. Through spatiotemporal analysis of a 50-node system under traveling wave inputs, we confirm that local phase transitions effectively purge anomalous information while preserving the computational integrity of the remaining nodes. The results suggest that the intrinsic causality of the reservoir's topology provides a robust framework for autonomous reliability and fault-tolerant physical intelligence.

Figures & tables

Panel of Figure 1 — Figure 1: Physical mechanism of causal information filtering in an asymmetric bidirectional ring topology. (a) Schematic of the distributed monitoring unit. Three Duffing resonators are coupled in a ring with significant anisotropy, where forward coupling $k_{\rm fwd}$ (thick arrows) is dominant over reverse coupling $k_{\rm rev}$ (thin arrows). This topology establishes a directional information flow that enables the cumulative amplification of semantic drifts while physically offsetting common-mode environmental noise. (b) Evolution of the effective potential $U_{\rm eff}$ and deterministic symmetry breaking. As the cumulative bias $B_{\rm eff}$ increases from 0 to 0.45, the symmetric double-well structure of the resonator collapses. At the critical point $B_{c}\approx 0.35$ (solid red line), the stability of the computational state is lost, triggering a macroscopic saddle-node bifurcation (jump phenomenon). This phase transition serves as a deterministic physical interlock, purging anomalous information before global computational failure.

Figure 2: Detection robustness under environmental drift. (a) Comparative analysis between the conventional unidirectional ladder topology, which is susceptible to drift, and the proposed asymmetrical ring topology, which physically cancels common-mode noise through differential monitoring. (b) ROC curve evaluation demonstrating the superiority of the ring coupling, achieving a high discrimination accuracy ( $\text{AUC}=0.77$ ) even in high-noise environments, whereas the ladder topology fails to discriminate anomalies ( $\text{AUC}=0.53$ ).

Parameter	Symbol	Value	Physical Significance
Number of nodes	$N$	5	Network configuration (Includes 3 ring nodes
			and 2 external nodes)
Damping coefficient	$\gamma$	0.0035	Energy dissipation (Memory retention property)
Nonlinear coefficient	$\beta$	3.23	Restoring force nonlinearity (Source of computation)
Driving frequency	$\omega_{\text{drv}}$	1.10	Angular frequency of the external carrier wave
Bias intensity	$\varepsilon_{\text{bias}}$	0.01	Operating point offset intensity

Table 1: Simulation parameters for the physical reservoir (Duffing oscillators).

Parameter	Symbol	Value	Physical Significance
Forward coupling constant	$k_{\text{fwd}}$	0.08	Propagation of primary information (Strong coupling)
Reverse coupling constant	$k_{\text{rev}}$	0.008	Synchronization and error cancellation (Weak coupling)
Asymmetry ratio	$k_{\text{fwd}}/k_{\text{rev}}$	10	Condition for balancing stability and robustness
Chain coupling constant	$k_{\text{base}}$	0.01	Isotropic baseline coupling outside monitor units

Table 2: Parameters for internal asymmetrical bidirectional ring coupling.

Parameter	Symbol	Value	Physical Significance
Regularization coefficient	$\lambda$	$10^{-3}\text{--}10^{-4}$	Overfitting suppression (Ridge regression)
Training samples	$N_{\text{train}}$	350	Number of samples for weight determination
Anomaly threshold	$V_{\text{th}}$	0.4	Decision boundary for deviation signal $\|\Delta x\|$
Anomaly intensity	$\alpha$	0.015 / 0.03	Disturbance intensity (0.015 for Sec. III, 0.03 for Secs. IV–V)
Self-repair start point	$k_{\text{repair}}$	750–1100	Completion of re-learning (1100 for Sec. III, 750/800 for Secs. IV–V)

Table 3: Parameters for the learning and signal processing algorithm.

Item	Symbol	Value	Physical Significance
Total number of nodes	$N$	5 / 50	Computational resources (5 for Sec. III, 50 for Secs. IV–V)
Ring-configured nodes	$N_{\text{ring}}$	3	Minimum unit of the feedback loop via asymmetric coupling
Input injection node	$\text{Node}_{0}$	1	Injection point for external signal $u(t)$ and anomaly $\alpha$
Output extraction nodes	$\text{Nodes }0\text{--}4$	5	All state variables sent to the readout layer (Ridge regression)
Topology	-	Ring	Asymmetrical bidirectional coupling including a closed loop

Table 4: Details of network configuration and coupling topology.

Figure 3: Intelligent interlock mechanism for anomaly classification. The deviation signal $\Delta x$ is processed in parallel through differential (high-pass) and integral (low-pass) paths to identify abrupt failures and gradual degradations, respectively. This dual-path architecture enables autonomous decision-making tailored to the specific type of physical anomaly.

Figure 4: Self-healing cycle via autonomous re-learning. Following the cessation of an anomaly, the autonomous return of the physical layer to a stable equilibrium state triggers the re-learning process. After the repair phase (indicated by the green line), the mean squared error (MSE) is instantaneously reduced by approximately two orders of magnitude, confirming the effective restoration and enhancement of computational accuracy.

Figure 5: Architecture of the 50-node physical reservoir array with asymmetric coupling and differential monitoring. (a) 1D ring topology consisting of 50 coupled Duffing resonators with strong forward coupling $k_{\text{fwd}}$ and weak reverse coupling $k_{\text{rev}}$ ( $10:1$ ratio). Local differential readout circuitry is attached to distributed 3-node monitoring units (Units 1–3). Node 1 serves as the reference input, and a localized fault is introduced at Node 23. (b) Comparative symmetrical 2-rail ladder topology with isotropic coupling ( $k_{\text{rail}}=k_{\text{rung}}=k_{\text{iso}}=0.044$ ) used for noise cancellation benchmark.

Figure 6: Spatiotemporal dynamics and distributed anomaly detection under static (standing wave) input. The upper panel displays the spatiotemporal evolution of nodal displacements $x_{i}$ across the 50-node reservoir, showing localized state distortion when a bias anomaly is injected at Node 23 ( $k=400\text{--}800$ ). Horizontal dashed lines indicate the monitoring unit centers (Units 1–3). The lower panel shows the absolute differential offset $|\Delta x_{\text{monitor}}|$ captured by the three monitoring units. Only Unit 2 (downstream of Node 23) exhibits a sharp detection peak, confirming precise fault localization.

Figure 7: Spatiotemporal dynamics and monitoring signals under traveling wave inputs. The upper panel displays the displacement offset across 50 nodes, where a spatial gradient of information flow is established. Green dashed lines indicate the positions of the monitoring units. The lower panel shows the absolute difference signals $|\Delta x_{\text{monitor}}|$ at these units. The monitoring unit downstream of the anomaly node captures a significant signal deviation, demonstrating the causal filtering capability of the asymmetric coupling topology.

Figure 8: Spatiotemporal analysis of continuous anomaly management and block-wise self-healing. The upper panel shows the reservoir dynamics under sustained failure, while the lower panel demonstrates the restoration of computational accuracy (MSE, log scale) through dynamic reconfiguration and re-learning.

Figure 9: Autonomous fault detection and functional recovery cycle under chaotic Mackey-Glass (MG-17) time-series prediction task. (top panel) Spatiotemporal evolution of nodal states $x_{i}$ in the 50-node reservoir. Horizontal dashed lines represent distributed monitoring units (Units 1–3, in cyan, magenta, and dark orange). A physical fault (node stall $x_{23}=0$ ) is injected at $k=400$ (yellow dashed line) and cleared at $k=800$ (green solid line). (second panel) Differential monitoring signals $|\Delta x_{\text{monitor}}|$ . Unit 2 (magenta) captures a prominent signal spike during the fault, localizing the failure. (third panel) Comparison between chaotic target and readout output $y$ , featuring two insets highlighting prediction accuracy before repair ( $k=700\text{--}800$ ) and after self-healing ( $k=950\text{--}1050$ ). (bottom panel) Evolution of MSE on a log scale. Following the trigger at $k=800$ , re-learning rapidly drops the MSE from $10^{-3}$ to below $10^{-5}$ in the healed period (green region), quantitatively confirming complete functional recovery.

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