On-Board Real-Time AI Compensation System for Nonlinear Drift in Electrochemical Seismometers

Technical Summary: On-Board Real-Time AI Compensation System for Nonlinear Drift in Electrochemical Seismometers

Problem Statement
Electrochemical seismometers (MET) are emerging as promising next-generation sensors for geophysical exploration due to their high sensitivity, wide bandwidth, and robustness against tilt and impact. However, their practical application is hindered by significant magnitude-dependent nonlinearity. This nonlinearity arises from the complex coupling of mechanical, fluid, and electrochemical processes, leading to frequency response drift (shifts in natural frequency and sensitivity) that degrades seismic data quality and event detection reliability.

Traditional mitigation strategies face distinct limitations:

• Structural Optimization: While MEMS design improvements enhance consistency, they offer limited linearity improvements, typically restricting quasi-linear operation to small input magnitudes (e.g., < 0.13 m/s²).
• Force Feedback: Active feedback can suppress nonlinearity but introduces mechanical complexity and risks closed-loop oscillation at large magnitudes due to reduced phase margins.
• Data-Driven Deep Learning: While accurate, standard deep learning models (e.g., MLPs, RNNs) rely on intensive matrix operations that struggle to meet real-time processing requirements on resource-constrained microcontrollers. Furthermore, purely data-driven approaches often ignore specific physical frequency response characteristics.

Methodology: Wiener-KAN
To address the dual challenges of real-time execution on embedded hardware and the need for physical interpretability, the authors propose Wiener-KAN, a feedforward compensation system designed for on-board real-time execution. The architecture integrates three core components:

1. Wiener Model Decomposition: The system adopts the Wiener model concept, separating the compensation task into a linear dynamic part and a static nonlinear part.

   • Linear Layer (Wiener Linear Layer): Instead of learning dynamics from scratch, this layer is initialized with magnitude-conditioned local transfer functions. These are derived from measured frequency response slices at fixed magnitudes, injecting a physical prior that enables rapid extraction of frequency features.
   • Nonlinear Layer (KAN): A Kolmogorov-Arnold Network (KAN) with trainable spline activation functions handles the remaining magnitude-dependent nonlinearity. To align with the physical behavior of MET sensors, the KAN employs sign and symmetry constraints (odd symmetry and positivity) to ensure the mapping preserves directional consistency and monotonicity.

2. Amplitude-Frequency Loss (AFMAE): Recognizing that standard pointwise losses (MAE/MSE) are insufficient for preserving frequency response characteristics, the authors introduce a custom loss function. AFMAE quantifies the logarithmic amplitude discrepancy between the compensated output and the ideal reference in the frequency domain. This is combined with standard MAE to jointly optimize time-domain waveform fidelity and frequency-domain stability.

3. Real-Time Embedded Deployment (LUT): To overcome the computational bottleneck of evaluating spline functions on microcontrollers, the trained KAN activation functions are precomputed into lookup tables (LUTs) after training. During inference, complex floating-point spline evaluations are replaced by ultra-low-latency memory addressing and table lookups, enabling point-by-point processing on a continuous sampling stream.

Key Contributions

• Wiener-KAN Architecture: A novel hybrid model combining a physics-informed Wiener linear layer with a constrained KAN for nonlinear mapping, specifically tailored for sensor compensation.
• Three-Dimensional Frequency Response Dataset: The authors constructed and released a comprehensive dataset capturing amplitude-frequency coupling effects. It includes 300 operating conditions (12 frequencies × 25 magnitudes) with paired time-domain sequences, recording the transition from small-signal linearity to large-signal nonlinearity.
• AFMAE Loss Function: A specialized loss metric designed to explicitly target frequency response drift and sensitivity stability, rather than just time-domain error.
• Embedded Optimization: A deployment strategy converting spline-based neural networks into lookup tables, achieving high throughput on resource-constrained hardware without sacrificing model expressiveness.

Experimental Results
The system was evaluated on an STM32F405 microcontroller using a dataset covering input magnitudes from 0.24 to 6.0 m/s² (a >10× increase over previous benchmarks).

• Compensation Performance: Wiener-KAN significantly reduced key physical metrics compared to the uncompensated sensor:
  • Frequency Drift: Reduced from 34.50 Hz to 2.34 Hz.
  • Sensitivity Drift (at 100 Hz): Reduced from 84.56 V·s/m to 9.09 V·s/m.
  • Linearity Error: Reduced from 2.004% to 0.511%.
• Comparison with Baselines: Wiener-KAN outperformed various time-series baselines (1DCNN, TCN, RNN, GRU, LSTM, LSTM-Transformer) in both physical compensation accuracy and on-board efficiency. For instance, while TCN achieved similar linearity error, its throughput was significantly lower.
• Real-Time Feasibility: The LUT-based implementation achieved a throughput of 6986.7 points/s with a minimal RAM footprint of 5.6 KB, satisfying the real-time requirements for continuous seismic sampling.

Significance and Claims
The paper claims that Wiener-KAN provides an effective solution for real-time nonlinear compensation of electrochemical seismometers without the need for additional force-feedback hardware. By decomposing the problem into a physically initialized linear stage and a constrained nonlinear stage, the method preserves the interpretability of the sensor's small-signal model while handling analytically difficult large-signal nonlinearity.

The work demonstrates that feedforward software compensation can effectively suppress magnitude-dependent drift, thereby extending the dynamic range and improving the reliability of MET sensors in geophysical exploration. The proposed workflow—linking amplitude-frequency testing, physical priors, constrained KAN mapping, and LUT-based inference—offers a reproducible path for deploying real-time AI compensation on edge devices. The authors note that the results are specific to the tested sensor sample and environmental conditions, suggesting that the workflow is adaptable to other MET structures provided that local frequency responses are remeasured and the reference system is rebuilt.