Analog Kolmogorov-Arnold Networks for Efficient Function Approximation in Flexible Electronics

1. Introduction

This paper presents a novel, systematic methodology for function approximation in Flexible Electronics (FE) using analog implementations of Kolmogorov-Arnold Networks (KANs). The core challenge addressed is the inherent trade-off in FE between computational capability and strict constraints on physical size, power budget, and manufacturing cost. Traditional digital approaches become prohibitively expensive in area and power for FE applications like wearables and IoT sensors. The proposed solution leverages a library of Analog Building Blocks (ABBs) to construct spline-based KANs, offering a generic and hardware-efficient pathway for embedding intelligent, near-sensor processing directly onto flexible substrates.

2. Background & Motivation

2.1 Flexible Electronics Constraints

Flexible Electronics, often based on materials like Indium Gallium Zinc Oxide (IGZO), enable novel form factors for wearables, medical patches, and environmental sensors. However, they suffer from larger feature sizes compared to silicon CMOS, making complex digital circuits area-inefficient. Furthermore, applications demand ultra-low power consumption for extended battery life or energy harvesting compatibility. This creates a pressing need for computational paradigms that are inherently frugal in hardware resources.

2.2 Kolmogorov-Arnold Networks (KANs)

KANs, recently revitalized by Liu et al. (2024), offer a compelling alternative to traditional Multi-Layer Perceptrons (MLPs). Instead of fixed activation functions on nodes, KANs place learnable univariate functions (typically splines) on the edges (weights) of the network. The Kolmogorov-Arnold representation theorem underpins this, stating that any multivariate continuous function can be represented as a finite composition of continuous functions of a single variable and addition. This structure is naturally amenable to efficient analog implementation, as complex functions are broken down into simpler, composable operations.

3. Proposed Analog KAN Architecture

3.1 Analog Building Blocks (ABBs)

The foundation of the approach is a set of pre-characterized, low-power analog circuits that perform fundamental mathematical operations: Addition, Multiplication, and Squaring. These blocks are designed considering FE process variations and parasitics. Their modular nature allows for systematic composition.

3.2 Spline Construction with ABBs

Each learnable univariate function in the KAN (a spline) is constructed by combining ABBs. A spline, defined by piecewise polynomials between knots, can be implemented by selectively activating and summing the outputs of multiplier and squarer blocks configured with polynomial coefficients. This analog spline replaces a digital Look-Up Table (LUT) or arithmetic unit, saving significant area.

3.3 KAN Network Assembly

A complete KAN layer is assembled by connecting the input variables to a bank of analog spline blocks (one per edge/weight). The outputs of splines converging on the same node are summed using addition ABBs. This process is repeated to build the network depth. The parameters (spline coefficients) are determined offline via training and then hardwired into the analog circuit biases and gains.

4. Technical Implementation & Details

4.1 Mathematical Formulation

The core of a KAN layer transforms an input vector $\mathbf{x} \in \mathbb{R}^n$ to an output vector $\mathbf{y} \in \mathbb{R}^m$ through learnable univariate functions $\Phi_{q,p}$: $$\mathbf{y} = \left( y_1, y_2, ..., y_m \right)$$ $$y_q = \sum_{p=1}^{n} \Phi_{q,p}(x_p), \quad q = 1,...,m$$ In the analog implementation, each $\Phi_{q,p}(\cdot)$ is a physical spline circuit. The summation is performed by a current-mode or voltage-mode adder ABB.

4.2 Circuit Design & Parasitics

The multiplier ABB can be based on a Gilbert cell or a translinear principle for low-voltage operation. The squarer can be derived from a multiplier with tied inputs. Key non-idealities include: transistor mismatch ($\sigma_V_T$), which affects coefficient accuracy; finite output impedance, causing loading errors; and parasitic capacitances, limiting bandwidth. These factors collectively contribute to the measured approximation error.

5. Experimental Results & Analysis

5.1 Hardware Efficiency Metrics

The proposed analog KAN was benchmarked against an equivalent digital spline implementation with 8-bit precision in a FE-compatible process. The results are striking:

• Area: 125x reduction. The analog design eliminates large digital registers, multipliers, and memory for LUTs.
• Power: 10.59% saving. Analog computation avoids the high dynamic power of clocking and switching digital circuits.

5.2 Approximation Error Analysis

The trade-off for hardware efficiency is computational precision. The system introduces a maximum approximation error of 7.58%. This error stems from two main sources:

1. Design Error: The inherent error from using a finite number of spline pieces to approximate the target function.
2. Parasitic Error: Errors introduced by analog non-idealities (mismatch, noise, parasitics) in the ABBs.

6. Case Study & Framework Example

Scenario: Implementing a lightweight anomaly detector for a flexible heart-rate monitor. The device needs to compute a simple health index $H$ from two inputs: heart rate variability (HRV) $x_1$ and pulse waveform skewness $x_2$. A known empirical relationship $H = f(x_1, x_2)$ exists but is non-linear.

Framework Application:

1. Function Decomposition: Using the proposed framework, $f(x_1, x_2)$ is approximated by a 2-layer KAN with structure [2, 3, 1]. The network is trained offline on a dataset.
2. ABB Mapping: The trained univariate functions (splines) on the 6 edges of the first layer and 3 edges of the second layer are mapped to polynomial coefficients.
3. Circuit Instantiation: For each spline, the required number of piecewise polynomial segments is determined. The corresponding multiplier and squarer ABBs are configured with the coefficients (as bias voltages/currents) and interconnected with adder ABBs according to the KAN graph.
4. Deployment: This analog KAN circuit is fabricated directly on the flexible patch. It continuously consumes micro-watts of power, processing sensor data in real-time to flag anomalies without digitization or wireless transmission of raw data.

7. Application Outlook & Future Directions

Near-term Applications:

• Smart Biomedical Patches: On-patch signal processing for ECG, EEG, or EMG, enabling local feature extraction (e.g., QRS detection) before data transmission.
• Environmental Sensor Hubs: In-situ calibration and data fusion for temperature, humidity, and gas sensors in IoT nodes.
• Wearable Gesture Recognition: Ultra-low-power preprocessing of data from flexible strain or pressure sensor arrays.

1. Error-Resilient Training: Developing training algorithms that co-optimize the KAN parameters for both accuracy and robustness to analog circuit non-idealities (akin to hardware-aware neural network training).
2. Adaptive & Reconfigurable ABBs: Exploring circuits where spline coefficients can be slightly tuned post-fabrication to compensate for process variations or to adapt to different tasks.
3. Integration with Sensing: Designing ABBs that directly interface with specific sensor types (e.g., photodiodes, piezoresistive elements), moving towards true analog sensor-processor fusion.
4. Scalability to Deeper Networks: Investigating architectural techniques and circuit designs to manage noise and error accumulation in deeper analog KANs for more complex tasks.

8. References

1. Z. Liu et al., "KAN: Kolmogorov-Arnold Networks," arXiv:2404.19756, 2024. (The seminal paper reviving KANs).
2. Y. Chen et al., "Flexible Hybrid Electronics: A Review," Advanced Materials Technologies, vol. 6, no. 2, 2021.
3. M. Payvand et al., "In-Memory Computing with Emerging Memory Technologies: A Review," Proceedings of the IEEE, 2023. (Context on alternative efficient computing paradigms).
4. J. Zhu et al., "Analog Neural Networks: An Overview," in IEEE Circuits and Systems Magazine, 2021. (Background on analog ML hardware).
5. International Roadmap for Devices and Systems (IRDS™), "More than Moore" White Paper, 2022. (Discusses the role of heterogeneous integration and application-specific hardware like FE).
6. B. Murmann, "Mixed-Signal Computing for Deep Neural Network Inference," IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 2021. (Relevant for precision-efficiency trade-off analysis).

9. Original Analysis & Expert Commentary