⚡ electrical engineering

Physical neural networks with trainable nonlinear connections for continuous control

This paper introduces a physical neural network architecture that places trainable nonlinear functions on every connection using analogue band-pass filters, demonstrating that this approach achieves high-fidelity continuous control tasks with significantly fewer parameters and connections than traditional multilayer perceptrons while enabling ultra-low-power hardware implementations.

Original authors: Ian Vidamour, Fernando Aguirre, Thomas Hayward, Matthew Ellis, Charles Swindells, Alexander McDonnell, Martin Trefzer, Finley Robins, Luca Manneschi, Susan Stepney, Anthony Kenyon, Oliver Sutton, Jack

Published 2026-10-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ian Vidamour, Fernando Aguirre, Thomas Hayward, Matthew Ellis, Charles Swindells, Alexander McDonnell, Martin Trefzer, Finley Robins, Luca Manneschi, Susan Stepney, Anthony Kenyon, Oliver Sutton, Jack Gartside, Ivan Tyukin, Adnan Mehonic, Eleni Vasilaki

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of modern computing, machines have become incredibly good at recognizing patterns, from identifying faces in photos to translating languages. These systems, known as neural networks, are usually built on digital processors that crunch numbers one by one. However, a growing field of research is asking whether we can build these thinking machines using the raw physics of electronic components themselves. Instead of simulating a brain on a standard computer, researchers are trying to create "physical neural networks" where the hardware does the thinking directly. The challenge has been that most of these physical systems are forced to act like their digital cousins, using simple, fixed rules to process information. This often wastes the unique, complex behaviors that real electronic devices naturally possess, such as the way a specific type of switch might react differently depending on how fast a signal arrives or how strong it is. The question facing scientists is whether they can stop fighting against these natural quirks and instead use them as the very building blocks of intelligence.

A team of researchers from universities across the United Kingdom has taken a bold step toward answering this question. They developed a new type of physical network where every single connection between two points is not just a simple number, but a trainable, flexible function. Imagine a standard network where the wires between nodes are just pipes that can be opened or closed to let more or less water through. In this new system, the pipes themselves can change their shape and behavior, filtering and reshaping the water as it flows. The researchers built these connections using analogue electronic filters, which are circuits that respond differently to different frequencies of sound or electricity. By training these filters to act as the "brain" of the connection, they created a system where the physical device does the heavy lifting of calculation, rather than just passing a signal along.

The team tested this approach on several real-world tasks that require smooth, continuous adjustments, such as controlling the movement of a robotic arm, balancing a pole on a moving cart, and finding the most efficient power output from a solar panel. In each case, they compared their new physical network against standard digital networks. The results were striking. For tasks involving smooth, continuous movements, the physical network achieved the same level of accuracy as the digital one but with far fewer parts. In some instances, the physical network used only a fraction of the connections and parameters required by the standard digital model. This efficiency is crucial because fewer connections mean less hardware to build, program, and power. When the researchers projected how much energy a dedicated chip built with this technology would consume for a solar panel controller, they estimated it would use about 35 microwatts. This is roughly two orders of magnitude less than the tiny microcontrollers currently used for similar edge-computing tasks, suggesting a path toward incredibly low-power intelligent devices.

To ensure their findings were not just a fluke of one specific type of circuit, the researchers also simulated the same network using a different kind of electronic component called a memristor, which changes its resistance based on its history. Even with this completely different physical material, the network retained its advantage over standard digital models. This suggests that the benefit comes from the architecture itself—using trainable, nonlinear connections—rather than from any single type of hardware. The researchers also demonstrated that they could take the mathematical model of their network, translate the settings directly into the physical hardware, and have it work immediately without needing to retrain the system on the device itself. They tested this transfer on over 35,000 individual connections, and the physical hardware matched the simulated predictions with high precision.

However, the study also clarified where this new approach is not the best fit. When the researchers tested the network on tasks that involve making sharp, binary decisions, such as classifying images of clothing or choosing between two distinct actions, the physical network did not show a clear advantage over standard digital networks. The researchers explain that this is because their system is designed to excel at smooth, continuous changes, whereas binary decisions often rely on sharp boundaries that are easier for traditional networks to create. This distinction is important: the new technology is not a magic bullet for every computing problem, but rather a specialized tool that shines when the task involves navigating a smooth landscape of possibilities.

The work represents a significant shift in how we think about building intelligent machines. Instead of trying to force physical devices to behave like simple, predictable digital switches, the researchers have shown that we can harness the complex, natural behaviors of those devices to do the work for us. By placing a learnable, nonlinear function on every connection, they have turned the physical limitations of hardware into a computational strength. This approach offers a promising route to compact, low-power networks capable of handling the continuous, real-world control tasks that are essential for robotics, energy management, and autonomous systems. The results suggest that the future of efficient machine learning may lie not in making our hardware more like our software, but in letting our hardware be exactly what it is.

Technical Summary: Physical Neural Networks with Trainable Nonlinear Connections for Continuous Control

Problem Statement
Physical neural networks (PNNs) aim to perform machine learning inference directly in hardware using the native physics of analogue devices, offering potential pathways to ultra-low-power computing. However, most existing PNN architectures adhere to the conventional software-based assumption that connections between nodes are represented as scalar weights. In this standard model, devices with rich, native nonlinear responses (e.g., memristor current-voltage curves or filter frequency responses) are often constrained to act merely as programmable linear weights or fixed, identical activation functions. This approach discards the computational potential inherent in device nonlinearity and often necessitates a large number of physical connections and nodes to achieve expressivity, increasing hardware complexity and power consumption.

Methodology
The authors propose and implement a Physical Kolmogorov–Arnold-inspired Network (PhyKAN), an architecture that inverts the standard assumption by placing a trainable nonlinear function on every connection (edge) rather than a scalar weight.

Architecture: Inspired by the Kolmogorov–Arnold representation theorem, each edge computes a learnable univariate function. In the primary implementation, these edges are realized as banks of tuneable analogue band-pass filters on Field-Programmable Analogue Arrays (FPAAs).
Encoding and Computation: Network inputs are encoded as signal frequencies. Each edge consists of a sum of parallel band-pass filter units (high-pass and low-pass stages with amplification). The trainable parameters are the corner frequencies and gains of these filters.
Training and Hardware Transfer: The networks are trained using gradient descent through an analytical transfer-function model of the analogue filters. To bridge the gap between simulation and hardware, the authors employ a straight-through estimator (STE). This allows the optimization to proceed over continuous latent parameters while enforcing discrete, hardware-achievable values (quantized corner frequencies and gains) during the forward pass.
Sparsity: A regularisation term encourages sparsity, allowing the pruning of filter units that contribute little to the edge response, thereby reducing the effective number of connections.
Alternative Substrate: To verify that the architectural benefits are not specific to analogue filters, the authors also simulated a MemKAN variant, where edges are realized using simulated memristor circuits with programmable nonlinear current-voltage characteristics.

Key Results
The study evaluates PhyKANs against standard Multilayer Perceptrons (MLPs) across several tasks, focusing on parameter efficiency and hardware fidelity.

Hardware Fidelity: The authors successfully transferred trained networks to FPAA hardware. Across approximately 35,000 experimentally realized connections, 90% matched their simulated targets within a mean-squared error (MSE) of $7.92 \times 10^{-5}$ .
Continuous Control Tasks (Parameter Efficiency):
- Six-Axis Robotic Kinematics: For both forward kinematics and inverse kinematics control (learned from actuator-corrupted observations), PhyKANs achieved lower MSE than MLPs with substantially fewer trainable parameters and physical connections, particularly in networks with two hidden layers.
- Continuous-Action Reinforcement Learning (CartPole): In a modified CartPole task with continuous force outputs, PhyKANs solved the task with fewer parameters than MLPs. The PhyKANs demonstrated stable control, whereas MLPs often exhibited high-frequency oscillatory behaviors or failed to stabilize the pole under certain initial conditions.
- Photovoltaic Maximum Power Point Tracking (MPPT): PhyKANs outperformed MLPs in tracking the maximum power point under partial shading conditions across all tested network sizes (except the smallest). The networks demonstrated a superior ability to navigate local maxima in the power-voltage curve using environmental feedback.
Discrete/Decision Tasks: On tasks requiring sharp decision boundaries, such as Fashion-MNIST classification and binary-action CartPole, PhyKANs did not show a consistent parameter-efficiency advantage over MLPs. The authors note that the advantage of trainable nonlinear edges is most pronounced for smooth, continuously valued mappings.
Memristor Validation: Simulated MemKANs replicated the parameter-efficiency advantages of the filter-based PhyKANs on the three continuous tasks, supporting the hypothesis that the benefit stems from the architecture (trainable nonlinear edges) rather than a specific physical device.
Power Projection: Based on a CMOS-compatible transconductor design, the authors project that a dedicated hardware implementation of the photovoltaic controller would consume approximately 35 µW. This is two orders of magnitude lower than the 0.33–22 mW typically consumed by microcontrollers running comparable neural networks.

Significance and Claims
The paper claims that making device nonlinearity trainable, rather than suppressing it, offers a viable route to compact, low-power analogue networks for smooth, continuous inference and control tasks.

Architectural Shift: The work demonstrates that physical systems should not be forced to approximate software-style scalar weights. Instead, the native nonlinear responses of devices can serve as the primary computational primitives.
Task Suitability: The authors identify a clear condition for the success of this architecture: tasks requiring smooth, continuously valued mappings (e.g., kinematics, continuous control, regression) benefit significantly from the expressive power of individual edges, allowing for fewer nodes and connections.
Hardware Viability: The results show that the architectural advantages can survive the transfer from model to physical hardware without the need for extensive post-deployment fine-tuning (such as hardware-in-the-loop reinforcement learning), provided the training accounts for hardware discretization.
Low-Power Potential: By reducing the number of required physical connections and nodes, and by leveraging the natural low-current analogue regime of devices (as opposed to high-current linear regimes often used in memristive crossbars), PhyKANs offer a promising strategy for edge computing applications where power is a critical constraint.

The authors maintain a modest stance, noting that while the approach excels in continuous control, it does not universally outperform MLPs in discrete classification or binary decision tasks, and that scaling to larger networks may require more complete modeling of parasitic effects or post-transfer calibration.

Technical Summary: Physical Neural Networks with Trainable Nonlinear Connections for Continuous Control

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