PID-CNN: Control-Theoretic Neural Architecture

• PID-CNN is a neural network architecture that explicitly integrates proportional, integral, and derivative operations into convolutional layers to mimic classic PID controllers.
• It uses weighted convolution kernels and feature reuse via residual connections to efficiently model spatiotemporal dynamics, enabling accurate 3D motion perception.
• Empirical results show near-optimal error rates and real-time performance, highlighting the method's potential for controlled synthetic and future real-world scenarios.

A Proportional-Integral-Derivative (PID) Convolutional Neural Network (PID-CNN) is a neural network architecture for vision tasks in which the convolutional layers and their nonlinearities are explicitly interpreted and configured analogously to classical PID controllers. Such designs leverage the roles of proportional, integral, and derivative operations to model and fit spatiotemporal dynamics or manage information across feature hierarchies. The PID-CNN paradigm provides an architecture-level analogy between classic feedback control theory and deep convolutional networks, enabling new insights into feature extraction, memory, and information fusion in deep learning (Jiazhao et al., 25 Nov 2025).

1. PID-CNN: Conceptual Foundation

The core of PID-CNNs is the mathematical correspondence between discrete convolutions and the three core components of PID control. In a 1D or 2D discrete convolutional context, the following template applies for a feature map $f(x)$:

• Proportional (P): Identity or “current value,” implemented as $C_p*f(x) = f(x)$ with the kernel $[0,1,0]$.
• Integral (I): Local average, e.g., $C_i*f(x) = \frac{1}{3}(f(x-1) + f(x) + f(x+1))$, kernel $[\frac{1}{3},\frac{1}{3},\frac{1}{3}]$.
• Derivative (D): Discrete first difference, $C_d*f(x) = f(x+1) - f(x-1)$, kernel $[-1,0,1]$.

Weighted sums of these kernels, $C = k_p C_p + k_i C_i + k_d C_d$, allow a convolutional layer to model the effect of a classical PID controller, with the $k_p$, $k_i$, and $k_d$ terms as tunable gains. For higher-order fits (e.g., local second derivatives), compositions and further weighted combinations are possible.

Stacking such layers with nonlinear activations (e.g., PReLU) enables the network to hierarchically fit complex spatiotemporal phenomena, such as reconstructing coordinate, velocity, and acceleration from image sequences (Jiazhao et al., 25 Nov 2025).

2. Network Architecture and Feature Reuse

A representative PID-CNN instantiation, as detailed in "3D Motion Perception of Binocular Vision Target with PID-CNN" (Jiazhao et al., 25 Nov 2025), comprises the following:

• Input: Two-view, three-time-step tensors (shape: $2\times3\times256\times256$; single channel per view after background subtraction).
• Blocks: Seven convolutional blocks, each containing:
  • Two $3\times3$ convolutions (stride 1, padding 1), each followed by a batch normalization and PReLU, taking input channels doubled at each stage (by concatenation).
  • Concatenation of input and output along channels (feature reuse).
  • $2\times2$ average pooling (halves spatial dimensions, doubles channels).
• Block Channel Progression:
  • Blocks sequentially expand from $D=2$ channels up to $D=128$; thus, after 7 blocks, the feature tensor is $256\times3\times2\times2$.
• Estimation Head: After flattening features from final block for each time/view, three fully connected pathways output:
  • Coordinates $p\in\mathbb{R}^3$
  • Velocity $v\in\mathbb{R}^3$ (via residual over time differences)
  • Acceleration $a\in\mathbb{R}^3$ (using a secondary residual structure)
• Feature Reuse: Building-block input and output concatenation across layers, analogous to dense connections, aids gradient flow and multi-scale feature aggregation, facilitating optimal representation of both fine and coarse spatial-temporal correlations (Jiazhao et al., 25 Nov 2025).

3. Mathematical Formalism

Following the controller analogy, the per-layer transformation in PID-CNNs can be formalized as:

$$z(x) = k_p f(x) + k_i \sum_{i=x-1}^{x+1} f(i) + k_d [f(x+1) - f(x-1)]$$

with the nonlinearity $f_\text{out}(x) = g(z(x))$ (e.g., PReLU). For network heads:

• Each frame’s output: $p_i = \mathrm{FC}_\text{coord}(S_{t-2+i})$
• Velocity: $$v = (p_2 - p_1) + \mathrm{FC}_\text{vel}(S_{t-1} - S_{t-2})$$
• Acceleration: $$a = (v_2 - v_1) + \mathrm{FC}_\text{acc}(S_t - S_{t-1})$$

This architecture models position, velocity, and acceleration estimation as learning residual corrections atop finite-difference approximations, with the FC heads absorbing nonlinearities or model discrepancies inherent in binocular 3D triangulation (Jiazhao et al., 25 Nov 2025).

4. Training Procedures and Empirical Results

The PID-CNN was specifically evaluated for 3D motion estimation of a simulated single sphere viewed by two fixed cameras. Training was staged: 1. Single-frame: coordinate regression. 2. Two-frame: velocity estimation (pretrained from coordinate-only). 3. Three-frame: acceleration estimation (pretrained from velocity).

Key empirical findings:

• Test errors: Coordinate $\sigma\approx0.18$ px, velocity $\sigma\approx0.25$ px/frame, acceleration $\sigma\approx0.43$ px/frame$^2$.
• Max errors: Position $\approx0.70$, velocity $\approx0.98$, acceleration $\approx1.71$ (unit: pixels).
• Performance approaches the theoretical upper bound determined by input image resolution, indicating near-optimal parameter efficiency and representational capacity.
• Runtime: 4.05 ms/sample ($\approx247$ samples/s), suited for real-time processing on the test hardware.
• Ablations: Feature reuse and residual learning in motion heads improve both convergence and error rates; average pooling outperforms max pooling in the initial training phase.
• Nonlinearity: Choice among ReLU, PReLU, and LeakyReLU is immaterial, attributed to the linearity of the underlying triangulation problem (Jiazhao et al., 25 Nov 2025).

5. Comparison with Related PID-Inspired and Nonlocal Architectures

Several network designs incorporate PID-related motifs or naming conventions:

Model	PID Analogy	Primary Domain
PID-CNN (Jiazhao et al., 25 Nov 2025)	PID kernels, feature reuse, explicit residual estimation	3D motion (binocular) perception
PIDNet (Xu et al., 2022)	Three-branch (P=detail, I=context, D=boundary)	Real-time semantic segmentation
PIDNet (Sun et al., 2020)	Two-branch backbone for intrusion detection, PID refers to “Pedestrian Intrusion Detection”	Dynamic pedestrian/AoI detection
PID-Net (Zhang et al., 2022)	Pixel Interval Downsampling with max-pool fusion (“PID” refers to the operation, not control theory)	Dense tiny object segmentation/counting

Of these, only (Jiazhao et al., 25 Nov 2025) and (Xu et al., 2022) draw explicit architectural and functional analogies between PID control terms and convolutional/spatial operators. The three-branch “PIDNet” (Xu et al., 2022) extends the analogy by associating proportional, integral, and derivative operations to detail, context, and boundary branches, respectively—enabling more accurate segmentation by damping overshoot near object boundaries.

Architectures inspired by partial integro-differential equations (PIDEs) (Bohn et al., 2021) employ nonlocal layers for extending receptive fields, but are not PID-convolutional in the classical sense—the analogy there is between fractional/integral operators and network propagation, not feedback control.

6. High-Dimensional Convolutional Extensions and Feature Utilization

The PID-CNN framework anticipates extension to high-dimensional (ND) convolutions, where convolutional kernels operate along spatial, channel, and temporal axes simultaneously. In $N$ dimensions, a $3^N$ kernel allows each spatial-temporal-feature point to integrate information from its immediate hypercube neighborhood. This contrasts with standard fully connected feature mixing, which scales as $d\cdot3^2$ for $d$ channels (i.e., is less efficient for many axes).

General ND-convolutions could more efficiently exploit the geometric structure of feature spaces, yielding richer and less redundant representations. However, such operations are not universally available in current deep learning frameworks beyond three dimensions. Their development may enable significant advances in spatiotemporal modeling and computational efficiency in deep architectures (Jiazhao et al., 25 Nov 2025).

7. Limitations, Perspectives, and Future Directions

Current PID-CNN implementations, while achieving near-optimal accuracy in controlled environments, are limited by the following:

• Evaluation confined to synthetic scenes with a single sphere and fixed camera geometry.
• Nonlinearity/complexity of scene geometry is moderate; the architecture's benefits for more inherently nonlinear or multi-object scenarios remain to be established.
• Lack of demonstration on physical datasets or with moving/variable viewpoints.

Future research directions identified include:

• Deployment on real-world, multi-object, and dynamically varying settings.
• Integration of true high-dimensional convolutions for more sophisticated feature interactions.
• Exploiting the PID analogy in the design of memory (integral-like, for long-term accumulation) and attention (derivative-like, for detecting change/novelty) modules, possibly at low computational cost. This suggests plausible synergies with efficient recurrent or transformer-style architectures guided by PID-style information aggregation (Jiazhao et al., 25 Nov 2025).

In summary, the PID-CNN paradigm unifies explicit control-theoretic interpretation and modern convolutional architectures, yielding real-time, accurate 3D motion perception and offering avenues for principled improvement via high-dimensional and modular neural engineering.