Deep Neural Network-guided PSO for Tracking a Global Optimal Position in Complex Dynamic Environment

Stephen Raharja; Toshiharu Sugawara

cs.NEApr 15, 2026

Deep Neural Network-guided PSO for Tracking a Global Optimal Position in Complex Dynamic Environment

Authors: Stephen Raharja, Toshiharu Sugawara

Organizations: Waseda University Shinjuku City, Tokyo, Japan

Abstract

We propose novel particle swarm optimization (PSO) variants incorporated with deep neural networks (DNNs) for particles to pursue globally optimal positions in dynamic environments. PSO is a heuristic approach for solving complex optimization problems. However, canonical PSO and its variants struggle to adapt efficiently to dynamic environments, in which the global optimum moves over time, and to track them accurately. Many PSO algorithms improve convergence by increasing the swarm size beyond potential optima, which are global/local optima but are not identified until they are discovered. Additionally, in dynamic environments, several methods use multiple sub-population and re-diversification mechanisms to address outdated memory and local optima entrapment. To track the global optimum in dynamic environments with smaller swarm sizes, the DNNs in our methods determine particle movement by learning environmental characteristics and adapting dynamics to pursue moving optimal positions. This enables particles to adapt to environmental changes and predict the moving optima. We propose two variants: a swarm with a centralized network and distributed networks for all particles. Our experimental results show that both variants can track moving potential optima with lower cumulative tracking error than those of several recent PSO-based algorithms, with fewer particles than potential optima.

Figures & tables

Panel of Figure 1 — Figure 1. Example of $g(t)$ location relative to peaks $h_{1},h_{2}$ . The figure shows the example positions of a $g(t)$ relative to two peaks on 2-dimensional plane, with y equals to the probability density function. In the first figure, the first peak of $h_1$ has smaller standard deviation , resulting in the position of $g(t)$ close to the center of $h1$ . In the second figure, both peaks $h_1, h_2$ have the same standard deviation, and result in $g(t)$ to be positioned exactly between the two peaks.

Figure 2. Visualization of particle $p_{i}$ in NNGPSO with two example observation points Top view of a particle in \nngpso, where each particle has multiple surrounding observation points. The position and utility values of each observation point are used as input data for the deep neural network.

Symbol	Description	Value
$E_{\it bound}$	$E$ bounds for each dimension	$10$
$E_{\it factor}$	Sampling area factor for $E$	$0.8$
$E_{\it count}$	Number of environments per $\mathcal{E}_{i}$	$5$
$E_{\it run}$	Number of experiments per $E$	$3$
$\sigma_{\min}$	Lower bound of $\sigma$ for $H$	$0.25$
$\sigma_{\max}$	Upper bound of $\sigma$ for $H$	$1$

Table 1. List of experimental parameters

Env. group ( $\mathcal{E}_{i}$ )	No. of peaks ( $\|H\|$ )	No. of centers ( $\|C\|$ )
$\mathcal{E}_{1}$	100	50
$\mathcal{E}_{2}$	200	100
$\mathcal{E}_{3}$	200	200
$\mathcal{E}_{4}$	500	100

Table 2. Parameter settings for each environment group

\mathcal{E}_{i}

Parameter	Value
$m_{\mathrm{fac}}$	0.8
$\gamma_{\min}$	-10
$\gamma_{\max}$	10
$\bm{\theta}_{i}$	$0,\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4},\pi,\frac{5\pi}{4},\frac{3\pi}{2},\textrm{ or }\frac{7\pi}{4}$
$\bm{r}_{i}$	$0.5,1.0,1.5,\textrm{ or }2.0$
$\tau_{\mathrm{epoch}}^{c}$	1

Table 3. Parameter settings for NNGPSO variants.

Algorithm	$\|p\|$	$n$	$\mathrm{error}(\mathcal{E}_{i})$ per Environment Group				$\mathrm{error}(\bar{\mathcal{E}})$
			$\mathcal{E}_{1}$	$\mathcal{E}_{2}$	$\mathcal{E}_{3}$	$\mathcal{E}_{4}$
CNNPSO	5	-	$5.331\pm 4.464$	$5.827\pm 3.603$	$5.350\pm 3.753$	$5.471\pm 3.747$	$5.495\pm 3.911$
DNNPSO	5	-	4.552 $\pm$ 4.370	5.213 $\pm$ 3.879	4.786 $\pm$ 3.611	3.858 $\pm$ 3.798	4.602 $\pm$ 3.955
PSO	5	-	$8.595\pm 4.347$	$10.196\pm 4.576$	$8.526\pm 4.042$	$7.995\pm 3.926$	$8.828\pm 4.310$
PSO	165	-	$8.804\pm 4.893$	$9.486\pm 4.773$	$8.356\pm 4.092$	$8.583\pm 4.242$	$8.818\pm 4.423$
PSPSO	165	5	$16.889\pm 7.060$	$18.878\pm 8.310$	$18.085\pm 8.266$	$16.739\pm 8.919$	$18.503\pm 7.507$

Table 4. Tracking error comparison against baselines

Panel of Figure 3 — Figure 3. Utility values of actual global best position ${\bm{a}}(t)$ and detected global best position ${\bm{g}}(t)$ in $E_{3,2}^{1}$

Algorithm	$\|p\|$	$\mathrm{error}(\bar{\mathcal{E}})$
CNNPSO	5	$4.860\pm 3.847$
CNNPSO	7	$4.688\pm 3.749$
CNNPSO	10	$4.656\pm 4.005$
DNNPSO	5	$4.390\pm 3.531$
DNNPSO	7	$4.248\pm 3.801$
DNNPSO	10	4.129 $\pm$ 3.562

Table 5. NNGPSO tracking errors against particle counts

|p|

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Particle swarm optimization (PSO) is a widely used metaheuristic, prized for its simplicity and small parameter set. Although decades of research have produced numerous PSO variants that improve performance by modifying key components (e.g., parameter schedules, swarm topologies, or updating rules), two fundamental challenges persist. First, most existing approaches are problem-specific and hand-crafted, leading to poor cross-task generalization and forcing practitioners to navigate an impractically large design space, which also hinders systematic reuse of prior effective mechanisms. Second, mainstream implementations remain CPU-bound, constraining scalability and substantially increasing computational cost in real-world applications. To address these challenges, we propose AutoPSO, a highly automated meta-framework for constructing customized PSO algorithms. AutoPSO formulates PSO-based optimization as a bi-level process: an outer search explores the joint space of effective PSO components, while an inner loop instantiates candidate variants to solve the target task and provide feedback. The outer search operates over a curated, open-design component pool, supporting flexible replacement of the component set and the outer optimizer. Crucially, by leveraging EvoX for population tensorization and batched evaluations, AutoPSO can efficiently assess thousands of particles within practical time budgets. Comprehensive experiments on numerical benchmarks and neuroevolution robotic control tasks demonstrate that AutoPSO consistently discovers novel PSO variants that significantly outperform strong baselines. Ablation and scalability studies further highlight the contribution of individual algorithmic components and confirm that AutoPSO achieves increasing performance gains with larger swarm sizes. Code is available at {https://github.com/EMI-Group/autopso}.

Abstract

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