Controlling synchronization dynamics via physics-informed neural networks

TL;DR

The paper addresses the problem of regulating both the time scale and the final coherence level of synchronization in networked nonlinear systems. It introduces a physics-informed neural control (PINN) framework that jointly learns state trajectories and control inputs under the governing dynamics, enforcing synchronization objectives via a persistence constraint on the order parameter $R(t)$ after a target time $t^*$ and level $R^*$. Demonstrations on networks of Kuramoto oscillators show that the approach achieves smooth synchronization with reduced transient effort and competitive energy compared to baselines, and remains effective in non-gradient and frustrated dynamics such as the Kuramoto–Sakaguchi model. Overall, the method provides a trajectory-level design paradigm that integrates dynamical systems theory, control, and machine learning, with potential applicability to a wide class of networked nonlinear systems.

Abstract

Synchronization control in networked dynamical systems requires regulating not only whether coherence is achieved, but also when and to what extent it emerges. We propose a physics-informed neural network (PINN) framework for continuous-time synchronization regulation, in which system trajectories and control inputs are jointly parameterized and constrained by the governing dynamics. Macroscopic synchronization objectives are imposed directly at the trajectory level by enforcing persistence conditions on the order parameter after a prescribed target time. This formulation enables simultaneous control of synchronization time and coherence level without assuming any explicit feedback law or solving a strict optimal control problem. Numerical studies on networked Kuramoto oscillators demonstrate smooth synchronization with reduced transient control effort and competitive cumulative cost relative to analytical baselines. The framework remains effective in non-gradient and frustrated dynamics, highlighting physics-informed neural control as a flexible trajectory-level approach to synchronization regulation.

Figures (8)

• Figure 1: Schematic illustration of the physics-informed neural control framework. Time $t$ is used as the input to a neural network with multiple hidden layers, which jointly outputs a state network $N_x(t)$ and a control network $N_u(t)$. These outputs are combined with a shaping function $h(t)$ to parameterize the system state $\mathbf{x}(t)$ and control input $\mathbf{u}(t)$ while exactly enforcing the initial condition. The governing system dynamics $\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{p},\mathbf{x}(t),\mathbf{u}(t))$ are incorporated through physics-informed residuals, and training is performed by minimizing a composite loss function consisting of dynamical, initial-condition, control, and regularization terms.
• Figure 2: (a) Phase time series of the oscillators without control, represented as $\sin\theta_i(t)$; (b) phase time series under physics-informed neural control; (c) time evolution of the order parameter $R(t)$ for controlled and uncontrolled dynamics. Dashed lines indicate the target synchronization level $R^\ast$ and the target time $t^\ast$.
• Figure 3: (a) Time evolution of the control inputs $u_i(t)$ learned by the physics-informed neural network; (b) control cost density $P$ as a function of time, quantifying the instantaneous control effort required to enforce the synchronization constraint.
• Figure 4: Integrated control cost $E(t)$ under different intrinsic system conditions. (a) Dependence of $E(t)$ on the frequency heterogeneity width $\delta$. (b) Dependence of $E(t)$ on the coupling strength $K$. In both panels, the synchronization target $(R^\ast, t^\ast)$ and all other system parameters are fixed. Solid lines indicate mean values, while shaded regions represent variability across independent realizations; for each parameter value, results are averaged over $20$ independent runs with different random initial conditions and frequency realizations.
• Figure 5: Dependence of the integrated control cost $E(t)$ on synchronization targets. (a) Integrated control cost as a function of the target synchronization level $R^\ast$ with fixed target time $t^\ast$. (b) Integrated control cost as a function of the target synchronization time $t^\ast$ with fixed $R^\ast$. Solid lines denote mean values over $20$ independent runs, and shaded regions indicate the corresponding variability.