The efficient control of complex dynamical systems is crucial across various scientific disciplines. Optimal control problems, while mathematically formulated using variational calculus, often prove analytically and computationally intractable for complex systems due to limitations in both control energy and cost. This paper addresses this challenge by introducing Al Pontryagin, a novel control framework built upon neural ordinary differential equations (NODEs). Al Pontryagin learns control signals that guide high-dimensional dynamical systems towards specified target states within defined timeframes. The significance of this approach lies in its ability to handle complex systems where traditional optimal control methods struggle. The widespread applicability of this framework spans diverse fields, including the development of robust quantum devices, regulation of cellular networks, power grid management, stabilization of financial systems, and epidemic control. Existing methods, like those based on Kalman's work and the Popov-Belevitch-Hautus test, provide controllability conditions for linear systems, but they lack the capacity to directly address resource and energy constraints inherent in real-world scenarios. Structural controllability frameworks offer some solutions for controlling complex networks using a minimum set of control inputs, however, these methods struggle with high-dimensional systems, the computation for finding the minimum set of driver nodes is NP-hard, and the design of the control signal is not directly defined. While optimal control (OC) methods address resource and energy constraints by incorporating cost functions, solving general optimal control problems using Pontryagin's maximum principle or the Hamilton-Jacobi-Bellman (HJB) equation often remains computationally intractable for complex systems. Previous attempts to use artificial neural networks (ANNs) have focused on approximating solutions to the HJB equation or solving Pontryagin's maximum principle through differentiable programming, but these approaches have limitations in scalability and applicability to high-dimensional systems. Al Pontryagin offers a unique approach by learning control signals without explicitly solving the maximum principle or HJB equations. This method leverages recent advancements in automatic differentiation and physics-informed artificial neural networks, showing that approximating optimal control is possible without explicitly minimizing control energy. Through implicit energy regularization, achieved through the interaction of ANN initialization and gradient descent, Al Pontryagin minimizes control energy effectively.

The paper extensively reviews existing literature on controlling complex dynamical systems. It starts by highlighting early works on controllability of linear systems, such as Kalman's work and the Popov-Belevitch-Hautus test. It then discusses the limitations of structural controllability frameworks in addressing general network controllability problems, particularly their NP-hard nature for minimum driver node identification and the lack of specified control signal design. The review then transitions to optimal control methods, focusing on Pontryagin's maximum principle and the HJB equation, and their limitations when applied to complex, analytically intractable systems. Prior work using ANNs for approximating optimal control is critically assessed, pointing out limitations in scalability and the need for twice-differentiable dynamical systems. The advancements in automatic differentiation and physics-informed ANNs are also mentioned as contributing to improved control approaches. The review sets the stage for Al Pontryagin by emphasizing the limitations of existing techniques in handling high-dimensional and complex systems while highlighting the potential of a new approach.

Al Pontryagin's methodology involves two key steps. First, it leverages neural ordinary differential equations (NODEs) to approximate and solve the dynamical system. The control input u(t) is represented by an artificial neural network (ANN) with a weight vector w, resulting in a control input representation ˆ(t; w). Second, a gradient descent algorithm iteratively determines the weight vector w by minimizing a loss function J(x, x*), which is typically the mean squared error between the final state x(T) and the target state x*. Automatic differentiation methods are employed to calculate the gradients, where the gradients flow through a time-unfolded ANN integrated by ODE solvers. The loss function focuses on minimizing the distance between the final state and the target state, without explicitly including a control energy term in the loss function. The process begins with initializing the ANN with small weights representing a small initial control signal. The dynamical system is then integrated, and a gradient descent is performed on the weights. The closer the final state is to the target state, the smaller the change in the weights, and the system adapts the control signal to achieve the goal. The paper demonstrates the mechanism by which this implicit energy regularization works: a gradient descent in the ANN weights induces a gradient descent in the control input, providing an approximation of optimal control that minimizes the control energy. For linear systems, Al Pontryagin's performance is compared with analytical optimal control inputs derived using Pontryagin's maximum principle. For non-linear systems, the Kuramoto model is used as an example, and Al Pontryagin's performance is compared with the adjoint-gradient method (AGM), a method based on Pontryagin's maximum principle. The ANN architectures, hyperparameters, and numerical solvers are detailed in the 'Methods' section and supplementary material. The paper includes various experiments using different network topologies (complete graph, Erdős-Rényi, square lattice, Watts-Strogatz) and sizes, showing that Al Pontryagin consistently performs well in different conditions and offers improved runtime compared to traditional AGM.

The paper's key findings center on the effectiveness and efficiency of the Al Pontryagin framework. Firstly, it demonstrates that Al Pontryagin successfully controls both linear and nonlinear dynamical systems, achieving results comparable to traditional optimal control methods. For a two-node linear system, Al Pontryagin generates trajectories and control energy that closely match those of optimal control, illustrating the effectiveness of the implicit energy regularization. This implicit regularization is further confirmed by a strong positive correlation between changes in ANN weights and control inputs during training. Secondly, the paper extends the application to a nonlinear system, using the Kuramoto model of coupled oscillators. Here, Al Pontryagin demonstrates comparable performance to the Adjoint Gradient Method (AGM), a traditional optimal control approach, in achieving synchronization across different network topologies. Specifically, it shows that Al Pontryagin can achieve synchronization slightly faster than the AGM and with similar control energy. The study expands to significantly larger networks (2500 nodes) where Al Pontryagin demonstrates a runtime two orders of magnitude faster than the AGM. This improved speed stems from Al Pontryagin's avoidance of solving the adjoint system, a computationally expensive step in traditional methods. Finally, Al Pontryagin's versatility is shown by its ability to achieve diverse control goals; it can not only synchronize oscillators but can also steer them to specific, non-synchronized target states by using different loss functions. The findings strongly suggest that Al Pontryagin offers a superior alternative to traditional optimal control methods, particularly for high-dimensional and complex systems.

The findings directly address the research question of developing an efficient control framework for complex dynamical systems. Al Pontryagin effectively overcomes the limitations of traditional optimal control methods by avoiding the computationally expensive steps of solving the adjoint system or the HJB equation. The implicit energy regularization mechanism, analytically described and empirically verified, demonstrates a novel approach to approximating optimal control. The results on both linear and nonlinear systems, particularly the Kuramoto model, show that Al Pontryagin achieves comparable or superior performance in terms of control energy and convergence speed. The significant improvement in runtime compared to the AGM, especially with large-scale systems, highlights the practical advantages of Al Pontryagin. The ability to adapt to different target states by changing the loss function enhances the framework's versatility. This work contributes significantly to the field of control theory by offering a scalable, efficient, and versatile method for controlling complex systems, opening new avenues for applications in various domains where high-dimensional dynamical systems are involved.

Al Pontryagin, a novel control framework based on neural ordinary differential equations, successfully addresses the challenges of controlling complex dynamical systems. It achieves comparable performance to optimal control methods while offering significant advantages in computational efficiency and scalability. The implicit energy regularization mechanism represents a valuable contribution to control theory. Future research could explore its application to quantum control problems, improving the robustness of quantum systems, and its use in preventing cascading failures in power grids. Combining Al Pontryagin with physics-informed neural networks could further enhance its ability to control partially unknown systems.

While Al Pontryagin demonstrates significant advantages, certain limitations should be noted. The performance is highly dependent on the choice of ANN architecture and hyperparameters, requiring careful tuning. The implicit energy regularization mechanism, although effective, may not be optimal in all situations. Further investigation into the theoretical properties of this regularization is warranted. The current study focuses on specific types of dynamical systems and network structures; further research is needed to assess the framework's generalizability to a broader range of systems.