Smarter Paths, Safer Missions: Real-Time Trajectory Optimization Using QA-PINN

Optimization plays a central role in modern engineering and physics. Whether it’s minimizing energy consumption in chip design or charting efficient flight paths for spacecraft, these tasks share a common goal: achieving a desired outcome under physical constraints, as efficiently and precisely as possible.

Real-time trajectory optimization is one such task critical in aerospace applications, where spacecraft must navigate complex environments, such as gravity-assist maneuvers or rapid course corrections. This challenge is not just about computing speed but about finding valid and physically feasible solutions that also align with mission goals.

Limitations of Traditional Optimization Techniques

Historically, numerical methods like gradient descent and simplex algorithms have been used to solve trajectory and control problems. While effective in many cases, they struggle when:

• The solution space becomes large and multi-dimensional
• Constraints are nonlinear or dynamically changing
• Real-time decisions must be made with incomplete data

Even more advanced methods such as genetic algorithms (GA) and reinforcement learning (RL) face limitations in these settings. These bottom-up approaches often require extensive sampling or exploration and are sensitive to instability in high-stakes applications like trajectory planning.

Physics-Informed Learning: A New Approach with Quantum Assisted Physics Informed Neural Networks

Quantum-Assisted Physics-Informed Neural Networks (QA-PINNs) offer an alternative. These hybrid models combine physical laws with data-driven learning, allowing predictions and optimizations that obey governing dynamics while targeting specific mission objectives.

Initially developed for solving partial differential equations in physics, PINNs are now being adapted for optimization by incorporating cost or goal functions directly into their architecture. This shift enables the network not only to satisfy physical constraints but also to find optimal solutions—such as minimal time or minimal energy paths.

With quantum assistance, the learning process becomes more efficient, particularly in high- dimensional or data-scarce environments. Quantum-enhanced training helps the network converge faster and more reliably than classical approaches.

Methodology: Applying QA-PINN to Trajectory Optimization

‍1. Problem Setup and Physical Inspiration

In physics, two well-known problems represent shortest-time trajectory solutions:

• Fermat’s Principle: Light follows the path that minimizes travel time between two points.
• Brachistochrone Problem: A particle moving under gravity reaches the endpoint in the least time when following a cycloidal path.

These problems translate naturally into trajectory optimization scenarios, such as determining the quickest path for a spacecraft during re-entry or course adjustment.

2. Network Inputs and Outputs

In the QA-PINN framework:

• The input to the network is a normalized time variable t∈[0,1]t \in [0, 1]t∈[0,1], representing the progression of the trajectory.
• The outputs are the corresponding spatial coordinates x(t)x(t)x(t) and y(t)y(t)y(t), which define the trajectory of the object over time.

This parametric representation enables the network to learn a continuous path over time while keeping the architecture compact and interpretable.

‍1. Embedding Governing Equations:

The governing laws, such as conservation of mechanical energy, are directly included in the network via automatic differentiation. For example,

where T (travel time) is a trainable parameter, and (x0 ,y0 ), (x1 ,y1 ) are the start and end coordinates.

2. Setting Constraints and Goals:

Boundary conditions ensure the trajectory starts and ends at the specified points. The cost function L_goal=T represents the optimization target—minimizing the time taken.

3. Training and Convergence:

The model starts with a trial path and adjusts itself during training. As iterations progress, the path gradually conforms to both the physics and the goal, converging toward the true shortest-time curve.

This structure allows QA-PINN to find trajectories that not only obey physical constraints but also optimize mission objectives like minimal time, fuel, or deviation.

Broader Applications of QA-PINN in aerospace

Though this study focuses on a classical physics problem, the method applies to real-world mission scenarios:

• Spacecraft swing-bys and re-entry trajectories
• High-efficiency UAV path planning

What sets QA-PINN apart is its ability to solve these problems without requiring full data upfront, while still adhering to complex physical systems. With quantum assistance, it becomes even more viable for real-time use in mission-critical environments.

Optimization is no longer just a numerical challenge—it’s a multidisciplinary problem where physics, machine learning, and real-time constraints converge. QA-PINN provides a practical and scalable method to navigate this intersection, enabling more efficient decision-making in dynamic environments.