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Trajectory Optimization for Minimum Threat Exposure using Physics-Informed Neural Networks

Alexandra E. Ballentine, Raghvendra V. Cowlagi

TL;DR

This work casts minimum-threat exposure trajectory planning as a TPBVP from Pontryagin's Minimum Principle and demonstrates that physics-informed neural networks can solve the resulting boundary-value problem for a 2D vehicle with fixed speed. A dual-PINN architecture is proposed: one network predicts states, control, and final time, while a second network predicts co-states, with a comprehensive set of PMP-residual losses guiding training. The method is validated for time-invariant and time-varying threat fields and extended to learn multiple endpoint pairs, showing PMP-consistent trajectories with residuals on the order of $O(0.1)$ and losses near $O(10^{-3})$ in several cases. This approach offers a promising route toward real-time optimal control under spatiotemporal threats, with clear avenues for improving convergence and generalization to broader initial-condition sets.

Abstract

We apply a physics-informed neural network (PINN) to solve the two-point boundary value problem (BVP) arising from the necessary conditions postulated by Pontryagin's Minimum Principle for optimal control. Such BVPs are known to be numerically difficult to solve by traditional shooting methods due to extremely high sensitivity to initial guesses. In the light of recent successes in applying PINNs for solving high-dimensional differential equations, we develop a PINN to solve the problem of finding trajectories with minimum exposure to a spatiotemporal threat for a vehicle kinematic model. First, we implement PINNs that are trained to solve the BVP for a given pair of initial and final states for a given threat field. Next, we implement a PINN conditioned on the initial state for a given threat field, which eliminates the need for retraining for each initial state. We demonstrate that the PINN outputs satisfy the necessary conditions with low numerical error.

Trajectory Optimization for Minimum Threat Exposure using Physics-Informed Neural Networks

TL;DR

This work casts minimum-threat exposure trajectory planning as a TPBVP from Pontryagin's Minimum Principle and demonstrates that physics-informed neural networks can solve the resulting boundary-value problem for a 2D vehicle with fixed speed. A dual-PINN architecture is proposed: one network predicts states, control, and final time, while a second network predicts co-states, with a comprehensive set of PMP-residual losses guiding training. The method is validated for time-invariant and time-varying threat fields and extended to learn multiple endpoint pairs, showing PMP-consistent trajectories with residuals on the order of and losses near in several cases. This approach offers a promising route toward real-time optimal control under spatiotemporal threats, with clear avenues for improving convergence and generalization to broader initial-condition sets.

Abstract

We apply a physics-informed neural network (PINN) to solve the two-point boundary value problem (BVP) arising from the necessary conditions postulated by Pontryagin's Minimum Principle for optimal control. Such BVPs are known to be numerically difficult to solve by traditional shooting methods due to extremely high sensitivity to initial guesses. In the light of recent successes in applying PINNs for solving high-dimensional differential equations, we develop a PINN to solve the problem of finding trajectories with minimum exposure to a spatiotemporal threat for a vehicle kinematic model. First, we implement PINNs that are trained to solve the BVP for a given pair of initial and final states for a given threat field. Next, we implement a PINN conditioned on the initial state for a given threat field, which eliminates the need for retraining for each initial state. We demonstrate that the PINN outputs satisfy the necessary conditions with low numerical error.
Paper Structure (10 sections, 19 equations, 3 figures, 3 tables)

This paper contains 10 sections, 19 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Method of Solution
  4. Loss Functions
  5. Tuning Hyperparameters
  6. Results and Discussion
  7. Time-Invariant Threat Field Results
  8. Time-varying Threat Field Results
  9. Planning Multiple Paths in a Static Threat Field
  10. Conclusions and Future Work

Figures (3)

  • Figure 1: A sample PINN output for a single path between two fixed end points in a time-invariant threat field. Each plot compares the constraints to the calculated values according the PINN output.
  • Figure 2: PINN results for a single path between two fixed end points in a time-varying threat field. Each plot compares the constraints to the calculated values according the PINN output.
  • Figure 3: Sample trajectories (upper left), Hamiltonians (upper right), state dynamics (middle), and co-state dynamics (bottom). The superscripts of variables in the legend correspond to the trajectory number (upper left).