Physics-Informed Neural Optimal Control for Precision Immobilization Technique in Emergency Scenarios

We propose Physics-Informed Tracking (PIT), a video-based framework for tracking a single particle from video, where a neural network autoencoder localizes a particle as a heatmap peak (landmark) and a differentiable physics module embedded in the autoencoder constrains several landmarks over time (a trajectory) to satisfy known dynamics. The novel Physics-Informed Landmark Loss (PILL) compares this predicted trajectory back against the landmarks, enforcing physical consistency without labels. Its supervised variant (PILLS) instead compares the prediction against ground-truth position, velocity, and bounce from simulation, enabling end-to-end backpropagation. To support supervised and unsupervised learning, we use an autoencoder with a split bottleneck that separates A) tracking-related structure via landmark heatmaps from B) background noise and subsequent image reconstruction. We evaluate a replicated 26 factorial design (n = 4 replicates, 64 configurations), showing that PILLS consistently achieves sub-pixel tracking accuracy for the bilinear and physics-refined decoder outputs under both clean and noisy conditions.

This work presents a transparent and reproducible benchmark study of a direct dual-network Physics-Informed Neural Network (PINN) formulation for the optimal control of a mass-spring-damper system. The classical linear-quadratic optimal control problem is solved by two independent classical methods -- Pontryagin's Minimum Principle with single shooting, and direct transcription through trapezoidal collocation -- and recast as a constrained optimization problem solved by two feedforward neural networks: a state network whose boundary conditions are enforced exactly through a composite cubic-and-mask ansatz, and an unconstrained control network. The composite loss combines the physics residual at the collocation points with a trapezoidal approximation of the cost functional, weighted by a single scalar hyperparameter. On the benchmark considered, the PINN reproduces the classical optimal cost to four significant digits, satisfies the terminal state constraints exactly by construction, and produces pointwise state and control errors that fall within the spread of the two classical references. Training is approximately two orders of magnitude slower than classical shooting on this benchmark, which is honestly reported. The contribution is methodological clarity rather than methodological novelty: the formulation and the accompanying Google Colab implementation are intended to lower the barrier to entry for practitioners exploring PINN-based optimal control without prior exposure to adjoint methods or two-point boundary value problems.

We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations. We first recall the classical direct and indirect viewpoints for optimal control of PDEs, and then present two PINN formulations: a direct formulation based on minimizing the objective under the state constraint, and an indirect formulation based on the first-order optimality system. For a class of semilinear parabolic equations, we derive the state equation, the adjoint equation, and the stationarity condition in a form consistent with continuous-time Pontryagin-type optimality conditions. We then specialize the framework to an Allen-Cahn control problem and compare three numerical approaches: (i) a discretize-then-optimize adjoint method, (ii) a direct PINN, and (iii) an indirect PINN. Numerical results show that the PINN parameterization has an implicit regularizing effect, in the sense that it tends to produce smoother control profiles. They also indicate that the indirect PINN more faithfully preserves the PDE contraint and optimality structure and yields a more accurate neural approximation than the direct PINN.