| Abstract: |
| A stochastic model for theme-park ride waiting times is developed by
modeling the waiting time as a continuous-time, discrete-state Markov process
with state-dependent, time-varying transition rates. These transition rates are
interpreted as a time-dependent feedback control acting on the waiting-time process, allowing us to formulate the model calibration task as a data-driven optimal
control problem. To solve this problem efficiently, we construct a physics-informed
neural network (PINN) that embeds the controlled Kolmogorov forward equation into its architecture. Under mild assumptions, we prove the existence of
an optimal time-dependent feedback control, providing theoretical support for
the learning procedure. Numerical simulations are conducted to demonstrate
the effectiveness of the PINN-based solution. The framework provides an interpretable, physically consistent, and data-driven approach for modeling and
forecasting ride waiting times. |
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