An Initial Condition-Dependent Neural Network Approach for Optimal Control Problems

TL;DR

This work develops an indirect neural-network approach to solving optimal-control problems by embedding Pontryagin's Minimum Principle into the loss and training across a spectrum of initial conditions. It introduces two architectures: a time- and initial-condition–dependent NN and an initial-condition–dependent NN augmented with a Fourier layer, both designed to approximate $x^*(t)$, $\lambda^*(t)$, and $u^*(t)$ over a family of problems defined by $x_0\in\mathcal{X}_0$. Empirical results on three OCPs show high accuracy (RMSE/MAE around $10^{-4}$ in several cases) and significant efficiency gains from the Fourier layer, suggesting strong potential for scaling to larger state spaces. Overall, the paper combines PMP-based indirect optimization with neural surrogates and Fourier representations to advance numerical methods for parameterized OCPs.

Abstract

In this work, we investigate an indirect approach for the numerical solution of optimal control problems via neural networks. A customized neural network is constructed, where optimal state, co-state and control trajectories are approximated by minimizing the underlying parameterized Hamiltonian, relying on Pontryagin's Minimum Principle. Departing from previous results reported in the literature, we propose novel, modified networks with both time and trajectory initial condition as inputs. Numerical results demonstrate the ability of neural networks to integrate both time and initial condition information in solving optimal control problems. Finally, it is empirically demonstrated that approximation accuracy may be enhanced through a structural modification incorporating an intermediate layer of Fourier coefficients.

Figures (3)

• Figure 1: Schematic representation of the initial condition-dependent NN with a Fourier layer. For illustrative purposes, the network structure is shown only for the state approximation but it is identical for both control and co-state functions.
• Figure 2: Results for OCP 3 for $M=N=5$ and $I = 6$ (with the Fourier layer). (Left) Approximated and the exact control and state surfaces, respectively. (Right) Absolute Percentage Error $(APE)$ in the control and state approximations, respectively, $\forall(t, x_0) \in \mathcal{T} \times X_0$.
• Figure 3: Sample trajectories of the state surfaces shown in Figure (\ref{['fig:mainfig']}) for three distinct initial conditions.