<p>This study focuses on determining periodic orbits in the Circular Restricted Three-Body Problem (CR3BP) by introducing a novel Physics-informed Neural Network framework with hard constraints based on Dual-Step Training (HDT-PINN). The proposed approach integrates several key components: initial hard constraints, a dual-step training strategy, and an augmented parameter optimization. The initial hard constraints enforce consistency between the computed orbit and the target orbit for the initial conditions. The dual-step training strategy, including the pre-training step and final training step, effectively avoids convergence to local minima and is crucial for the successful identification of target periodic orbits. Furthermore, the augmented parameter optimization incorporates the initial velocity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\bar{\dot{y}}_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mrow> <mover accent="true"> <mi>y</mi> <mo>˙</mo> </mover> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> and the orbital period <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t_f$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>t</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> as trainable parameters, which are used to solve planar symmetry periodic orbits. For three-dimensional halo orbits, an additional optimization parameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\bar{x}_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mrow> <mi>x</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> is added, significantly enhancing the accuracy of the solutions. The effectiveness and robustness of the method are demonstrated across four distinct families of periodic orbits: Lyapunov orbits around the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> points, distant retrograde orbits (DROs), and halo orbits. Furthermore, we conducted a convergence analysis of the initial velocity <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\bar{\dot{y} }_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mrow> <mover accent="true"> <mi>y</mi> <mo>˙</mo> </mover> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> under various levels of disturbance and determined the convergence region for the initial velocity error <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\delta _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> associated with the periodic orbit considered in the example. The method proposed in this study exhibits a significantly broader convergence region in initial velocity error and precise target periodic orbit.</p>

Physics-informed neural network with hard constraints based on dual-step training strategy for periodic orbits in the CR3BP

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发表日期：

2026年6月22日

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Yong Chen,
Ming Cui,
Ying-Jing Qian,
Ye-Feng Cheng,
Wen-Xue Chen

摘要

This study focuses on determining periodic orbits in the Circular Restricted Three-Body Problem (CR3BP) by introducing a novel Physics-informed Neural Network framework with hard constraints based on Dual-Step Training (HDT-PINN). The proposed approach integrates several key components: initial hard constraints, a dual-step training strategy, and an augmented parameter optimization. The initial hard constraints enforce consistency between the computed orbit and the target orbit for the initial conditions. The dual-step training strategy, including the pre-training step and final training step, effectively avoids convergence to local minima and is crucial for the successful identification of target periodic orbits. Furthermore, the augmented parameter optimization incorporates the initial velocity $\bar{\dot{y}}_0$ ${\bar{\dot{y}}}_{0}$ and the orbital period $t_f$ $t_{f}$ as trainable parameters, which are used to solve planar symmetry periodic orbits. For three-dimensional halo orbits, an additional optimization parameter $\bar{x}_0$ ${\bar{x}}_{0}$ is added, significantly enhancing the accuracy of the solutions. The effectiveness and robustness of the method are demonstrated across four distinct families of periodic orbits: Lyapunov orbits around the $L_1$ $L_{1}$ and $L_2$ $L_{2}$ points, distant retrograde orbits (DROs), and halo orbits. Furthermore, we conducted a convergence analysis of the initial velocity $\bar{\dot{y} }_0$ ${\bar{\dot{y}}}_{0}$ under various levels of disturbance and determined the convergence region for the initial velocity error $\delta _1$ $δ_{1}$ associated with the periodic orbit considered in the example. The method proposed in this study exhibits a significantly broader convergence region in initial velocity error and precise target periodic orbit.