用高阶隐式龙格-库塔神经网络求解非线性微分方程相空间解。
Learning phase-space flows using time-discrete implicit Runge-Kutta PINNs
- 基于隐式龙格-库塔构造物理信息神经网络，处理坐标依赖的动态系统。
- 成功求解中心力场与周期电场中粒子的运动方程，精度高且稳定。
- 适合研究保守系统、周期性场中的动力学问题，尤其适合高精度仿真。
我们提出一种计算框架，利用高阶隐式龙格-库塔物理信息神经网络（IRK-PINNs）求解多维相空间中非线性耦合微分方程组的解。在原始工作基础上，将坐标作为函数处理，使方法适用于粒子在外场中的运动方程求解。该方案特别适用于显式时间无关和周期性场。我们成功应用该方法求解了质点在中心力场及带电粒子在周期电场中的运动方程，验证了其高效性和准确性。
原文摘要 · Abstract (English)
We present a computational framework for obtaining multidimensional phase-space solutions of systems of non-linear coupled differential equations, using high-order implicit Runge-Kutta Physics- Informed Neural Networks (IRK-PINNs) schemes. Building upon foundational work originally solving differential equations for fields depending on coordinates [J. Comput. Phys. 378, 686 (2019)], we adapt the scheme to a context where the coordinates are treated as functions. This modification enables us to efficiently solve equations of motion for a particle in an external field. Our scheme is particularly useful for explicitly time-independent and periodic fields. We apply this approach to successfully solve the equations of motion for a mass particle placed in a central force field and a charged particle in a periodic electric field.
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