Stochastic dynamical systems provide essential mathematical frameworks for modeling complex real-world phenomena. The Fokker-Planck-Kolmogorov (FPK) equation governs the evolution of probability density functions associated with stochastic system trajectories. Developing robust numerical methods for solving the FPK equation is critical for understanding and predicting stochastic behavior. Here, we introduce the distribution self-adaptive normalized physics-informed neural network (DSN-PINNs) for solving time-dependent FPK equations through the integration of soft normalization constraints with adaptive resampling strategies. Specifically, we employ a normalization-enhanced PINN model in a pretraining phase to establish the solution's global structure and scale, generating a reliable prior distribution. Subsequently, guided by this prior, we dynamically reallocate training points via weighted kernel density estimation, concentrating computational resources on regions most representative of the underlying probability distribution throughout the learning process. The key innovation lies in our method's ability to exploit the intrinsic structural properties of stochastic dynamics while maintaining computational accuracy and implementation simplicity. We demonstrate the framework's effectiveness through comprehensive numerical experiments and comparative analyses with existing methods, including validation on real-world economic datasets.

翻译：随机动力系统为复杂现实世界现象的建模提供了必要的数学框架。Fokker-Planck-Kolmogorov（FPK）方程支配着与随机系统轨迹相关的概率密度函数的演化过程。开发求解FPK方程的稳健数值方法对于理解和预测随机行为至关重要。本文提出了一种分布自适应的归一化物理信息神经网络（DSN-PINNs），通过将软归一化约束与自适应重采样策略相结合，用于求解时间相关的FPK方程。具体而言，我们在预训练阶段采用归一化增强的PINN模型来建立解的全局结构和尺度，从而生成可靠的先验分布。随后，在此先验分布的引导下，我们通过加权核密度估计动态重新分配训练点，在整个学习过程中将计算资源集中在最能代表底层概率分布的区域。该方法的核心创新在于能够利用随机动力学的内在结构特性，同时保持计算精度和实现简洁性。我们通过全面的数值实验、与现有方法的对比分析（包括在真实经济数据集上的验证），证明了该框架的有效性。