Solving Fokker-Planck-Kolmogorov Equation by Distribution Self-adaptation Normalized Physics-informed Neural Networks

2025-10-10

AI Summary

This work addresses the lack of robustness and insufficient resolution in critical regions when solving time-varying Fokker–Planck–Kolmogorov (FPK) equations for probability density evolution. We propose a Distribution-Adaptive Normalized Physics-Informed Neural Network (DA-N-PINN), which integrates soft normalization constraints with a prior-guided adaptive resampling strategy to establish global structural awareness during pretraining and dynamically concentrate sampling on high-gradient and high-probability-density regions. Furthermore, DA-N-PINN couples normalized-enhanced PINNs, weighted kernel density estimation, and a physics-constraint-driven joint optimization mechanism. Extensive validation on benchmark numerical experiments and real-world macroeconomic datasets demonstrates significant improvements in solution accuracy and computational efficiency. Notably, the method exhibits superior generalizability and robustness under sparse-data and strongly nonlinear regimes.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Probabilistic Programming
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• Solving time-dependent Fokker-Planck-Kolmogorov equations using neural networks
• Developing adaptive sampling for probability density function evolution
• Enhancing computational accuracy for stochastic dynamical systems modeling

Innovation

• Soft normalization constraints enhance PINN training
• Adaptive resampling concentrates on key distribution regions
• Weighted kernel density estimation guides point allocation