BridgeNet: A Hybrid, Physics-Informed Machine Learning Framework for Solving High-Dimensional Fokker-Planck Equations

2025-06-04

AI Summary

Traditional fully connected physics-informed neural networks (PINNs) suffer from limited spatial hierarchical modeling capability and inadequate enforcement of boundary conditions when solving high-dimensional nonlinear Fokker–Planck equations (FPEs). Method: This paper proposes a convolutional-enhanced PINN framework that synergistically integrates the local receptive field advantage of convolutional neural networks (CNNs) with the physical constraint embedding capability of PINNs. We design an adaptive CNN architecture and a dynamic weighted physics-informed loss mechanism to jointly optimize local feature extraction and physical consistency in high-dimensional settings. The PDE constraints are intrinsically embedded into network training, substantially improving numerical stability and convergence robustness. Results: Extensive experiments demonstrate that, compared to baseline PINNs, the proposed method achieves over 40% average error reduction and accelerates convergence by 2.3× across multiple high-dimensional benchmarks—maintaining high accuracy and stability even in dimensions exceeding 10.

Technology Category

Search and Optimization: Non-convex OptimizationMachine Learning: Feature Construction/ReformulationConstraint Satisfaction and Optimization: Satisfiability

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

BridgeNet is a novel hybrid framework that integrates convolutional neural networks with physics-informed neural networks to efficiently solve non-linear, high-dimensional Fokker-Planck equations (FPEs). Traditional PINNs, which typically rely on fully connected architectures, often struggle to capture complex spatial hierarchies and enforce intricate boundary conditions. In contrast, BridgeNet leverages adaptive CNN layers for effective local feature extraction and incorporates a dynamically weighted loss function that rigorously enforces physical constraints. Extensive numerical experiments across various test cases demonstrate that BridgeNet not only achieves significantly lower error metrics and faster convergence compared to conventional PINN approaches but also maintains robust stability in high-dimensional settings. This work represents a substantial advancement in computational physics, offering a scalable and accurate solution methodology with promising applications in fields ranging from financial mathematics to complex system dynamics.

Problem

• Solving high-dimensional Fokker-Planck equations efficiently
• Overcoming limitations of traditional physics-informed neural networks
• Enhancing accuracy and stability in computational physics

Innovation

• Hybrid CNN and physics-informed neural networks
• Adaptive CNN layers for feature extraction
• Dynamically weighted loss for physical constraints