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

2025/06/04 by Elmira Mirzabeigi, Mirzabeigi, Elmira, Rezvan Salehi +3
Computer Science · Materials Science · Physics and Astronomy · #Analysis of PDEs (math.AP) #Computational Physics (physics.comp-ph) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Generative Adversarial Networks and Image Synthesis #Machine Learning (cs.LG) #Machine Learning in Materials Science #Mathematical Physics (math-ph) #Model Reduction and Neural Networks

paper · pdf · doi:10.48550/arxiv.2506.04354

openalex publication_date 2025/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Tensor neural networks for high-dimensional Fokker-Planck equations
• The use of physics-informed neural network approach to image restoration via nonlinear PDE tools
• A fast surrogate cross validation algorithm for meshfree RBF collocation approaches
• Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models
• Probability flow solution of the Fokker-Planck equation
• Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport
• Legendre Deep Neural Network (LDNN) and its application for approximation of nonlinear Volterra Fredholm Hammerstein integral equations
• Symbolically Solving Partial Differential Equations using Deep Learning
• Designing of knowledge-based potentials via B-spline basis functions for native proteins detection
• A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• ImageNet classification with deep convolutional neural networks

Related

• A deep learning method for solving Fokker-Planck equations
• Tensor neural networks for high-dimensional Fokker-Planck equations
• Solving Fokker-Planck equations using the zeros of Fokker-Planck operators and the Feynman-Kac formula
• From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning
• Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equations