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

TL;DR

BridgeNet introduces a hybrid physics-informed neural network framework combining CNNs and PINNs to efficiently solve complex, high-dimensional Fokker-Planck equations with improved accuracy and stability.

Contribution

It presents a novel hybrid architecture that enhances PINNs with CNNs and a dynamic loss function, significantly improving solution accuracy and convergence in high-dimensional problems.

Findings

Lower error metrics compared to traditional PINNs

Faster convergence in numerical experiments

Robust stability in high-dimensional settings

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.