用混合神经网络高效求解高维福克-普朗克方程，精度和速度双提升。
BridgeNet: A Hybrid, Physics-Informed Machine Learning Framework for Solving High-Dimensional Fokker-Planck Equations
- 结合卷积网络与物理信息神经网络，捕捉复杂空间结构
- 相比传统方法误差更低、收敛更快，高维场景更稳定
- 适合需要高精度模拟的金融建模与复杂系统研究
BridgeNet 是一种新型混合框架，将卷积神经网络与物理信息神经网络结合，高效求解非线性高维福克-普朗克方程（FPEs）。传统 PINN 通常依赖全连接结构，难以捕捉复杂的空间层级并精确施加边界条件。BridgeNet 采用自适应卷积层进行局部特征提取，并引入动态加权损失函数，严格满足物理约束。大量数值实验表明，该方法在多种测试案例中不仅显著降低误差指标、加快收敛速度，且在高维设置下保持强稳定性。本工作为计算物理提供了可扩展、高精度的解决方案，具有在金融数学与复杂系统动力学等领域的广泛应用前景。
原文摘要 · Abstract (English)
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.
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