标题： BridgeNet：求解高维福克-普朗克方程的混合物理信息机器学习框架 显示英文标题

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

摘要： BridgeNet 是一种新颖的混合框架，它将卷积神经网络与物理信息神经网络相结合，以高效求解非线性、高维福克-普朗克方程（FPEs）。 传统的PINNs通常依赖全连接架构，往往难以捕捉复杂的空间层次结构并强制执行复杂的边界条件。 相比之下，BridgeNet 利用自适应CNN层进行有效的局部特征提取，并结合动态加权损失函数来严格遵守物理约束。 广泛的数值实验表明，BridgeNet 不仅在各种测试用例中实现了显著更低的误差指标和更快的收敛速度，而且在高维设置中也保持了稳健的稳定性，这比传统的PINN方法更为优越。 这项工作在计算物理学领域代表了重大进展，提供了一种可扩展且准确的解决方案方法，在金融数学到复杂系统动力学等领域具有广阔的应用前景。 显示英文摘要

摘要： 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.