基于能量守恒的神经网络，能稳定模拟带噪声的物理系统。
Stochastic Port-Hamiltonian Neural Networks: Universal Approximation with Passivity Guarantees
- 用神经网络参数化能量函数，强制满足物理系统的对称性和耗散性。
- 在有限时间区间内可逼近任意目标系统的动态行为，误差可控。
- 适合需要长期稳定模拟的物理系统建模，如机械振动、电路等。
随机端口-哈密顿系统以能量视角描述带有耗散、输入和随机扰动的开放动力系统。本文提出随机端口-哈密顿神经网络（SPH-NN），通过前馈网络参数化哈密顿量，并强制互连矩阵斜对称、耗散矩阵半正定。针对伊藤型动态，在紧集上停止过程的显式生成器条件下，建立了期望意义下的弱无源不等式。进一步证明了普适逼近性：在任意紧集与有限时域内，SPH-NN 可以 $C^2$ 精度逼近目标系统的系数，且耦合解在均方意义下于退出时间前保持接近。在含噪质量-弹簧、杜芬及范德波尔振子上的实验表明，相比多层感知机基线，其长期预测更准确，能量误差显著降低。
原文摘要 · Abstract (English)
Stochastic port-Hamiltonian systems represent open dynamical systems with dissipation, inputs, and stochastic forcing in an energy based form. We introduce stochastic port-Hamiltonian neural networks, SPH-NNs, which parameterize the Hamiltonian with a feedforward network and enforce skew symmetry of the interconnection matrix and positive semidefiniteness of the dissipation matrix. For Itô dynamics we establish a weak passivity inequality in expectation under an explicit generator condition, stated for a stopped process on a compact set. We also prove a universal approximation result showing that, on any compact set and finite horizon, SPH-NNs approximate the coefficients of a target stochastic port-Hamiltonian system with $C^2$ accuracy of the Hamiltonian and yield coupled solutions that remain close in mean square up to the exit time. Experiments on noisy mass spring, Duffing, and Van der Pol oscillators show improved long horizon rollouts and reduced energy error relative to a multilayer perceptron baseline.
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