用随机权重网络快速学习动力系统算子的谱特性
How deep is your network? Deep vs. shallow learning of transfer operators
- 隐藏层权重随机初始化，仅训练输出层，大幅降低计算开销
- 可直接获得算子特征函数的闭式解，且支持不确定性估计
- 适用于复杂系统分析，如蛋白质折叠与量子谐振子
我们提出一种名为 RaNNDy 的随机神经网络方法，用于从数据中学习转移算子及其谱分解。神经网络的隐藏层权重随机设定，仅训练输出层。该方法在不显著降低精度的前提下，显著减少训练时间和资源消耗，避免了深度学习常见的超参数敏感和收敛慢问题。此外，该框架可推导输出层的闭式解，直接表示算子的特征函数，并通过集成学习估计谱性质的不确定性。实验涵盖柯尔普曼算子、佩隆-弗罗贝尼乌斯算子及薛定谔算子，应用于多个随机动力系统、蛋白质折叠过程和量子谐振子。数值结果展示了该方法的优势与局限性。
原文摘要 · Abstract (English)
We propose a randomized neural network approach called RaNNDy for learning transfer operators and their spectral decompositions from data. The weights of the hidden layers of the neural network are randomly selected and only the output layer is trained. The main advantage is that without a noticeable reduction in accuracy, this approach significantly reduces the training time and resources while avoiding common problems associated with deep learning such as sensitivity to hyperparameters and slow convergence. Additionally, the proposed framework allows us to compute a closed-form solution for the output layer which directly represents the eigenfunctions of the operator. Moreover, it is possible to estimate uncertainties associated with the computed spectral properties via ensemble learning. We present results for different dynamical operators, including Koopman and Perron-Frobenius operators, which have important applications in analyzing the behavior of complex dynamical systems, and the Schrödinger operator. The numerical examples, which highlight the strengths but also weaknesses of the proposed framework, include several stochastic dynamical systems, protein folding processes, and the quantum harmonic oscillator.
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