论文标题
部分可观测时空混沌系统的无模型预测
On the Convergence of Optimizing Persistent-Homology-Based Losses
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
Topological loss based on persistent homology has shown promise in various applications. A topological loss enforces the model to achieve certain desired topological property. Despite its empirical success, less is known about the optimization behavior of the loss. In fact, the topological loss involves combinatorial configurations that may oscillate during optimization. In this paper, we introduce a general purpose regularized topology-aware loss. We propose a novel regularization term and also modify existing topological loss. These contributions lead to a new loss function that not only enforces the model to have desired topological behavior, but also achieves satisfying convergence behavior. Our main theoretical result guarantees that the loss can be optimized efficiently, under mild assumptions.
