We show the similarities between our method, Neural ODE, and differentiable physics in Figure 4. All the three approaches have a differentiable system governed by some kinds of differential equations. Our method parametrizes the dynamics using continuous basis functions; Neural ODE uses neural networks; and Differentiable physics describes the dynamics system using physics equations like Newton's Second Law, Navier-Stokes equations. Let Uv(t2,t1) be as defined in Theorem 3.2. Let Lbe defined as (4), and H(v,t) = P jfj(v,t)Hj.

We formulate the first differentiable analog quantum computing framework with specific parameterization design at the analog signal (pulse) level to better exploit near-term quantum devices via variational methods. We further propose a scalable approach to estimate the gradients of quantum dynamics using a forward pass with Monte Carlo sampling, which leads to a quantum stochastic gradient descent algorithm for scalable gradient-based training in our framework. Applying our framework to quantum optimization and control, we observe a significant advantage of differentiable analog quantum computing against SOTAs based on parameterized digital quantum circuits by orders of magnitude.

A.1 Proof of Theorem 1 Before proofing Theorem 1, We first demonstrate the superiority of even-hop neighbors over odd-hop neighbors from the perspective of random walks. In a binary node classification task, denote the probability of a random walk of length k that starts and ends with nodes of the same label as pk,k > 0. Suppose the edge homophily level his a random variable that belongs to a uniform distribution in [0,1] and p1 = h, then: Lemma 1. If k is odd, Eh[pk] = 12. If k is even, Eh[pk] 12. Proof. We now provide a brief discussion of the superiority of even-hop neighbors in multi-class node classification tasks following [14].

Graph Neural Networks (GNNs) have received extensive research attention for their promising performance in graph machine learning. Despite their extraordinary predictive accuracy, existing approaches, such as GCN and GPRGNN, are not robust in the face of homophily changes on test graphs, rendering these models vulnerable to graph structural attacks and with limited capacity in generalizing to graphs of varied homophily levels. Although many methods have been proposed to improve the robustness of GNN models, the majority of these techniques are restricted to the spatial domain and employ complicated defense mechanisms, such as learning new graph structures or calculating edge attention. In this paper, we study the problem of designing simple and robust GNN models in the spectral domain. We propose EvenNet, a spectral GNN corresponding to an even-polynomial graph filter. Based on our theoretical analysis in both spatial and spectral domains, we demonstrate that EvenNet outperforms full-order models in generalizing across homophilic and heterophilic graphs, implying that ignoring odd-hop neighbors improves the robustness of GNNs. We conduct experiments on both synthetic and real-world datasets to demonstrate the effectiveness of EvenNet. Notably, EvenNet outperforms existing defense models against structural attacks without introducing additional computational costs and maintains competitiveness in traditional node classification tasks on homophilic and heterophilic graphs.

In this paper, we introduce a novel approach to fine-grained cross-view geolocalization.