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Deep learning models have gained great success in many real-world applications. However, most existing networks are typically designed in heuristic manners, thus lack of rigorous mathematical principles and derivations. Several recent…
Network scientists often use complex dynamic processes to describe network contagions, but tools for fitting contagion models typically assume simple dynamics. Here, we address this gap by developing a nonparametric method to reconstruct a…
In this study, an efficient reanalysis strategy for dynamic topology optimization is proposed. Compared with other related studies, an online successive dynamic reanalysis method and POD-based approximate dynamic displacement strategy are…
Reconstructing network dynamics from data is crucial for predicting the changes in the dynamics of complex systems such as neuron networks; however, previous research has shown that the reconstruction is possible under strong constraints…
Real-world network systems are inherently dynamic, with network topologies undergoing continuous changes over time. Previous works often focus on static networks or rely on complete prior knowledge of evolving topologies, whereas real-world…
We build on the dynamical systems approach to deep learning, where deep residual networks are idealized as continuous-time dynamical systems, from the approximation perspective. In particular, we establish general sufficient conditions for…
Recent progress in deep learning has been driven by increasingly larger models. However, their computational and energy demands have grown proportionally, creating significant barriers to their deployment and to a wider adoption of deep…
Neural networks hold great potential to act as approximate models of nonlinear dynamical systems, with the resulting neural approximations enabling verification and control of such systems. However, in safety-critical contexts, the use of…
A useful approach to the mathematical analysis of large-scale biological networks is based upon their decompositions into monotone dynamical systems. This paper deals with two computational problems associated to finding decompositions…
Neural networks offer high-accuracy solutions to a range of problems, but are costly to run in production systems because of computational and memory requirements during a forward pass. Given a trained network, we propose a techique called…
Numerical methods of approximate solution of the Cauchy problem for coupled systems of evolution equations are considered. Separating simpler subproblems for individual components of the solution achieves simplification of the problem at a…
Multiscale dynamics are ubiquitous in applications of modern science. Because of time scale separation between relatively small set of slowly evolving variables and (typically) much larger set of rapidly changing variables, direct numerical…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…
Differential equations are a ubiquitous tool to study dynamics, ranging from physical systems to complex systems, where a large number of agents interact through a graph with non-trivial topological features. Data-driven approximations of…
This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach is formalised as attempting to solve a low-rank…
Discrete dynamic models are a powerful tool for the understanding and modeling of large biological networks. Although a lot of progress has been made in developing analysis tools for these models, there is still a need to find approaches…
Images of natural systems may represent patterns of network-like structure, which could reveal important information about the topological properties of the underlying subject. However, the image itself does not automatically provide a…
In this paper, a class of Decentralized Approximate Newton (DEAN) methods for addressing convex optimization on a networked system are developed, where nodes in the networked system seek for a consensus that minimizes the sum of their…
The use of separable approximations is proposed to mitigate the curse of dimensionality related to the approximation of high-dimensional value functions in optimal control. The separable approximation exploits intrinsic decaying sensitivity…
Numerically computing global policies to optimal control problems for complex dynamical systems is mostly intractable. In consequence, a number of approximation methods have been developed. However, none of the current methods can quantify…