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We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose…

In the Network Flow Interdiction problem an adversary attacks a network in order to minimize the maximum s-t-flow. Very little is known about the approximatibility of this problem despite decades of interest in it. We present the first…

Inspired by normalizing flows, we analyze the bilinear control of neural transport equations by means of time-dependent velocity fields restricted to fulfill, at any time instance, a simple neural network ansatz. The L^1 approximate…

We propose Characteristic-Neural Ordinary Differential Equations (C-NODEs), a framework for extending Neural Ordinary Differential Equations (NODEs) beyond ODEs. While NODEs model the evolution of a latent variables as the solution to an…

Fluid limit techniques have become a central tool to analyze queueing networks over the last decade, with applications to performance analysis, simulation and optimization. In this paper, some of these techniques are extended to a general…

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…

The power flow equations are fundamental to power system planning, analysis, and control. However, the inherent non-linearity and non-convexity of these equations present formidable obstacles in problem-solving processes. To mitigate these…

In recent years, deep learning has been connected with optimal control as a way to define a notion of a continuous underlying learning problem. In this view, neural networks can be interpreted as a discretization of a parametric Ordinary…

The study of high dimensional data sets often rely on their low dimensional projections that preserve the local geometry of the original space. While numerous methods have been developed to summarize this space as variations of tree-like…

How can we understand gradient-based training over non-convex landscapes? The edge of stability phenomenon, introduced in Cohen et al. (2021), indicates that the answer is not so simple: namely, gradient descent (GD) with large step sizes…

We develop efficient algorithms for a fundamental network design problem arising in potential-based flow models, which are central to many energy transport networks (e.g., hydrogen and electricity). In contrast to classical network flow…

Flow matching has emerged as a simulation-free alternative to diffusion-based generative modeling, producing samples by solving an ODE whose time-dependent velocity field is learned along an interpolation between a simple source…

Neural ordinary differential equations (neural ODE) are powerful continuous-time machine learning models for depicting the behavior of complex dynamical systems, but their verification remains challenging due to limited reachability…

Neural Controlled Differential Equations (NCDEs) are a state-of-the-art tool for supervised learning with irregularly sampled time series (Kidger, 2020). However, no theoretical analysis of their performance has been provided yet, and it…

Normalizing flows are a powerful technique for obtaining reparameterizable samples from complex multimodal distributions. Unfortunately, current approaches are only available for the most basic geometries and fall short when the underlying…

This work proposes an extension of neural ordinary differential equations (NODEs) by introducing an additional set of ODE input parameters to NODEs. This extension allows NODEs to learn multiple dynamics specified by the input parameter…

Optimal Power Flow (OPF) is a fundamental problem in power systems. It is computationally challenging and a recent line of research has proposed the use of Deep Neural Networks (DNNs) to find OPF approximations at vastly reduced runtimes…

Neural networks are regularly employed in adaptive control of nonlinear systems and related methods of reinforcement learning. A common architecture uses a neural network with a single hidden layer (i.e. a shallow network), in which the…

Data-driven modeling of constrained multibody dynamics remains challenged by (i) the training cost of Neural ODEs, which typically require backpropagation through an ODE solver, and (ii) error accumulation in rollout predictions. We…

Theoretical treatments of tunneling in electronic devices are often based on one-dimensional (1D) approximations. Here we show that for many nanoscale devices, such as widely studied semiconductor gate-defined quantum dots, 1D…