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Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…

Machine learning pipelines often rely on optimization procedures to make discrete decisions (e.g., sorting, picking closest neighbors, or shortest paths). Although these discrete decisions are easily computed, they break the…

Traditional approaches to stabilizing hyperbolic PDEs, such as PDE backstepping, often encounter challenges when dealing with high-dimensional or complex nonlinear problems. Their solutions require high computational and analytical costs.…

In this paper we consider the training of single hidden layer neural networks by pseudoinversion, which, in spite of its popularity, is sometimes affected by numerical instability issues. Regularization is known to be effective in such…

The unprecedented success of deep learning (DL) makes it unchallenged when it comes to classification problems. However, it is well established that the current DL methodology produces universally unstable neural networks (NNs). The…

Private machine learning involves addition of noise while training, resulting in lower accuracy. Intuitively, greater stability can imply greater privacy and improve this privacy-utility tradeoff. We study this role of stability in private…

When an online learning algorithm is used to estimate the unknown parameters of a model, the signals interacting with the parameter estimates should not decay too quickly for the optimal values to be discovered correctly. This requirement…

Improving the robustness of deep neural networks (DNNs) to adversarial examples is an important yet challenging problem for secure deep learning. Across existing defense techniques, adversarial training with Projected Gradient Decent (PGD)…

Learning how complex dynamical systems evolve over time is a key challenge in system identification. For safety critical systems, it is often crucial that the learned model is guaranteed to converge to some equilibrium point. To this end,…

Network security applications, including intrusion detection systems of deep neural networks, are increasing rapidly to make detection task of anomaly activities more accurate and robust. With the rapid increase of using DNN and the volume…

Robustness in deep neural networks and machine learning algorithms in general is an open research challenge. In particular, it is difficult to ensure algorithmic performance is maintained on out-of-distribution inputs or anomalous instances…

Deep learning methods have demonstrated significant potential for addressing complex nonlinear control problems. For real-world safety-critical tasks, however, it is crucial to provide formal stability guarantees for the designed…

Probabilistic solvers for ordinary differential equations (ODEs) provide efficient quantification of numerical uncertainty associated with simulation of dynamical systems. Their convergence rates have been established by a growing body of…

PDE-Constrained Optimization (PDECO) problems can be accelerated significantly by employing gradient-based methods with surrogate models like neural operators compared to traditional numerical solvers. However, this approach faces two key…

Modern power systems require fast and accurate dynamic simulations for stability assessment, digital twins, and real-time control, but classical ODE solvers are often too slow for large-scale or online applications. We propose a…

Scientific machine learning is an emerging field that broadly describes the combination of scientific computing and machine learning to address challenges in science and engineering. Within the context of differential equations, this has…

This research introduces an extended application of neural networks for solving nonlinear partial differential equations (PDEs). A neural network, combined with a pseudo-arclength continuation, is proposed to construct bifurcation diagrams…

Solving ill-posed inverse problems necessitates effective regularization strategies to stabilize the inversion process against measurement noise. While classical methods like Tikhonov regularization require heuristic parameter tuning, and…

Recent work has demonstrated that deep neural networks are vulnerable to adversarial examples---inputs that are almost indistinguishable from natural data and yet classified incorrectly by the network. In fact, some of the latest findings…

Ordinary differential equations (ODEs) provide a powerful framework for modeling dynamic systems arising in a wide range of scientific domains. However, most existing ODE methods focus on a single system, and do not adequately address the…