Over the past three decades, a fruitful approach for analysis and data-driven modeling of dynamical systems has been to consider the action of (nonlinear) dynamics in state space on linear spaces of observables. These methods leverage the linearity of the associated evolution operators, namely the Koopman and transfer operators, to carry out tasks such as…

Linear complementarity problems (LCP) generalizes some fundamental problems of mathematical optimization like linear programming (LP) problem, linearly constrained quadratic programming (LQP) problem and some others. It admits an enormous number of applications in economics, engineering, science, and many other fields. After all these, it is not surprising that LCPs are usually NP-complete problems (S.J. Chung,…

Deep neural networks (DNNs) have been a successful model across diverse machine learning tasks, increasingly capturing the interest for their potential in engineering problems where PDEs have long been the dominant model. This talk delves into efficient training for PDE operator learning in both the forward and the inverse problems setting. Firstly, we address the curse…

It is well-known that nearly any function can be approximated arbitrarily-well by a neural network with non-linear activations. However, one cannot guarantee that the weights in the neural network remain bounded in norm as the approximation error goes to zero, which is an important consideration when practically training neural networks. This raises the question: What…