We prove a theorem concerning the approximation of generalized bandlimited multivariate functions by deep ReLU networks for which the curse of the dimensionality is overcome. Our theorem is based on a result by Maurey and on the ability of deep ReLU networks to approximate Chebyshev polynomials and analytic functions efficiently.

We propose a novel algorithm, based on physics-informed neural networks (PINNs) to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara, Camassa-Holm and Benjamin-Ono equations. The stability of solutions of these dispersive PDEs is leveraged to prove rigorous bounds on the resulting error. We present several numerical experiments to demonstrate that PINNs can approximate solutions of these dispersive PDEs very accurately.

Randomize-then-optimize (RTO) is widely used for sampling from posterior distributions in Bayesian inverse problems. However, RTO can be computationally intensive for complexity problems due to repetitive evaluations of the expensive forward model and its gradient. In this work, we present a novel goal-oriented deep neural networks (DNN) surrogate approach to substantially reduce the computation burden of RTO. In particular, we propose to draw the training points for the DNN-surrogate from a local approximated posterior distribution — yielding a flexible and efficient sampling algorithm that converges to the direct RTO approach. We present a Bayesian inverse problem governed by elliptic PDEs to demonstrate the computational accuracy and efficiency of our DNN-RTO approach, which shows that DNN-RTO can significantly outperform the traditional RTO.

We analyze the convergence of the weighted nonlocal Laplacian (WNLL) on the high dimensional randomly distributed point cloud. Our analysis reveals the importance of the scaling weight, $\mu \sim |P|/|S|$ with $|P|$ and $|S|$ being the number of entire and labeled data, respectively, in WNLL. The established result gives a theoretical foundation of the WNLL for high dimensional data interpolation.

A new understanding of adversarial examples and adversarial robustness is proposed by decoupling the data generator and the label generator (which we call the teacher). In our framework, adversarial robustness is a conditional concept — the student model is not absolutely robust, but robust with respect to the teacher. Based on the new understanding, we claim that adversarial examples exist because the student cannot obtain sufficient information of the teacher from the training data. Various ways of achieving robustness is compared. Theoretical and numerical evidence shows that to efficiently attain robustness, a teacher that actively provides its information to the student may be necessary.

This paper discusses a numerical method for computing the evolution of large interacting system of quantum particles. The idea of the random batch method is to replace the total interaction of each particle with the $N-1$ other particles by the interaction with $p\ll N$ particles chosen at random at each time step, multiplied by $(N-1)/p$. This reduces the computational cost of computing the interaction potential per time step from $O(N^2)$ to $O(N)$. For simplicity, we consider only in this work the case $p=1$ — in other words, we assume that $N$ is even, and that at each time step, the $N$ particles are organized in $N/2$ pairs, with a random reshuffling of the pairs at the beginning of each time step. We obtain a convergence estimate for the Wigner transform of the single-particle reduced density matrix of the particle system at time $t$ that is both uniform in $N>1$ and independent of the Planck constant $\hbar$. The key idea is to use a new type of distance on the set of quantum states that is reminiscent of the Wasserstein distance of exponent $1$ (or Monge-Kantorovich-Rubinstein distance) on the set of Borel probability measures on $\mathbf{R}^d$ used in the context of optimal transport.