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Neural networks have been widely used as predictive models to fit data distribution, and they could be implemented through learning a collection of samples. In many applications, however, the given dataset may contain noisy samples or…

Recent work has introduced a simple numerical method for solving partial differential equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the method while applying it to analyze one of the most fundamental…

Deep Neural Networks are well known for efficiently fitting training data, yet experiencing poor generalization capabilities whenever some kind of bias dominates over the actual task labels, resulting in models learning "shortcuts". In…

Deep neural networks (DNN) have outstanding performance in various applications. Despite numerous efforts of the research community, out-of-distribution (OOD) samples remain a significant limitation of DNN classifiers. The ability to…

Neural ordinary differential equations (Neural ODEs) is a class of machine learning models that approximate the time derivative of hidden states using a neural network. They are powerful tools for modeling continuous-time dynamical systems,…

Residual neural networks are state-of-the-art deep learning models. Their continuous-depth analog, neural ordinary differential equations (ODEs), are also widely used. Despite their success, the link between the discrete and continuous…

This paper considers a noisy data structure recovery problem. The goal is to investigate the following question: Given a noisy observation of a permuted data set, according to which permutation was the original data sorted? The focus is on…

We present a computational technique for modeling the evolution of dynamical systems in a reduced basis, with a focus on the challenging problem of modeling partially-observed partial differential equations (PDEs) on high-dimensional…

We propose a technique for reformulation of state and parameter estimation problems as that of matching explicitly computable definite integrals with known kernels to data. The technique applies for a class of systems of nonlinear ordinary…

PDE discovery shows promise for uncovering predictive models of complex physical systems but has difficulty when measurements are sparse and noisy. We introduce a new approach for PDE discovery that uses two Rational Neural Networks and a…

In the last decade, the scientific community has devolved its attention to the deployment of data-driven approaches in scientific research to provide accurate and reliable analysis of a plethora of phenomena. Most notably, Physics-informed…

Augmenting mechanistic ordinary differential equation (ODE) models with machine-learnable structures is an novel approach to create highly accurate, low-dimensional models of engineering systems incorporating both expert knowledge and…

It is a challenging topic in applied mathematics to solve high-dimensional nonlinear partial differential equations (PDEs). Standard approximation methods for nonlinear PDEs suffer under the curse of dimensionality (COD) in the sense that…

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…

Modeling biological dynamical systems is challenging due to the interdependence of different system components, some of which are not fully understood. To fill existing gaps in our ability to mechanistically model physiological systems, we…

In this paper, we consider robust nonparametric regression using deep neural networks with ReLU activation function. While several existing theoretically justified methods are geared towards robustness against identical heavy-tailed noise…

The neural Ordinary Differential Equation (ODE) model has shown success in learning complex continuous-time processes from observations on discrete time stamps. In this work, we consider the modeling and forecasting of time series data that…

Background. Commonly, Deep Neural Networks (DNNs) generalize well on samples drawn from a distribution similar to that of the training set. However, DNNs' predictions are brittle and unreliable when the test samples are drawn from a…

Neural Ordinary Differential Equations (NODEs) are a novel neural architecture, built around initial value problems with learned dynamics which are solved during inference. Thought to be inherently more robust against adversarial…

Neural CDEs provide a natural way to process the temporal evolution of irregular time series. The number of function evaluations (NFE) is these systems' natural analog of depth (the number of layers in traditional neural networks). It is…