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Diffusion-based generative models use stochastic differential equations (SDEs) and their equivalent ordinary differential equations (ODEs) to establish a smooth connection between a complex data distribution and a tractable prior…

We develop fast and scalable methods for computing reduced-order nonlinear solutions (RONS). RONS was recently proposed as a framework for reduced-order modeling of time-dependent partial differential equations (PDEs), where the modes…

Machine learning algorithms have been successfully used to approximate nonlinear maps under weak assumptions on the structure and properties of the maps. We present deep neural networks using dense and convolutional layers to solve an…

We address the problem of Bayesian inference for parameters in ordinary differential equation (ODE) models based on observational data. Conventional approaches in this setting typically rely on numerical solvers such as the Euler or…

Differential-algebraic equations (DAEs) with state-dependent events arise in systems whose continuous dynamics are constrained by algebraic equations and interrupted by mode changes, switching logic, impacts, or state reinitializations.…

In this paper we investigate adaptive discretization of the iteratively regularized Gauss- Newton method IRGNM. All-at-once formulations considering the PDE and the measurement equation simultaneously allow to avoid (approximate) solution…

Gradient methods are experiencing a growth in methodological and theoretical developments owing to the challenges posed by optimization problems arising in data science. However, such gradient methods face diverging optimality gaps or…

We introduce Neural Parameter Regression (NPR), a novel framework specifically developed for learning solution operators in Partial Differential Equations (PDEs). Tailored for operator learning, this approach surpasses traditional DeepONets…

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…

Inferring parameters of macro-kinetic growth models, typically represented by Ordinary Differential Equations (ODE), from the experimental data is a crucial step in bioprocess engineering. Conventionally, estimates of the parameters are…

Many problems encountered in science and engineering can be formulated as estimating a low-rank object (e.g., matrices and tensors) from incomplete, and possibly corrupted, linear measurements. Through the lens of matrix and tensor…

Stochastic Gradient Descent (SGD) methods see many uses in optimization problems. Modifications to the algorithm, such as momentum-based SGD methods have been known to produce better results in certain cases. Much of this, however, is due…

We investigate the problem of sequential linear data prediction for real life big data applications. The second order algorithms, i.e., Newton-Raphson Methods, asymptotically achieve the performance of the "best" possible linear data…

Probabilistic ordinary differential equation (ODE) solvers have been introduced over the past decade as uncertainty-aware numerical integrators. They typically proceed by assuming a functional prior to the ODE solution, which is then…

The solution of systems of non-autonomous linear ordinary differential equations is crucial in a variety of applications, such us nuclear magnetic resonance spectroscopy. A new method with spectral accuracy has been recently introduced in…

This paper presents an algorithmic framework for solving unconstrained stochastic optimization problems using only stochastic function evaluations. We employ central finite-difference based gradient estimation methods to approximate the…

Gradient methods are widely used in optimization problems. In practice, while the smoothness parameter can be estimated utilizing techniques such as backtracking, estimating the strong convexity parameter remains a challenge; moreover, even…

We propose a new stochastic optimization framework for empirical risk minimization problems such as those that arise in machine learning. The traditional approaches, such as (mini-batch) stochastic gradient descent (SGD), utilize an…

In the present work, a high order finite element type residual distribution scheme is designed in the framework of multidimensional compressible Euler equations of gas dynamics. The strengths of the proposed approximation rely on the…

The Path-dependent Neural Jump ODE (PD-NJ-ODE) is a model for online prediction of generic (possibly non-Markovian) stochastic processes with irregular (in time) and potentially incomplete (with respect to coordinates) observations. It is a…