Related papers: Data-Driven, ML-assisted Approaches to Problem Wel…
In this paper, we analyze nonlinear differential equations subject to generalized boundary conditions. More specifically, we provide a framework from which we can provide conditions, which are straightforward to check, for the solvability…
Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to derive the relations between the involved physical quantities…
The performance of machine learning algorithms heavily relies on the availability of a large amount of training data. However, in reality, data usually reside in distributed parties such as different institutions and may not be directly…
In this paper, we investigate the well-posedness and asymptotic behavior of difference equations of the form $x(t) = A x(t - \tau(t))$, $t \geq 0$, where the unknown function $x$ takes values in $\mathbb R^d$ for some positive integer $d$,…
Computer algebra can answer various questions about partial differential equations using symbolic algorithms. However, the inclusion of data into equations is rare in computer algebra. Therefore, recently, computer algebra models have been…
Approximate solutions of partial differential equations (PDEs) obtained by neural networks are highly affected by hyper parameter settings. For instance, the model training strongly depends on loss function design, including the choice of…
We discuss the issue of finding a good mathematical programming solver configuration for a particular instance of a given problem, and we propose a two-phase approach to solve it. In the first phase we learn the relationships between the…
An algebraic criterion that is sufficient to establish the existence of certain a priori estimates for the solution of first-order homogeneous linear characteristic problems is derived. Estimates of such kind ensure the stability of the…
The theory of one-sided coupled operator matrices, recently introduced by K.-J. Engel, is an abstract framework for concrete initial value problems and allows complete information on well-posedness, and stability of solutions. These notes…
We consider the task of training machine learning models with data-dependent constraints. Such constraints often arise as empirical versions of expected value constraints that enforce fairness or stability goals. We reformulate…
We investigate degenerate saddle point problems, which can be viewed as limit cases of standard mixed formulations of symmetric problems with large jumps in coefficients. We prove that they are well-posed in a standard norm despite the…
This paper addresses the local well-posedness of the Cauchy problem for a one-dimensional diffusion equation equipped with a dynamic boundary condition and an additional boundary condition that renders the one-dimensional Laplace operator…
When testing conditions differ from those represented in training data, so-called out-of-distribution (OOD) inputs can mar the reliability of learned components in the modern robot autonomy stack. Therefore, coping with OOD data is an…
Partial differential equations (PDEs) that fit scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects, such as physics and finance. The data-driven discovery of PDEs from…
We present a new approach to the problems of evaluating and learning personalized decision policies from observational data of past contexts, decisions, and outcomes. Only the outcome of the enacted decision is available and the historical…
We consider the long time well-posedness of the Cauchy problem with large Sobolev data for a class of nonlinear Schr\"odinger equations (NLS) on $\mathbb{R}^2$ with power nonlinearities of arbitrary odd degree. Specifically, the method in…
We establish the local Hadamard well-posedness of a certain third-order nonlinear Schr\"odinger equation with a multi-term linear part and a general power nonlinearity known as the higher-order nonlinear Schr\"odinger equation, formulated…
In theory, boundary and initial conditions are important for the wellposedness of partial differential equations (PDEs). Numerically, these conditions can be enforced exactly in classical numerical methods, such as finite difference method…
This research focuses on an innovative task of extracting equations from incomplete data, moving away from traditional methods used for complete solutions. The study addresses the challenge of extracting equations from data, particularly in…
Standard meta-learning for representation learning aims to find a common representation to be shared across multiple tasks. The effectiveness of these methods is often limited when the nuances of the tasks' distribution cannot be captured…