Related papers: Joint Bayesian Inference of Parameter and Discreti…

We present a new approach for estimating parameters in rational ODE models from given (measured) time series data. In typical existing approaches, an initial guess for the parameter values is made from a given search interval. Then, in a…

Recently, Neural Ordinary Differential Equations has emerged as a powerful framework for modeling physical simulations without explicitly defining the ODEs governing the system, but instead learning them via machine learning. However, the…

Computer simulations of differential equations require a time discretization, which inhibits to identify the exact solution with certainty. Probabilistic simulations take this into account via uncertainty quantification. The construction of…

We introduce an unfitted finite element method with Lagrange-multipliers to study an Eulerian time stepping scheme for moving domain problems applied to a model problem where the domain motion is implicit to the problem. We consider a…

Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs), also known as ODE filters, have been established as efficient methods for quantifying numerical uncertainty in the solution of ODEs. In practical…

In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of an optimal control problem governed by a simplified linear gradient enhanced damage model. The model equations are of a…

The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical…

Practical diffusion sampling is a numerical approximation problem: under a fixed inference budget, one must simulate a reverse-time ODE or SDE using only a limited number of denoising steps, so discretization error is often the dominant…

Parameter inference in ordinary differential equations is an important problem in many applied sciences and in engineering, especially in a data-scarce setting. In this work, we introduce a novel generative modeling approach based on…

This paper studies Bayesian ranking and selection (R&S) problems with correlated prior beliefs and continuous domains, i.e. Bayesian optimization (BO). Knowledge gradient methods [Frazier et al., 2008, 2009] have been widely studied for…

We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs). We show that the accumulation of rounding errors results in a solution that is inherently random and we obtain the…

Traditional partial differential equations with constant coefficients often struggle to capture abrupt changes in real-world phenomena, leading to the development of variable coefficient PDEs and Markovian switching models. Recently,…

In this article, we consider computing expectations w.r.t. probability measures which are subject to discretization error. Examples include partially observed diffusion processes or inverse problems, where one may have to discretize time…

This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for…

This paper suggests a framework for the learning of discretizations of expensive forward models in Bayesian inverse problems. The main idea is to incorporate the parameters governing the discretization as part of the unknown to be estimated…

Mechanistic models with differential equations are a key component of scientific applications of machine learning. Inference in such models is usually computationally demanding, because it involves repeatedly solving the differential…

Diffusion and flow matching models generate high-fidelity data by simulating paths defined by Ordinary or Stochastic Differential Equations (ODEs/SDEs), starting from a tractable prior distribution. The probability flow ODE formulation…

Ordinary Differential Equations are a simple but powerful framework for modeling complex systems. Parameter estimation from times series can be done by Nonlinear Least Squares (or other classical approaches), but this can give…

Ordinary differential equations (ODEs) are commonly used to model dynamic behavior of a system. Because many parameters are unknown and have to be estimated from the observed data, there is growing interest in statistics to develop…

A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a…