Related papers: Algebraic identifiability of partial differential …
Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation…
Mathematical modeling with Ordinary Differential Equations (ODEs) has proven to be extremely successful in a variety of fields, including biology. However, these models are completely deterministic given a certain set of initial conditions.…
The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modeled into the equations as random coefficients. However, very often the variability of…
We explore the derivation of distributed parameter system evolution laws (and in particular, partial differential operators and associated partial differential equations, PDEs) from spatiotemporal data. This is, of course, a classical…
We present a deep learning emulator for stochastic and chaotic spatio-temporal systems, explicitly conditioned on the parameter values of the underlying partial differential equations (PDEs). Our approach involves pre-training the model on…
The paper represents the method for construction of the families of particular solutions to some new classes of $(n+1)$ dimensional nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic…
In recent years a large literature on deep learning based methods for the numerical solution partial differential equations has emerged; results for integro-differential equations on the other hand are scarce. In this paper we study deep…
Since the seminal work of [9] and their Physics-Informed neural networks (PINNs), many efforts have been conducted towards solving partial differential equations (PDEs) with Deep Learning models. However, some challenges remain, for…
Differential Equations are among the most important Mathematical tools used in creating models in the science, engineering, economics, mathematics, physics, aeronautics, astronomy, dynamics, biology, chemistry, medicine, environmental…
Foundation models for partial differential equations (PDEs) have emerged as powerful surrogates pre-trained on diverse physical systems, but adapting them to new downstream tasks remains challenging due to limited task-specific data and…
The dynamics of systems biological processes are usually modeled by a system of ordinary differential equations (ODEs) with many unknown parameters that need to be inferred from noisy and sparse measurements. Here, we introduce…
Learning underlying dynamics from data is important and challenging in many real-world scenarios. Incorporating differential equations (DEs) to design continuous networks has drawn much attention recently, however, most prior works make…
This article is a review of theoretical advances in the research field of algebraic geometry and Bayesian statistics in the last two decades. Many statistical models and learning machines which contain hierarchical structures or latent…
Modeling real-world problems with partial differential equations (PDEs) is a prominent topic in scientific machine learning. Classic solvers for this task continue to play a central role, e.g. to generate training data for deep learning…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…
When a mathematical or computational model is used to analyse some system, it is usual that some parameters resp.\ functions or fields in the model are not known, and hence uncertain. These parametric quantities are then identified by…
The Partial Integral Equation (PIE) framework was developed to computationally analyze linear Partial Differential Equations (PDEs) where the PDE is first converted to a PIE and then the analysis problem is solved by solving operator-valued…
The data-driven models allow one to define the model structure in cases when a priori information is not sufficient to build other types of models. The possible way to obtain physical interpretation is the data-driven differential equation…
We investigate the use of models from the theory of regularity structures as features in machine learning tasks. A model is a polynomial function of a space-time signal designed to well-approximate solutions to partial differential…
Partial differential equation parameter estimation is a mathematical and computational process used to estimate the unknown parameters in a partial differential equation model from observational data. This paper employs a greedy sampling…