Related papers: Formalising the Continuous/Discrete Modeling Step

This paper is more an essay than a report. There is a gentle introduction to some issues in modeling, followed by the use of steepest descent methods to develop a model as contrasted to using such methods to solve one already in hand, as in…

Recurrent neural networks (RNN) as used in machine learning are commonly formulated in discrete time, i.e. as recursive maps. This brings a lot of advantages for training models on data, e.g. for the purpose of time series prediction or…

End-to-end learning of dynamical systems with black-box models, such as neural ordinary differential equations (ODEs), provides a flexible framework for learning dynamics from data without prescribing a mathematical model for the dynamics.…

Temporal causal representation learning methods assume that causal mechanisms switch instantaneously between discrete domains, yet real-world systems often exhibit continuous mechanism transitions. For example, a vehicle's dynamics evolve…

Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of…

In this paper we present a method of discrete modeling and analysis of multi-level dynamics of complex large-scale hierarchical dynamic systems subject to external dynamic control mechanism. In a model each state describes parallel dynamics…

Data-driven modeling of dynamical systems is a crucial area of machine learning. In many scenarios, a thorough understanding of the model's behavior becomes essential for practical applications. For instance, understanding the behavior of a…

This book explores an alternative to the current dominant paradigm where a discrete computer model is constructed as an attempt to approximate some continuum theory. We focus on a class of discrete computer models that are based on simple…

A key appeal of the recently proposed Neural Ordinary Differential Equation (ODE) framework is that it seems to provide a continuous-time extension of discrete residual neural networks. As we show herein, though, trained Neural ODE models…

Complex system design often proceeds in an iterative fashion, starting from a high-level model and adding detail as the design matures. This process can be assisted by metamodeling techniques that automate some model manipulations and check…

We propose a method of classifying the operation of a system into finitely many modes. Each mode has its own objectives for the system's behaviour and its own mathematical models and algorithms designed to accomplish its objectives. A…

The existing Neural ODE formulation relies on an explicit knowledge of the termination time. We extend Neural ODEs to implicitly defined termination criteria modeled by neural event functions, which can be chained together and…

Transitions between steady dynamical regimes in diverse applications are often modelled using discontinuities, but doing so introduces problems of uniqueness. No matter how quickly a transition occurs, its inner workings can affect the…

The aim of the present paper is to efficiently describe the membrane potential dynamics of neural populations formed by species having a high density difference in specific brain areas. We propose a hybrid model whose main ingredients are a…

In model-driven development, an ordered model transformation is a nested set of transformations between source and target classes, in which each transformation is governed by its own pre and post- conditions, but structurally dependent on…

Diffusion models generate high-dimensional data such as images by learning a process that gradually removes noise from corrupted data. Recent studies have shown that the backward dynamics of diffusion models exhibit two characteristic…

Mathematical models of biological populations commonly use discrete structure classes to capture trait variation among individuals (e.g. age, size, phenotype, intracellular state). Upscaling these discrete models into continuum descriptions…

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…

There are many different models--both continuous and discrete--used to describe gene mutation fixation. In particular, the Moran process, the Kimura equation and the replicator dynamics are all well known models, that might lead to…

Software is now ubiquitous and involved in complex interactions with the human users and the physical world in so-called cyber-physical systems where the management of time is a major issue. Separation of concerns is a key asset in the…