Related papers: Structure-Preserving Neural Ordinary Differential …
Nonlinear phenomena can be analyzed via linear techniques using operator-theoretic approaches. Data-driven method called the extended dynamic mode decomposition (EDMD) and its variants, which approximate the Koopman operator associated with…
Modal methods are a long-standing approach to physical modelling synthesis. Extensions to nonlinear problems are possible, leading to coupled nonlinear systems of ordinary differential equations. Recent work in scalar auxiliary variable…
Continuous deep learning models, referred to as Neural Ordinary Differential Equations (Neural ODEs), have received considerable attention over the last several years. Despite their burgeoning impact, there is a lack of formal analysis…
Learning complex trajectories from demonstrations in robotic tasks has been effectively addressed through the utilization of Dynamical Systems (DS). State-of-the-art DS learning methods ensure stability of the generated trajectories;…
We propose a framework to analyze stability for a class of linear non-autonomous hybrid systems, where the continuous evolution of solutions is governed by an ordinary differential equation and the instantaneous changes are governed by a…
Stiff dynamical systems represent a central challenge in multi scale modeling across combustion, chemical kinetics, and nonlinear dynamical systems. Neural operator learning has recently emerged as a promising approach to approximate…
Reduced-order models (ROMs) are essential for rapid simulation of complex biomechanical systems and for bridging the gap between high fidelity models and clinical application. However, ROMs for tissue growth and remodeling (G&R) remain…
Learning stable dynamical systems from data is crucial for safe and reliable robot motion planning and control. However, extending stability guarantees to trajectories defined on Riemannian manifolds poses significant challenges due to the…
Stability guarantees are crucial when ensuring a fully autonomous robot does not take undesirable or potentially harmful actions. Unfortunately, global stability guarantees are hard to provide in dynamical systems learned from data,…
Despite their impressive performance, contemporary neural networks often lack structural safeguards that promote stable learning and interpretable behavior. In this work, we introduce a reformulation of layer-level transformations that…
We present a statistical learning framework for robust identification of partial differential equations from noisy spatiotemporal data. Extending previous sparse regression approaches for inferring PDE models from simulated data, we address…
Neural Networks (NNs) have been identified as a potentially powerful tool in the study of complex dynamical systems. A good example is the NN differential equation (DE) solver, which provides closed form, differentiable, functional…
Physics-informed neural network architectures have emerged as a powerful tool for developing flexible PDE solvers which easily assimilate data, but face challenges related to the PDE discretization underpinning them. By instead adapting a…
Neural controlled differential equations (NCDEs), which are continuous analogues to recurrent neural networks (RNNs), are a specialized model in (irregular) time-series processing. In comparison with similar models, e.g., neural ordinary…
Random ordinary differential equations (RODEs), i.e. ODEs with random parameters, are often used to model complex dynamics. Most existing methods to identify unknown governing RODEs from observed data often rely on strong prior knowledge.…
A class of neural networks that gained particular interest in the last years are neural ordinary differential equations (neural ODEs). We study input-output relations of neural ODEs using dynamical systems theory and prove several results…
This short, self-contained article seeks to introduce and survey continuous-time deep learning approaches that are based on neural ordinary differential equations (neural ODEs). It primarily targets readers familiar with ordinary and…
Data-driven machine learning approaches are being increasingly used to solve partial differential equations (PDEs). They have shown particularly striking successes when training an operator, which takes as input a PDE in some family, and…
We present a unified theoretical framework for analyzing the stability and consistency of Physics-Informed Neural Networks (PINNs), grounded in operator coercivity, variational formulations, and non-asymptotic perturbation theory. PINNs…
Stochastic regularization of neural networks (e.g. dropout) is a wide-spread technique in deep learning that allows for better generalization. Despite its success, continuous-time models, such as neural ordinary differential equation (ODE),…