Related papers: Error Estimation and Correction from within Neural…

We propose rigorous lower and upper error bounds for neural network (NN) approximations to PDEs by efficiently computing the Riesz representations of suitable extension and restrictions of the NN residual towards geometrically simpler…

Deep Neural Networks (DNNs) often rely on statistical correlations rather than causal reasoning, limiting their robustness and interpretability. While testing methods can identify failures, effective debugging and repair remain challenging.…

We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretisation errors…

Ordinary differential equations (ODEs), via their induced flow maps, provide a powerful framework to parameterize invertible transformations for the purpose of representing complex probability distributions. While such models have achieved…

Inverse problems arise anywhere we have indirect measurement. As, in general they are ill-posed, to obtain satisfactory solutions for them needs prior knowledge. Classically, different regularization methods and Bayesian inference based…

Implicit layer deep learning techniques, like Neural Differential Equations, have become an important modeling framework due to their ability to adapt to new problems automatically. Training a neural differential equation is effectively a…

The success of deep neural networks (DNNs) is attributable to three factors: increased compute capacity, more complex models, and more data. These factors, however, are not always present, especially for edge applications such as autonomous…

The growing body of research shows how to replace classical partial differential equation (PDE) integrators with neural networks. The popular strategy is to generate the input-output pairs with a PDE solver, train the neural network in the…

In this paper, we construct approximated solutions of Differential Equations (DEs) using the Deep Neural Network (DNN). Furthermore, we present an architecture that includes the process of finding model parameters through experimental data,…

Deep neural networks (DNNs) have achieved remarkable success in numerous domains, and their application to PDE-related problems has been rapidly advancing. This paper provides an estimate for the generalization error of learning Lipschitz…

Although Neural Differential Equations have shown promise on toy problems such as MNIST, they have yet to be successfully applied to more challenging tasks. Inspired by variational methods for image restoration relying on partial…

All discretized numerical models contain modelling errors - this reality is amplified when reduced-order models are used. The ability to accurately approximate modelling errors informs statistics on model confidence and improves…

Neural networks are among the most accurate supervised learning methods in use today. However, their opacity makes them difficult to trust in critical applications, especially when conditions in training may differ from those in practice.…

Neural ordinary differential equations (NODE) have been recently proposed as a promising approach for nonlinear system identification tasks. In this work, we systematically compare their predictive performance with current state-of-the-art…

Deep neural networks can be roughly divided into deterministic neural networks and stochastic neural networks.The former is usually trained to achieve a mapping from input space to output space via maximum likelihood estimation for the…

We explore unique considerations involved in fitting ML models to data with very high precision, as is often required for science applications. We empirically compare various function approximation methods and study how they scale with…

State of the art Deep Neural Networks (DNN) can now achieve above human level accuracy on image classification tasks. However their outstanding performances come along with a complex inference mechanism making them arduously interpretable…

Neural ordinary differential equations (NODEs) -- parametrizations of differential equations using neural networks -- have shown tremendous promise in learning models of unknown continuous-time dynamical systems from data. However, every…

We develop a framework for estimating unknown partial differential equations from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network,…

Solving Linear Ordinary Differential Equations (ODEs) plays an important role in many applications. There are various numerical methods and solvers to obtain approximate solutions. However, few work about global error estimation can be…