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To quantify uncertainties in inverse problems of partial differential equations (PDEs), we formulate them into statistical inference problems using Bayes' formula. Recently, well-justified infinite-dimensional Bayesian analysis methods have…
Neural ordinary differential equations (NODEs) treat computation of intermediate feature vectors as trajectories of ordinary differential equation parameterized by a neural network. In this paper, we propose a novel model, delay…
A Neural Network (NN) based numerical method is formulated and implemented for solving Boundary Value Problems (BVPs) and numerical results are presented to validate this method by solving Laplace equation with Dirichlet boundary condition…
Neural Ordinary Differential Equations (ODEs) represent a significant advancement at the intersection of machine learning and dynamical systems, offering a continuous-time analog to discrete neural networks. Despite their promise, deploying…
We introduce the framework of continuous--depth graph neural networks (GNNs). Graph neural ordinary differential equations (GDEs) are formalized as the counterpart to GNNs where the input-output relationship is determined by a continuum of…
Incorporating neural networks for the solution of Ordinary Differential Equations (ODEs) represents a pivotal research direction within computational mathematics. Within neural network architectures, the integration of the intrinsic…
An adaptation of the arbitrary high order ADER-DG numerical method with local DG predictor for solving the IVP for a first-order non-linear ODE system is proposed. The proposed numerical method is a completely one-step ODE solver with…
Numerical integration (NI) packages commonly used in scientific research are limited to returning the value of a definite integral at the upper integration limit, also commonly referred to as numerical quadrature. These quadrature…
A wide range of practically important problems is nowadays efficiently solved using artificial neural networks. This gave momentum to intensive development of their optical implementations, among which, the so-called diffractive neural…
We describe a method for the identification of models for dynamical systems from observational data. The method is based on the concept of symbolic regression and uses genetic programming to evolve a system of ordinary differential…
Solving initial value problems and boundary value problems of Linear Ordinary Differential Equations (ODEs) plays an important role in many applications. There are various numerical methods and solvers to obtain approximate solutions…
Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equation. We apply this principle by finding dilatations and…
Neural Ordinary Differential Equations (N-ODEs) are a powerful building block for learning systems, which extend residual networks to a continuous-time dynamical system. We propose a Bayesian version of N-ODEs that enables well-calibrated…
This paper focuses on proposing a deep learning initialized iterative method (Int-Deep) for low-dimensional nonlinear partial differential equations (PDEs). The corresponding framework consists of two phases. In the first phase, an…
A concept of using Neural Ordinary Differential Equations(NODE) for Transfer Learning has been introduced. In this paper we use the EfficientNets to explore transfer learning on CIFAR-10 dataset. We use NODE for fine-tuning our model. Using…
Domain decomposition provides an effective way to tackle the dilemma of physics-informed neural networks (PINN) which struggle to accurately and efficiently solve partial differential equations (PDEs) in the whole domain, but the lack of…
Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). In this work, the classical problem of pricing European and American financial options, based on the corresponding PDE…
Stiff ordinary differential equations (ODEs) are common in many science and engineering fields, but standard neural ODE approaches struggle to accurately learn these stiff systems, posing a significant barrier to widespread adoption of…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…
Parametric optimal control problems governed by partial differential equations (PDEs) are widely found in scientific and engineering applications. Traditional grid-based numerical methods for such problems generally require repeated…