Related papers: Conditionally Parameterized, Discretization-Aware …

Many different deep networks have been used to approximate, accelerate or improve traditional image operators. Among these traditional operators, many contain parameters which need to be tweaked to obtain the satisfactory results, which we…

Complex physical systems are often described by partial differential equations (PDEs) that depend on parameters such as the Reynolds number in fluid mechanics. In applications such as design optimization or uncertainty quantification,…

I provide an introduction to the application of deep learning and neural networks for solving partial differential equations (PDEs). The approach, known as physics-informed neural networks (PINNs), involves minimizing the residual of the…

Simulations with an explicit description of intermolecular forces using electronic structure methods are still not feasible for many systems of interest. As a result, empirical methods such as force fields (FF) have become an established…

Parametric models abstract part of the specification of dynamical models by integral parameters. They are for example used in computational systems biology, notably with parametric regulatory networks, which specify the global architecture…

The inverse problem of supervised reconstruction of depth-variable (time-dependent) parameters in a neural ordinary differential equation (NODE) is considered, that means finding the weights of a residual network with time continuous…

Deep Neural Networks (DNNs) are usually over-parameterized, causing excessive memory and interconnection cost on the hardware platform. Existing pruning approaches remove secondary parameters at the end of training to reduce the model size;…

Deep learning approaches for partial differential equations (PDEs) have received much attention in recent years due to their mesh-freeness and computational efficiency. However, most of the works so far have concentrated on time-dependent…

Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering, and mathematical problems involving functions of several variables, such as the propagation of heat…

Partial differential equations (PDEs) with Dirichlet boundary conditions defined on boundaries with simple geometry have been succesfuly treated using sigmoidal multilayer perceptrons in previous works. This article deals with the case of…

Neural differential equations offer a powerful approach for learning dynamics from data. However, they do not impose known constraints that should be obeyed by the learned model. It is well-known that enforcing constraints in surrogate…

Singularly perturbed partial differential equations arise in many applications, including magnetohydrodynamic duct flows, chemical reaction transport systems, and Poisson Boltzmann electrostatics. These problems are characterized by sharp…

We investigate numerous structural connections between numerical algorithms for partial differential equations (PDEs) and neural architectures. Our goal is to transfer the rich set of mathematical foundations from the world of PDEs to…

Deep learning models have proven to be exceptionally useful in performing many machine learning tasks. However, for each new dataset, choosing an effective size and structure of the model can be a time-consuming process of trial and error.…

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…

Accurate representations of unknown and sub-grid physical processes through parameterizations (or closure) in numerical simulations with quantified uncertainty are critical for resolving the coarse-grained partial differential equations…

Physics Informed Neural Networks is a numerical method which uses neural networks to approximate solutions of partial differential equations. It has received a lot of attention and is currently used in numerous physical and engineering…

We present a novel optimization strategy for training neural networks which we call "BitNet". The parameters of neural networks are usually unconstrained and have a dynamic range dispersed over all real values. Our key idea is to limit the…

Modeling the traffic dynamics is essential for understanding and predicting the traffic spatiotemporal evolution. However, deriving the partial differential equation (PDE) models that capture these dynamics is challenging due to their…

Differential equations are used to model and predict the behaviour of complex systems in a wide range of fields, and the ability to solve them is an important asset for understanding and predicting the behaviour of these systems.…