Related papers: Can Neural Networks learn Finite Elements?

To reduce memory footprint and run-time latency, techniques such as neural network pruning and binarization have been explored separately. However, it is unclear how to combine the best of the two worlds to get extremely small and efficient…

Artificial Neuronal Networks are models widely used for many scientific tasks. One of the well-known field of application is the approximation of high-dimensional problems via Deep Learning. In the present paper we investigate the Deep…

In a general setting, we study a posteriori estimates used in finite element analysis to measure the error between a solution and its approximation. The latter is not necessarily generated by a finite element method. We show that the error…

We present a novel approach to modelling and learning vector fields from physical systems using neural networks that explicitly satisfy known linear operator constraints. To achieve this, the target function is modelled as a linear…

We propose a set of precise criteria for saying a neural net learns and uses a "world model." The goal is to give an operational meaning to terms that are often used informally, in order to provide a common language for experimental…

The neural network-based approach to solving partial differential equations has attracted considerable attention due to its simplicity and flexibility in representing the solution of the partial differential equation. In training a neural…

One of the arguments to explain the success of deep learning is the powerful approximation capacity of deep neural networks. Such capacity is generally accompanied by the explosive growth of the number of parameters, which, in turn, leads…

Graph Neural Networks (GNNs) are a powerful representational tool for solving problems on graph-structured inputs. In almost all cases so far, however, they have been applied to directly recovering a final solution from raw inputs, without…

We investigate the complexity of deep neural networks through the lens of functional equivalence, which posits that different parameterizations can yield the same network function. Leveraging the equivalence property, we present a novel…

We study an alternative use of machine learning. We train neural nets to provide the parameter estimate of a given (structural) econometric model, for example, discrete choice or consumer search. Training examples consist of datasets…

Training deep neural networks for solving machine learning problems is one great challenge in the field, mainly due to its associated optimisation problem being highly non-convex. Recent developments have suggested that many training…

One of the most important parts of Artificial Neural Networks is minimizing the loss functions which tells us how good or bad our model is. To minimize these losses we need to tune the weights and biases. Also to calculate the minimum value…

Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is…

Modern deep learning algorithms tend to optimize an objective metric, such as minimize a cross entropy loss on a training dataset, to be able to learn. The problem is that the single metric is an incomplete description of the real world…

We present a novel approach that integrates unfitted finite element methods and neural networks to approximate partial differential equations on complex geometries. Easy-to-generate background meshes (e.g., a simple Cartesian mesh) that cut…

$\partial\mathbb{B}$ nets are differentiable neural networks that learn discrete boolean-valued functions by gradient descent. $\partial\mathbb{B}$ nets have two semantically equivalent aspects: a differentiable soft-net, with real weights,…

Deep neural networks (DNNs) have shown great success in many machine learning tasks. Their training is challenging since the loss surface of the network architecture is generally non-convex, or even non-smooth. How and under what…

Graph Neural Networks (GNNs) are the first choice for learning algorithms on graph data. GNNs promise to integrate (i) node features as well as (ii) edge information in an end-to-end learning algorithm. How does this promise work out…

The use of neural networks to approximate partial differential equations (PDEs) has gained significant attention in recent years. However, the approximation of PDEs with localised phenomena, e.g., sharp gradients and singularities, remains…

We investigate the potential of applying (D)NN ((deep) neural networks) for approximating nonlinear mappings arising in the finite element discretization of nonlinear PDEs (partial differential equations). As an application, we apply the…