Related papers: De Rham compatible Deep Neural Network FEM
There has been rapid progress recently on the application of deep networks to the solution of partial differential equations, collectively labelled as Physics Informed Neural Networks (PINNs). In this paper, we develop Physics Informed…
We investigate deep morphological neural networks (DMNNs). We demonstrate that despite their inherent non-linearity, "linear" activations are essential for DMNNs. To preserve their inherent sparsity, we propose architectures that constraint…
Approximation theorists have established best-in-class optimal approximation rates of deep neural networks by utilizing their ability to simultaneously emulate partitions of unity and monomials. Motivated by this, we propose partition of…
We present a neural network-based method for solving linear and nonlinear partial differential equations, by combining the ideas of extreme learning machines (ELM), domain decomposition and local neural networks. The field solution on each…
In this article we present new results on neural networks with linear threshold activation functions. We precisely characterize the class of functions that are representable by such neural networks and show that 2 hidden layers are…
Feed-forward ReLU neural networks partition their input domain into finitely many "affine regions" of constant neuron activation pattern and affine behaviour. We analyze their mathematical structure and provide algorithmic primitives for an…
We consider the use of extreme learning machines (ELM) for computational partial differential equations (PDE). In ELM the hidden-layer coefficients in the neural network are assigned to random values generated on $[-R_m,R_m]$ and fixed,…
In this paper, we present a finite element method (FEM) framework enhanced by an operator-adapted wavelet decomposition algorithm designed for the efficient analysis of multiscale electromagnetic problems. Usual adaptive FEM approaches,…
Deep neural networks (DNNs) enable high performance across domains but remain vulnerable to adversarial perturbations, limiting their use in safety-critical settings. Here, we introduce two quantum-optimization-based models for robust…
Physics-informed neural networks (PINNs) and their variants have recently emerged as alternatives to traditional partial differential equation (PDE) solvers, but little literature has focused on devising accurate numerical integration…
We propose a hybrid method, the Neural Enrichment Finite Element Method (NEFEM), designed for problems involving strong oscillations or interface problems with weak discontinuities. This method is based on the stable generalized finite…
We present a complete theoretical and empirical framework establishing feedforward neural networks as universal finite-state machines (N-FSMs). Our results prove that finite-depth ReLU and threshold networks can exactly simulate…
In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The…
We present a framework for solving a broad class of ill-posed inverse problems governed by partial differential equations (PDEs), where the target coefficients of the forward operator are recovered through an iterative regularization scheme…
Dimensionality reduction (DR) plays a vital role in the visual analysis of high-dimensional data. One main aim of DR is to reveal hidden patterns that lie on intrinsic low-dimensional manifolds. However, DR often overlooks important…
Neural Networks (NNs) are the method of choice for building learning algorithms. Their popularity stems from their empirical success on several challenging learning problems. However, most scholars agree that a convincing theoretical…
Deep neural networks typically treat nonlinearities as fixed primitives (e.g., ReLU), limiting both interpretability and the granularity of control over the induced function class. While recent additive models (like KANs) attempt to address…
It is a challenging topic in applied mathematics to solve high-dimensional nonlinear partial differential equations (PDEs). Standard approximation methods for nonlinear PDEs suffer under the curse of dimensionality (COD) in the sense that…
The effectiveness of deep neural architectures has been widely supported in terms of both experimental and foundational principles. There is also clear evidence that the activation function (e.g. the rectifier and the LSTM units) plays a…
In deformable registration, the geometric framework - large deformation diffeomorphic metric mapping or LDDMM, in short - has inspired numerous techniques for comparing, deforming, averaging and analyzing shapes or images. Grounded in…