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Found 11 papers, 6 papers with code
Shallow ReLU neural networks and finite elements
We point out that (continuous or discontinuous) piecewise linear functions on a convex polytope mesh can be represented by two-hidden-layer ReLU neural networks in a weak sense.
Learning solution operators of PDEs defined on varying domains via MIONet
In this work, we propose a method to learn the solution operators of PDEs defined on varying domains via MIONet, and theoretically justify this method.
A hybrid iterative method based on MIONet for PDEs: Theory and numerical examples
We propose a hybrid iterative method based on MIONet for PDEs, which combines the traditional numerical iterative solver and the recent powerful machine learning method of neural operator, and further systematically analyze its theoretical properties, including the convergence condition, the spectral behavior, as well as the convergence rate, in terms of the errors of the discretization and the model inference.
Experimental observation on a low-rank tensor model for eigenvalue problems
Here we utilize a low-rank tensor model (LTM) as a function approximator, combined with the gradient descent method, to solve eigenvalue problems including the Laplacian operator and the harmonic oscillator.
On Numerical Integration in Neural Ordinary Differential Equations
The combination of ordinary differential equations and neural networks, i. e., neural ordinary differential equations (Neural ODE), has been widely studied from various angles.
MIONet: Learning multiple-input operators via tensor product
Based on our theory and a low-rank approximation, we propose a novel neural operator, MIONet, to learn multiple-input operators.
Approximation capabilities of measure-preserving neural networks
Measure-preserving neural networks are well-developed invertible models, however, their approximation capabilities remain unexplored.
Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks
We propose the Poisson neural networks (PNNs) to learn Poisson systems and trajectories of autonomous systems from data.
SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems
We propose new symplectic networks (SympNets) for identifying Hamiltonian systems from data based on a composition of linear, activation and gradient modules.
DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data.
Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness
To derive a meaningful bound, we study the generalization error of neural networks for classification problems in terms of data distribution and neural network smoothness.
