no code implementations • 9 Mar 2024 • Pengzhan Jin
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.
no code implementations • 23 Feb 2024 • Shanshan Xiao, Pengzhan Jin, Yifa Tang
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.
no code implementations • 11 Feb 2024 • Jun Hu, Pengzhan Jin
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.
no code implementations • 1 Feb 2023 • Jun Hu, Pengzhan Jin
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.
1 code implementation • 15 Jun 2022 • Aiqing Zhu, Pengzhan Jin, Beibei Zhu, Yifa Tang
The combination of ordinary differential equations and neural networks, i. e., neural ordinary differential equations (Neural ODE), has been widely studied from various angles.
2 code implementations • 12 Feb 2022 • Pengzhan Jin, Shuai Meng, Lu Lu
Based on our theory and a low-rank approximation, we propose a novel neural operator, MIONet, to learn multiple-input operators.
no code implementations • 21 Jun 2021 • Aiqing Zhu, Pengzhan Jin, Yifa Tang
Measure-preserving neural networks are well-developed invertible models, however, their approximation capabilities remain unexplored.
1 code implementation • 5 Dec 2020 • Pengzhan Jin, Zhen Zhang, Ioannis G. Kevrekidis, George Em Karniadakis
We propose the Poisson neural networks (PNNs) to learn Poisson systems and trajectories of autonomous systems from data.
1 code implementation • 11 Jan 2020 • Pengzhan Jin, Zhen Zhang, Aiqing Zhu, Yifa Tang, George Em. Karniadakis
We propose new symplectic networks (SympNets) for identifying Hamiltonian systems from data based on a composition of linear, activation and gradient modules.
4 code implementations • 8 Oct 2019 • Lu Lu, Pengzhan Jin, George Em. Karniadakis
This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data.
1 code implementation • 27 May 2019 • Pengzhan Jin, Lu Lu, Yifa Tang, George Em. Karniadakis
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.