3.4k citations · 3.4k across the 19 of their papers we have counts for
22 papers
Exact Representation of Piecewise Polynomial Finite Element Functions by Two-Hidden-Layer Multilayer Perceptrons
Jun Hu, Pengzhan Jin, Shengyang Xu
We construct exact representations of finite element functions of arbitrary polynomial degree on simplicial meshes in any spatial dimension by two-hidden-layer multilayer perceptro…
Learning Operators of Geometry with an Interface Autoencoder
Aiqing Zhu, Pengzhan Jin
Geometry-dependent PDEs define operators whose inputs and outputs may each consist of an oriented interface and a function on that interface, whereas most existing neural operators…
MENO: Memory-Efficient Neural Operator
Shengyang Xu, Weijun Zhang, Jun Hu +1
We propose the Memory-Efficient Neural Operator (MENO) as a high-performance PDE neural solver based on the Manifold Function Encoder (MFE). MENO features three primary advantages:…
CSympNet-ID: conformal-symplectic map learning for linearly damped Hamiltonian systems
Jiale Gong, Pengzhan Jin, Dongyang Kuang +2
Learning dissipative dynamics from discrete observations is essential for reliable long-horizon prediction and physically meaningful parameter identification. For linearly damped H…
Manifold Function Encoder: Identifying Different Functions Defined on Different Manifolds
Jun Hu, Pengzhan Jin, Weijun Zhang
We propose the Manifold Function Encoder (MFE) for identifying different functions defined on different manifolds. Both a manifold in Euclidean space and a function defined on this…
A deformation-based framework for learning solution mappings of PDEs defined on varying domains
Shanshan Xiao, Pengzhan Jin, Yifa Tang
In this work, we establish a deformation-based framework for learning solution mappings of PDEs defined on varying domains. The union of functions defined on varying domains can be…
Learning solution operators of PDEs defined on varying domains via MIONet
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. We first extend the approxim…
Two-hidden-layer ReLU neural networks and finite elements
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. In…
A hybrid iterative method based on MIONet for PDEs: Theory and numerical examples
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 o…
Experimental observation on a low-rank tensor model for eigenvalue problems
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 a…
Tensor Neural Network and Its Numerical Integration
Yifan Wang, Pengzhan Jin, Hehu Xie
In this paper, we introduce a type of tensor neural network. For the first time, we propose its numerical integration scheme and prove the computational complexity to be the polyno…
On Numerical Integration in Neural Ordinary Differential Equations
Aiqing Zhu, Pengzhan Jin, Beibei Zhu +1
The combination of ordinary differential equations and neural networks, i.e., neural ordinary differential equations (Neural ODE), has been widely studied from various angles. Howe…
MIONet: Learning multiple-input operators via tensor product
Pengzhan Jin, Shuai Meng, Lu Lu
As an emerging paradigm in scientific machine learning, neural operators aim to learn operators, via neural networks, that map between infinite-dimensional function spaces. Several…
Optimal unit triangular factorization of symplectic matrices
Pengzhan Jin, Zhangli Lin, Bo Xiao
We prove that any symplectic matrix can be factored into no more than 5 unit triangular symplectic matrices, moreover, 5 is the optimal number. This result improves the existing tr…
Approximation capabilities of measure-preserving neural networks
Aiqing Zhu, Pengzhan Jin, Yifa Tang
Measure-preserving neural networks are well-developed invertible models, however, their approximation capabilities remain unexplored. This paper rigorously analyses the approximati…
Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks
Pengzhan Jin, Zhen Zhang, Ioannis G. Kevrekidis +1
We propose the Poisson neural networks (PNNs) to learn Poisson systems and trajectories of autonomous systems from data. Based on the Darboux-Lie theorem, the phase flow of a Poiss…
Inverse modified differential equations for discovery of dynamics
Aiqing Zhu, Pengzhan Jin, Beibei Zhu +1
The combination of numerical integration and deep learning, i.e., ODE-net, has been successfully employed in a variety of applications. In this work, we introduce inverse modified…
Deep Hamiltonian networks based on symplectic integrators
Aiqing Zhu, Pengzhan Jin, Yifa Tang
HNets is a class of neural networks on grounds of physical prior for learning Hamiltonian systems. This paper explains the influences of different integrators as hyper-parameters o…
SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems
Pengzhan Jin, Zhen Zhang, Aiqing Zhu +2
We propose new symplectic networks (SympNets) for identifying Hamiltonian systems from data based on a composition of linear, activation and gradient modules. In particular, we def…
Unit triangular factorization of the matrix symplectic group
Pengzhan Jin, Yifa Tang, Aiqing Zhu
In this work, we prove that any symplectic matrix can be factored into no more than 9 unit triangular symplectic matrices. This structure-preserving factorization of the symplectic…
DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
Lu Lu, Pengzhan Jin, George Em Karniadakis
While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a sin…
Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness
Pengzhan Jin, Lu Lu, Yifa Tang +1
The accuracy of deep learning, i.e., deep neural networks, can be characterized by dividing the total error into three main types: approximation error, optimization error, and gene…