Related papers: PDE-constrained Models with Neural Network Terms: …

Segmentation of microscopy images constitutes an ill-posed inverse problem due to measurement noise, weak object boundaries, and limited labeled data. Although deep neural networks provide flexible nonparametric estimators, unconstrained…

Neural networks are discrete entities: subdivided into discrete layers and parametrized by weights which are iteratively optimized via difference equations. Recent work proposes networks with layer outputs which are no longer quantized but…

Physics-informed neural networks (PINNs) have recently become a popular method for solving forward and inverse problems governed by partial differential equations (PDEs). By incorporating the residual of the PDE into the loss function of a…

The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modeled into the equations as random coefficients. However, very often the variability of…

Finding the optimal configuration of parameters in ResNet is a nonconvex minimization problem, but first-order methods nevertheless find the global optimum in the overparameterized regime. We study this phenomenon with mean-field analysis,…

Solving high-dimensional partial differential equations (PDEs) is a major challenge in scientific computing. We develop a new numerical method for solving elliptic-type PDEs by adapting the Q-learning algorithm in reinforcement learning.…

In this paper, we present a new framework how a PDE with constraints can be formulated into a sequence of PDEs with no constraints, whose solutions are convergent to the solution of the PDE with constraints. This framework is then used to…

Real-world systems are often formulated as constrained optimization problems. Techniques to incorporate constraints into Neural Networks (NN), such as Neural Ordinary Differential Equations (Neural ODEs), have been used. However, these…

We show that an optimal finite element mesh refinement algorithm for a prototypical elliptic PDE can be learned by a recurrent neural network with a fixed number of trainable parameters independent of the desired accuracy and the input…

Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…

Motivated by recent research on Physics-Informed Neural Networks (PINNs), we make the first attempt to introduce the PINNs for numerical simulation of the elliptic Partial Differential Equations (PDEs) on 3D manifolds. PINNs are one of the…

Physics-informed neural networks (PINNs) as a means of discretizing partial differential equations (PDEs) are garnering much attention in the Computational Science and Engineering (CS&E) world. At least two challenges exist for PINNs at…

Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…

Neural networks can be used to learn the solution of partial differential equations (PDEs) on arbitrary domains without requiring a computational mesh. Common approaches integrate differential operators in training neural networks using a…

We present a numerical framework for deep neural network (DNN) modeling of unknown time-dependent partial differential equations (PDE) using their trajectory data. Unlike the recent work of [Wu and Xiu, J. Comput. Phys. 2020], where the…

Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable…

We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent…

Physics-informed neural networks (PINNs) are an increasingly powerful way to solve partial differential equations, generate digital twins, and create neural surrogates of physical models. In this manuscript we detail the inner workings of…

Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by…

Modeling the traffic dynamics is essential for understanding and predicting the traffic spatiotemporal evolution. However, deriving the partial differential equation (PDE) models that capture these dynamics is challenging due to their…