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Discovering hidden physical laws and identifying governing system parameters from sparse observations are central challenges in computational science and engineering. Existing data-driven methods, such as physics-informed neural networks…
Approximate solutions of partial differential equations (PDEs) obtained by neural networks are highly affected by hyper parameter settings. For instance, the model training strongly depends on loss function design, including the choice of…
Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order…
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…
Neural operators have emerged as powerful tools for learning solution operators of partial differential equations. However, in time-dependent problems, standard training strategies such as teacher forcing introduce a mismatch between…
Modern power systems require fast and accurate dynamic simulations for stability assessment, digital twins, and real-time control, but classical ODE solvers are often too slow for large-scale or online applications. We propose a…
Neural operators, as a powerful approximation to the non-linear operators between infinite-dimensional function spaces, have proved to be promising in accelerating the solution of partial differential equations (PDE). However, it requires a…
Deep Neural Networks (DNNs) have become very popular for prediction in many areas. Their strength is in representation with a high number of parameters that are commonly learned via gradient descent or similar optimization methods. However,…
Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…
The estimation of distributed parameters in partial differential equations (PDE) from measures of the solution of the PDE may lead to under-determination problems. The choice of a parameterization is a usual way of adding a-priori…
Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which \textcolor{black}{predicts the PDE solution with variable PDE…
We propose quadratic residual networks (QRes) as a new type of parameter-efficient neural network architecture, by adding a quadratic residual term to the weighted sum of inputs before applying activation functions. With sufficiently high…
Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a…
The notion of an Evolutional Deep Neural Network (EDNN) is introduced for the solution of partial differential equations (PDE). The parameters of the network are trained to represent the initial state of the system only, and are…
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…
Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most…
Randomized neural network (RaNN) methods have been proposed for solving various partial differential equations (PDEs), demonstrating high accuracy and efficiency. However, initializing the fixed parameters remains challenging. Additionally,…
We propose tensorial neural networks (TNNs), a generalization of existing neural networks by extending tensor operations on low order operands to those on high order ones. The problem of parameter learning is challenging, as it corresponds…
More competent learning models are demanded for data processing due to increasingly greater amounts of data available in applications. Data that we encounter often have certain embedded sparsity structures. That is, if they are represented…
This paper establishes an approximation theorem for randomized neural networks (RaNNs) whose hidden-layer parameters are uniformly sampled from a prescribed bounded domain. Our analysis shows that, for RaNNs of the form $\mathop{\sum}_i W_i…