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We propose a new method to deal with the essential boundary conditions encountered in the deep learning-based numerical solvers for partial differential equations. The trial functions representing by deep neural networks are…

In theory, boundary and initial conditions are important for the wellposedness of partial differential equations (PDEs). Numerically, these conditions can be enforced exactly in classical numerical methods, such as finite difference method…

This paper proposes a domain decomposition subspace neural network method for efficiently solving linear and nonlinear partial differential equations. By combining the principles of domain decomposition and subspace neural networks, the…

Deep learning models have been successfully used in computer vision and many other fields. We propose an unorthodox algorithm for performing quantization of the model parameters. In contrast with popular quantization schemes based on…

Deep neural network approaches show promise in solving partial differential equations. However, unlike traditional numerical methods, they face challenges in enforcing essential boundary conditions. The widely adopted penalty-type methods,…

Deterministic Lateral Displacement (DLD) devices enable liquid biopsy for cancer detection by separating circulating tumor cells (CTCs) from blood samples based on size, but designing these microfluidic devices requires computationally…

As function approximators, deep neural networks have served as an effective tool to represent various signal types. Recent approaches utilize multi-layer perceptrons (MLPs) to learn a nonlinear mapping from a coordinate to its corresponding…

The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed…

We present a Fourier neural network (FNN) that can be mapped directly to the Fourier decomposition. The choice of activation and loss function yields results that replicate a Fourier series expansion closely while preserving a…

Current physics-informed (standard or deep operator) neural networks still rely on accurately learning the initial and/or boundary conditions of the system of differential equations they are solving. In contrast, standard numerical methods…

Deep Neural Networks (DNNs) deliver impressive performance but their black-box nature limits deployment in high-stakes domains requiring transparency. We introduce Compositional Function Networks (CFNs), a novel framework that builds…

Deep learning has gained attention for solving PDEs, but the black-box nature of neural networks hinders precise enforcement of boundary conditions. To address this, we propose a boundary condition-guaranteed evolutionary Kolmogorov-Arnold…

Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…

We present a subspace method based on neural networks (SNN) for solving the partial differential equation with high accuracy. The basic idea of our method is to use some functions based on neural networks as base functions to span a…

In this paper, we propose a novel machine learning method based on an adaptive tensor neural network subspace for solving quasiperiodic elliptic problems. To this end, we first provide a theoretical analysis of the associated quasiperiodic…

The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…

Deep neural networks (DNNs) have emerged as a powerful tool with a growing body of literature exploring Lyapunov-based approaches for real-time system identification and control. These methods depend on establishing bounds for the second…

In this paper, an innovative Physical Model-driven Neural Network (PMNN) method is proposed to solve time-fractional differential equations. It establishes a temporal iteration scheme based on physical model-driven neural networks which…

In this paper, we introduce a new approach based on distance fields to exactly impose boundary conditions in physics-informed deep neural networks. The challenges in satisfying Dirichlet boundary conditions in meshfree and particle methods…

With the increasing adoption of Deep Neural Network (DNN) models as integral parts of software systems, efficient operational testing of DNNs is much in demand to ensure these models' actual performance in field conditions. A challenge is…