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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…

Physics-Informed Neural Networks (PINNs) and Neural Ordinary Differential Equations (NODEs) represent two distinct machine learning frameworks for modeling nonlinear neuronal dynamics. This study systematically evaluates their performance…

To combine a feedforward neural network (FNN) and Lie group (symmetry) theory of differential equations (DEs), an alternative artificial NN approach is proposed to solve the initial value problems (IVPs) of ordinary DEs (ODEs). Introducing…

The framework Doplicher-Fredenhagen-Roberts (DFR) of a noncommutative (NC) space-time is considered as a alternative approach to study the NC space-time of the early Universe. In this formalism, the parameter of noncommutative…

Form a pure mathematical point of view, common functional forms representing different physical phenomena can be defined. For example, rates of chemical reactions, diffusion and heat transfer are all governed by exponential-type…

Ordinary and partial differential equations (DE) are used extensively in scientific and mathematical domains to model physical systems. Current literature has focused primarily on deep neural network (DNN) based methods for solving a…

Extracting macroscopic properties of a system from microscopic interactions has always been an interesting topic with the most diverse applications. Here, we use the quantum Boltzmann equation to investigate the density matrix evolution of…

While Hamiltonian mechanics provides a powerful inductive bias for neural networks modeling dynamical systems, Hamiltonian Neural Networks and their variants often fail to capture complex temporal dynamics spanning multiple timescales. This…

A physics-informed neural network is developed to solve conductive heat transfer partial differential equation (PDE), along with convective heat transfer PDEs as boundary conditions (BCs), in manufacturing and engineering applications where…

Neural differential equations are a promising new member in the neural network family. They show the potential of differential equations for time series data analysis. In this paper, the strength of the ordinary differential equation (ODE)…

Quantum computing promises exponential improvements in solving large systems of partial differential equations (PDE), which forms a bottleneck in high-resolution computational fluid dynamics (CFD) simulations, in, among others, aerospace…

Partial differential equations (PDEs) are central to describing complex physical system simulations. Their expensive solution techniques have led to an increased interest in deep neural network based surrogates. However, the practical…

In the realm of computational science and engineering, constructing models that reflect real-world phenomena requires solving partial differential equations (PDEs) with different conditions. Recent advancements in neural operators, such as…

Solving Singularly Perturbed Differential Equations (SPDEs) poses computational challenges arising from the rapid transitions in their solutions within thin regions. The effectiveness of deep learning in addressing differential equations…

Neural ordinary differential equations (NODE) have been recently proposed as a promising approach for nonlinear system identification tasks. In this work, we systematically compare their predictive performance with current state-of-the-art…

Physics-informed neural networks (PINNs) are demonstrating remarkable promise in integrating physical models with gappy and noisy observational data, but they still struggle in cases where the target functions to be approximated exhibit…

We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. In this second…

Machine learning techniques have been successfully applied to classifying an extensive range of phenomena in quantum theory. From detecting quantum phase transitions to identifying Bell non-locality, it has been established that classical…

We develop the statistical mechanics of the Hopfield model in a transverse field to investigate how quantum fluctuations affect the macroscopic behavior of neural networks. When the number of embedded patterns is finite, the Trotter…

Physics-informed neural networks (PINNs) have emerged as a promising approach to solving partial differential equations (PDEs) using neural networks, particularly in data-scarce scenarios, due to their unsupervised training capability.…