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Pseudo-Hamiltonian neural networks (PHNN) were recently introduced for learning dynamical systems that can be modelled by ordinary differential equations. In this paper, we extend the method to partial differential equations. The resulting…

Physics-Informed Neural Networks (PINNs) offer a promising approach to solving differential equations and, more generally, to applying deep learning to problems in the physical sciences. We adopt a recently developed transfer learning…

The potential of neural networks (NN) in engineering is rooted in their capacity to understand intricate patterns and complex systems, leveraging their universal nonlinear approximation capabilities and high expressivity. Meanwhile,…

Artificial intelligence is revolutionizing our lives at an ever increasing pace. At the heart of this revolution is the recent advancements in deep neural networks (DNN), learning to perform sophisticated, high-level tasks. However,…

In this study, novel physics-informed neural network (PINN) methods for coupling neighboring support points and their derivative terms which are obtained by automatic differentiation (AD), are proposed to allow efficient training with…

A classic approach for solving differential equations with neural networks builds upon neural forms, which employ the differential equation with a discretisation of the solution domain. Making use of neural forms for time-dependent…

Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for…

We present a novel approach that integrates unfitted finite element methods and neural networks to approximate partial differential equations on complex geometries. Easy-to-generate background meshes (e.g., a simple Cartesian mesh) that cut…

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…

We develop a randomized Newton's method for solving differential equations, based on a fully connected neural network discretization. In particular, the randomized Newton's method randomly chooses equations from the overdetermined nonlinear…

Verification of Neural Networks (NNs) that approximate the solution of Partial Differential Equations (PDEs) is a major milestone towards enhancing their trustworthiness and accelerating their deployment, especially for safety-critical…

The efficiency of modern machine intelligence depends on high accuracy with minimal computational cost. In spiking neural networks (SNNs), synaptic delays are crucial for encoding temporal structure, yet existing models treat them as fully…

The finite element method, finite difference method, finite volume method and spectral method have achieved great success in solving partial differential equations. However, the high accuracy of traditional numerical methods is at the cost…

Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional…

Probabilistic theory and differential equation are powerful tools for the interpretability and guidance of the design of machine learning models, especially for illuminating the mathematical motivation of learning latent variable from…

Physics informed neural networks (PINNs) represent a very popular class of neural solvers for partial differential equations. In practice, one often employs stochastic gradient descent type algorithms to train the neural network. Therefore,…

Training a neural network (NN) typically relies on some type of curve-following method, such as gradient descent (GD) (and stochastic gradient descent (SGD)), ADADELTA, ADAM or limited memory algorithms. Convergence for these algorithms…

We develop a minimax rate analysis to describe the reason that deep neural networks (DNNs) perform better than other standard methods. For nonparametric regression problems, it is well known that many standard methods attain the minimax…

Given the facts of the extensiveness of multi-material diffusion problems and the inability of the standard PINN(Physics-Informed Neural Networks) method for such problems, in this paper we present a novel PINN method that can accurately…

Physics-informed neural networks (PINNs), owing to their mesh-free nature, offer a powerful approach for solving high-dimensional partial differential equations (PDEs) in complex geometries, including irregular domains. This capability…