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Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In…

The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…

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…

In this article, we present an efficient deep learning method called coupled deep neural networks (CDNNs) for coupled physical problems. Our method compiles the interface conditions of the coupled PDEs into the networks properly and can be…

Physics-informed neural networks (PINNs) [4, 10] are an approach for solving boundary value problems based on differential equations (PDEs). The key idea of PINNs is to use a neural network to approximate the solution to the PDE and to…

We present DeepFDM, a differentiable finite-difference framework for learning spatially varying coefficients in time-dependent partial differential equations (PDEs). By embedding a classical forward-Euler discretization into a convolutional…

Physics-informed neural networks (PINNs) have recently emerged as an alternative way of solving partial differential equations (PDEs) without the need of building elaborate grids, instead, using a straightforward implementation. In…

Mixed dimensional partial differential equations (PDEs) are equations coupling unknown fields defined over domains of differing topological dimension. Such equations naturally arise in a wide range of scientific fields including geology,…

Many partial differential equations (PDEs) such as Navier--Stokes equations in fluid mechanics, inelastic deformation in solids, and transient parabolic and hyperbolic equations do not have an exact, primal variational structure. Recently,…

We introduce a sampling based machine learning approach, Monte Carlo physics informed neural networks (MC-PINNs), for solving forward and inverse fractional partial differential equations (FPDEs). As a generalization of physics informed…

Deep learning-based partial differential equation(PDE) solvers have received much attention in the past few years. Methods of this category can solve a wide range of PDEs with high accuracy, typically by transforming the problems into…

This work is concerned with discovering the governing partial differential equation (PDE) of a physical system. Existing methods have demonstrated the PDE identification from finite observations but failed to maintain satisfying results…

Physics-Informed Neural Networks (PINNs) represent a significant advancement in Scientific Machine Learning (SciML), which integrate physical domain knowledge into an empirical loss function as soft constraints and apply existing machine…

The automatic differentiation (AD) in the vanilla physics-informed neural networks (PINNs) is the computational bottleneck for the high-efficiency analysis. The concept of derivative discretization in smoothed particle hydrodynamics (SPH)…

We use a space-time discretization based on physics informed deep learning (PIDL) to approximate solutions of a class of rate-dependent strain gradient plasticity models. The differential equation governing the plastic flow, the so-called…

Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their…

In this paper we establish a connection between non-convex optimization methods for training deep neural networks and nonlinear partial differential equations (PDEs). Relaxation techniques arising in statistical physics which have already…

Throughout many fields, practitioners often rely on differential equations to model systems. Yet, for many applications, the theoretical derivation of such equations and/or accurate resolution of their solutions may be intractable. Instead,…

Data-driven identification of differential equations is an interesting but challenging problem, especially when the given data are corrupted by noise. When the governing differential equation is a linear combination of various differential…

Deep neural networks have garnered widespread attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this study, we probe into solving a class of elliptic partial differential…