Log In Sign Up

A deep learning approach to solve forward differential problems on graphs

by   Yuanyuan Zhao, et al.

We propose a novel deep learning (DL) approach to solve one-dimensional non-linear elliptic, parabolic, and hyperbolic problems on graphs. A system of physics-informed neural network (PINN) models is used to solve the differential equations, by assigning each PINN model to a specific edge of the graph. Kirkhoff-Neumann (KN) nodal conditions are imposed in a weak form by adding a penalization term to the training loss function. Through the penalization term that imposes the KN conditions, PINN models associated with edges that share a node coordinate with each other to ensure continuity of the solution and of its directional derivatives computed along the respective edges. Using individual PINN models for each edge of the graph allows our approach to fulfill necessary requirements for parallelization by enabling different PINN models to be trained on distributed compute resources. Numerical results show that the system of PINN models accurately approximate the solutions of the differential problems across the entire graph for a broad set of graph topologies.


page 16

page 18

page 19

page 22

page 23

page 35

page 37

page 38


Physics-informed Neural Networks approach to solve the Blasius function

Deep learning techniques with neural networks have been used effectively...

Stochastic Scaling in Loss Functions for Physics-Informed Neural Networks

Differential equations are used in a wide variety of disciplines, descri...

Enhanced physics-informed neural networks for hyperelasticity

Physics-informed neural networks have gained growing interest. Specifica...

DeepPhysics: a physics aware deep learning framework for real-time simulation

Real-time simulation of elastic structures is essential in many applicat...

Reduced-PINN: An Integration-Based Physics-Informed Neural Networks for Stiff ODEs

Physics-informed neural networks (PINNs) have recently received much att...

An unsupervised deep learning approach in solving partial-integro differential equations

We investigate solving partial integro-differential equations (PIDEs) us...

On functions computed on trees

Any function can be constructed using a hierarchy of simpler functions t...