A deep learning approach to solve forward differential problems on graphs

TL;DR

This paper introduces a deep learning method using physics-informed neural networks to solve complex differential equations on graphs, ensuring continuity at nodes and enabling parallel computation.

Contribution

The novel approach assigns individual PINNs to graph edges with weak enforcement of Kirchhoff-Neumann conditions, facilitating accurate solutions and parallel training.

Findings

Accurately approximates solutions on various graph topologies

Enables parallel training of PINN models on distributed resources

Maintains solution continuity and derivative consistency at nodes

Abstract

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.