Learning continuous-time partial differential equations from sparse data using graph neural networks

Reprint: Deephub Imba

This is a paper published in ICLR in 2020. The paper uses a graph neural network to learn continuous-time partial differential equations from sparse data. The main innovation of the model proposed in the paper is to allow arbitrary space and time discretization, that is, when solving When the partial differential mesh is used, the mesh can be non-uniform. Since the governing equation to be solved is unknown, the author uses a message-passing graph neural network for parameterization when representing the governing equation.

Partial differential equations are crucial in many systems. But solving most partial differential equations has long been a difficult task, often requiring complex numerical solving skills, especially when the parameters or boundary conditions of the equations are partially unknown.

Graph neural networks (GNNs) offer novel and exciting concepts for solving partial differential equations because of their broad applicability in modeling non-Euclidean systems.

In this article, we will review a method using graph neural networks to represent the important time derivative components of partial differential equations.

A common partial differential equation is defined as,

where the time evolution of the system with respect to the spatial coordinate x and time t depends on itself and its first or higher derivative with respect to the spatial coordinate x.

This form of partial differential equations is the basis for a large class of scientific problems and has wide applications in systems with propagation properties such as acoustic waves, fluids, and thermal diffusion.

[1] proposed to use GNN to approximate the function F of the discrete point grid, discretize the original equation by the method of straight lines (MOL), and select N nodes in the system domain Ω, so the function F is discretized on these space nodes, which can be Expressed as

where N(i) is a set of indices of adjacent nodes at xi, and x {N(i)} and u {N(i)} are the positions and states of nodes in N(i).

We will denote an undirected graph by G=(V, E), where V is the set of vertices and E is the set of edges. To construct this graph, a Delaunay triangulation is first used for discrete points, and two nodes are considered adjacent if they are on the same side of at least one triangle, as shown in the following figure

Delaunay triangulation of a set of points. Green and orange points are considered neighbors because they share the same edge. Source[1]

The function F is then modeled by using Message Passing Neural Networks (MPNN) to propagate the hidden state through K graphs, each layer k first collects messages for each node i and then updates the corresponding node state,

where φ and γ are differentiable functions parameterized by DNN,

Then use the last layer of graph to calculate the PDE,

To supervise the learning of the model, mean squared error is used to observe the difference between the state and the estimated state.

Compared with pure discrete-time models with widely spaced data, the advantage of this approach is that it can predict the state of the system in continuous time, while learning the state of the system in discrete time.

a) Relative test error of the heat conduction equation. b) Real and learned system dynamics. Source[1]

This evolutionary mechanism is mathematically described by partial differential equations, and graph neural networks abstract these mechanisms as the flow of information between nodes (or edges). The paper mentions that graph neural networks will further advance scientific research and socioeconomics because of their relevance to describing the natural structure of non-Euclidean data or systems that are widespread in nature and human society.

Paper information:

1.Valerii Iakovlev, et. al., “Learning Continuous-time PDEs From Sparse Data with Graph Neural Networks”, arXiv:2006.08956.

Author: Madali Nabil

Learning continuous-time partial differential equations from sparse data using graph neural networks

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