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Title: Physics-aware Difference Graph Networks for Sparsely-Observed Dynamics

Paper content:
Sparsely available data points cause a numerical error on finite differences which hinder to modeling the dynamics of physical systems.
The discretization error becomes even larger when the sparse data are irregularly distributed so that the data defined on an unstructured grid, making it hard to build deep learning models to handle physics-governing observations on the unstructured grid.
In this paper, we propose a novel architecture named Physics-aware Difference Graph Networks (PA-DGN) that exploits neighboring information to learn finite differences inspired by physics equations.
PA-DGN further leverages data-driven end-to-end learning to discover underlying dynamical relations between the spatial and temporal differences in given observations.
We demonstrate the superiority of PA-DGN in the approximation of directional derivatives and the prediction of graph signals on the synthetic data and the real-world climate observations from weather stations.
Modeling real world phenomena, such as climate observations, traffic flow, physics and chemistry simulation (Li et al., 2018; Geng et al., 2019; Long et al., 2018; de Bezenac et al., 2018; SanchezGonzalez et al., 2018; Gilmer et al., 2017) , is important but extremely challenging.
While deep learning has achieved remarkable successes in prediction tasks by learning latent representations from data-rich applications such as image recognition (Krizhevsky et al., 2012 ), text understanding (Wu et al., 2016) , and speech recognition , we are confronted with many challenging scenarios in modeling natural phenomena by deep neural networks when a limited number of observations are only available.
Particularly, the sparsely available data points cause substantial numerical error and the limitation requires a more principled way to redesign deep learning models.
Although many works have been proposed to model physics-simulated observations using deep learning, many of them are designed under the assumption that input is on a continuous domain.
For example, Raissi et al. (2017a; proposed Physics-informed neural networks (PINNs) to learn nonlinear relations between input (spatial-and temporal-coordinates (x, t)) and output simulated by a given PDE.
Since Raissi et al. (2017a; use the coordinates as input and compute derivatives based on the coordinates to represent a given PDE, the setting is only valid when the data are continuously observed over spatial and temporal space.
Under the similar direction of PINNs, Chen et al. (2015) proposed a method to leverage the nonlinear diffusion process for image restoration.
de Bezenac et al. (2018) incorporated the transport physics (advection-diffusion equation) with deep neural networks for forecasting sea surface temperature by extracting the motion field.
Lutter et al. (2019) introduced Deep Lagrangian Networks specialized to learn Lagrangian mechanics with learnable parameters.
Seo & Liu (2019) proposed a physicsinformed regularizer to impose data-specific physics equations.
In common, the methods in Chen et al. (2015) ; de Bezenac et al. (2018) ; Lutter et al. (2019) are not efficiently applicable to sparsely discretized input as only a small number of data points are available and continuous properties on given space are not easily recovered.
It is inappropriate to directly use continuous differential operators to provide local behaviors because it is hard to approximate the continuous derivatives precisely with the sparse points (Amenta & Kil, 2004; Luo et al., 2009; Shewchuk, 2002) .
Furthermore, they are only applicable when the specific physics equations are explicitly given and still hard to be generalized to incorporate other types of equations.
As another direction to modeling physics-simulated data, Long et al. (2018) proposed PDE-Net which uncovers the underlying hidden PDEs and predicts the dynamics of complex systems.
Ruthotto & Haber (2018) derived new CNNs: parabolic and hyperbolic CNNs based on ResNet (He et al., 2016) architecture motivated by PDE theory.
While Long et al. (2018) ; Ruthotto & Haber (2018) are flexible to uncover hidden physics from the constrained kernels, it is still restrictive to a regular grid where the proposed constraints on the learnable filters are easily defined.
Reasoning physical dynamics of discrete objects has been actively studied Battaglia et al., 2016; Chang et al., 2016) as the appearance of graph-based neural networks (Kipf & Welling, 2017; Santoro et al., 2017; Gilmer et al., 2017) .
Although these models can handle sparsely located data points without explicitly given physics equations, they are purely data-driven so that the physics-inspired inductive bias, exploiting finite differences, is not considered at all.
In contrast, our method consists of physics-aware modules allowing efficiently leveraging the inductive bias to learn spatiotemporal data from the physics system.
In this paper, we propose Physics-aware Difference Graph Networks (PA-DGN) whose architecture is motivated to leverage differences of sparsely available data from the physical systems.
The differences are particularly important since most of the physics-related dynamic equations (e.g., Navier-Stokes equations) handle differences of physical quantities in spatial and temporal space instead of using the quantities directly.
Inspired by the property, we first propose Spatial Difference Layer (SDL) to efficiently learn the local representations by aggregating neighboring information in the sparse data points.
The layer is based on Graph Networks (GN) as it easily leverages structural features to learn the localized representations and the parameters for computing the localized features are shared.
Then, the layer is connected with Recurrent Graph Networks (RGN) to be combined with temporal difference which is another core component of physics-related dynamic equations.
PA-DGN is applicable to various tasks and we provide two representative tasks; the approximation of directional derivatives and the prediction of graph signals.
In this paper, we introduce a novel architecture (PA-DGN) that approximates spatial derivatives to use them to represent PDEs which have a prominent role for physics-aware modeling.
PA-DGN effectively learns the modulated derivatives for predictions and the derivatives can be used to discover hidden physics describing interactions between temporal and spatial derivatives.

assistant: text: We propose physics-aware difference graph networks designed to effectively learn spatial differences to modeling sparsely-observed dynamics.