Source Identification and Field Reconstruction of
Advection-Diffusion Process from Sparse Sensor
Measurements
Arka Daw∗
Virginia TechBlacksburg, United States.
darka@vt.edu

Kyongmin Yeo
IBM T.J. Watson Research Center
Yorktown Heights, United States.
kyeo@us.ibm.com

Anuj Karpatne
Virginia TechBlacksburg, United States.
karpatne@vt.edu

Levente Klein
IBM T.J. Watson Research Center
Yorktown Heights, United States.
kleinl@us.ibm.com

Abstract
Inferring the source information of greenhouse gases, such as methane, fromspatially sparse sensor observations is an essential element in mitigating climatechange. While it is well understood that the complex behavior of the atmosphericdispersion of such pollutants is governed by the Advection-Diffusion equation, it isdifficult to directly apply the governing equations to identify the source information because of the spatially sparse observations, i.e., the pollution concentration
is known only at the sensor locations. Here, we develop a multi-task learning
framework that can provide high-fidelity reconstruction of the concentration field
and identify emission characteristics of the pollution sources such as their location,
emission strength, etc. from sparse sensor observations. We demonstrate that our
proposed framework is able to achieve accurate reconstruction of the methane
concentrations from sparse sensor measurements as well as precisely pin-point thelocation and emission strength of these pollution sources.

1

Introduction

Methane is one of the potent greenhouse gasses [4, 3, 10] that is emitted into the atmosphere throughleakages in natural gas systems, raising livestocks, or via natural sources such as wetlands. Thesemethane emissions caused by human activities have further been identified as a major contributor
to climate change [1, 9, 8]. Thus, inferring the location and emission strength of pollution sources
such as methane leaks are both essential in monitoring the air-quality as well as mitigating theclimate change. However, one of the major challenges in localization of these emission sources is theunavailability of high-resolution methane concentration maps, and these individual pollution sources
are to be estimated from the limited concentration measurements from a sparse sensor network.
Atmospheric inverse models that aim to either reconstruct the concentration field [7] or identify
pollution source information [5] of airborne pollutants have been extensively studied in statistics andapplied mathematics communities. However, due to the ill-posed nature of the inverse problem, it
still remains as a challenging topic. Recently, potential of deep learning approaches for the inverse
problem has been demonstrated [2].
*Work done during internship at IBM T.J.Watson Research Center.

Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

We develop a multi-task learning framework that can provide a high-fidelity reconstruction of thespatio-temporal concentration field and identify the emission characteristics of the pollution sources
such as their location, and emission strengths from a time series of sensor measurements. We propose
a novel 3D diffusive-masked convolution to gradually propagate the information from the sensor
locations to the unobserved field using a diffusion process. We demonstrate that our multi-task
model is able to achieve accurate reconstruction of the pollution concentrations from sparse sensor
measurements as well as precisely pin-point the location and emission strength of these pollution
sources.

2

Background and Problem Setup

Let there be Ns number of potential emission sources and No number of sensors. We assume that theatmospheric dispersion is determined by the following Advection-Diffusion Equation:NsX∂ϕ+ u∇ϕ − K ∇2 ϕ =qi (x, t); x ∈ R2 , t ∈ R+
∂t
i=1

(1)

where ϕ is the concentration, u is the wind velocity field, K is the turbulent diffusivity, and qi (x, t)
is the emission strength of the i − th source.
Also, let the time-series of measurements from the No sensors be Φ = {Φ1 , Φ2 , ..., ΦNo }, where Φitdenotes the (time varying) sensor measurement of the i-th sensor, i.e., Φi = [Φ1i , Φ2i , ..., ΦNi ], withΦ Φ
Nt denoting the number of time steps. The location of the i-th sensors are denoted by xΦ
=(xii , yi ).
Also, note that the wind velocity field u is available on a Nx × Ny uniform-grid, such that Nx
and Ny are the number of discretizations of the input domain in x and y directions respectively.
We further assume that our pollution sources are point sources, and can be realized as qi (x, t) =ci δ(x − xsi , y − yis ), where ci is the emission strength of the i-th emission source, (xsi , yis ) is thelocation of the source, and δ is the Dirac delta function.
Problem Statement: Given the time series measurements of the sensor networks Φ ∈ RNo ×Nt ,ΦΦtheir locations xΦ = {xΦ
1 , x2 , ..., xNo }, and the wind velocity field u, the spatio-temporal fieldreconstruction problem can be formulated as estimating the methane concentration ϕ on the regulargrid Nt × Nx × Ny . Additionally, we are also interested in estimating the emission characteristics of
the Ns potential emission sources, such as their constant emission strengths c = [c1 , c2 , ..., cNs ], andtheir locations xs = {xs1 , xs2 , ..., xsNs }.

3

Proposed Method

3.1

Multi-task learning framework

Multi-task learning [12, 11] is a learning paradigm, where the knowledge from one task can beutilized to improve the performance of the model on other similar tasks. Here, we aim to leverage thestrong correlation between the two tasks of identifying the source information and reconstructing
the concentration field. Note that, through the advection-diffusion equation, the concentration field
is strongly coupled with the source characteristics given u. We propose to learn a shared encoderg(θ) : [Φ, xΦ , u] → z which aims to learn a latent representation z. Then, this latent representationz, is fed into task-specific decoders f1 (ω) : z → ϕ and f2 (ψ) : z → [c, xs ] to find the solution of
the inverse problem and obtain the reconstructed field.
The multi-task learning framework can then be optimized using the objective function L(θ, ω, ψ) =Lrecon (θ, ω) + Linverse (θ, ψ), where Lrecon (θ, ω) is the loss function for reconstructing the spatiotemporal concentration field, and Linverse (θ, ψ) is the loss function for the inverse problem.
3.2

Field Reconstruction using Diffusive Masked Convolution

We employ a series of masked convolutions for the reconstruction of the spatio-temporal concentration
field from the sparse observations. We first define a “masked” convolution operation by restricting theconvolution operation on the masked region centered around the sensor locations where we actually2have the observations. The masked convolution operation is defined as,(sum(1)W T (X ⊙ M ) sum(M′) + b, if sum(M ) > 0x =0,otherwise(2)

where W and b, respectively, denote the weights and bias of a convolutional filter, X is the current
feature (pixel) values for the convolutional sliding window, M is a corresponding mask, and ⊙represents element-wise multiplication operation. Note that the masked convolution is defined similar
to the Partial Convolutions [6] introduced for an inpainting problem.
Then, the mask, M , is updated using a diffusion process after each masked convolution layer. The
diffusion process is modeled by a Spatial Gaussian Convolutional Kernel of size k × k denoted byW̃k ∈ Rk×k
as follows:
+
1 h
k−1 2
k − 1 2iW̃k (i, j) = exp(− 2 (i −) + (j −) ); i, j ∈ {0, 1, .., k − 1},
(3)
2σk
2
2
where i, j are the i-th row and j-th column of the k × k kernel, σk is the only learnable parameterof the kernel. In other words, if σk is large then the spatial-convolution kernel W̃k would diffuse
the mask values over a larger spatial region. Thus, the learnable-parameter σk controls the rate ofdiffusion between the different layers.
The spatial kernel W̃k can be repeated across time once (since the size of the kernel along time is
one to ensure time-invariance) and the input Cin and output Cout channels to form the time-invariant
Diffusion Kernel Wkdiff , and the mask is updated using the diffusion as M ′ = (Wkdiff )T M .
After the mask update we additionally clip the mask values greater than one. We first initialize themask at the input layer M0 by the sparse sensor-network locations, as shown in Equation 4.

1, if x ∈ xΦM0 (x) =(4)0, otherwiseThen, we stack multiple Diffusive Masked Convolution layers so that the mask M0 gradually grows
after each convolution layer and ultimately we can reconstruct the full field from the sparse sensor
measurements. Note that this formulation using the Diffusive Masked Convolution can encode
arbitrarily placed sensors as the input mask. Thus, by training such a model with different sparse
sensor measurements, it is possible to predict the concentration fields on unseen sensor-network
configurations.
3.2.1

Gaussian Negative Log-likelihood Formulation for Uncertainty Quantification

Here, we employ a Gaussian model, where ϕ ∼ N (ϕµ , σ 2 ) to quantify the uncertainty in the predictedconcentration field. The negative log-likelihood loss function is given as,
Lrecon =1(ϕtrue − ϕµ )2
Ex∼pdata [log σ 2 +
]
2
σ2

(5)

where ϕµ and σ 2 and denote the predicted mean and variance of the concentration field respectively.
Later, we demonstrate that the Gaussian negative log-likelihood loss formulation prevents overfitting on the ground truth concentration field and offers a smooth estimation of the predicted mean
concentration ϕµ .
3.3

Emission Characteristics Estimation

Next, we consider the task of solving the inverse problem of estimating the emission characteristics.
One of the challenges is that the number of potential sources can vary, thus, the decoder f2 should be
able to handle this varying output size. We propose to divide the 2D spatial domain (Nx × Ny ) intoa S × S grid. For each cell in the S × S grid we predict the following source characteristic vector[pi , xi , yi , ci ], where i ∈ {1, 2, ..., S 2 }, pi is the probability of containing a source in the grid cell i,xi and yi represent the relative location of the source with respect to the top left corner of the cell,and ci represents the emission strength of the source.
3

The objective function for estimating the emission characteristics can be evaluated as follows:
2

L

inverse

=λsrc

S
X

2

s
2
s
2
1src
ij [(xi − x̂i ) + (yi − ŷi ) ] + λsrc

i=1

λnosrcSX

2
1src
ij [(σ(p̂i ) − 1) ]+

i=1

S2
X

1nosrc[(σ(p̂i ))2 ] + λsrc
ij

i=1

S2
X

2
1src
ij [(ci − Softplus(ĉi )) ]

(6)

i=1

where, 1src
denotes if the source occurs in the grid cell i, and 1src
i
ij denotes if the j-th source
characteristic vector in grid cell i is “responsible” for that prediction.
The neural network architecture and implementation details of the proposed framework using
diffusive-masked convolution are provided in the Appendix A.

4Results

Experiment Setup: We performed our experiments on 4000 simulations of the forward problem of
the advection-diffusion equation, with varying source locations and their emission strengths understochastically generated wind conditions. In each simulation, the number of sources is randomly
selected between one to four. These set of simulations were there divided into training and test sets totrain our proposed multi-task learning framework.
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Figure 1: Demonstrating the reconstruction of the global field from sparse sensor measurements on arepresentative test example. The white crosses denote the position of the sensors. Top row: shows
the ground truth concentration fields at various time stamps, Bottom row: the predicted mean of theconcentration fields
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10SourcesFigure 2: Localization of the pollution sources (left and middle) and their emission strength estimation(right) on a representative test example.
Evaluating Proposed Framework on Simulations: The proposed multi-task learning framework
was trained on 80% of the simulations and tested on the rest. To evaluate the performance on the
global field reconstruction task, we compute the Mean Squared Error (MSE) between the predictedconcentration field and the ground truth concentration field on the test set. We are able to obtainhigh fidelity reconstructions of the global concentration field using our proposed diffusive masked
convolution with a MSE of 0.051. An example of the reconstructed concentration field on a test case isshown in Figure 1. To evaluate the performance on the inverse problem, i.e., localization and emission
4

characteristics estimation of the pollution sources, we compute the precision, recall and relative
location MSE, which provides us an estimate of the detection efficiency of the pollution sources. We
also compute the MSE of the source magnitude strengths to quantify the error in approximating theemission strengths. We are able to demonstrate a fairly high recall of 94.5%, and a precision of 71.0%.
This suggests that the model is able to recover 94.5% of the pollution sources that were present in
the simulations. However, the low precision suggests that the model also generates about 30% false
positives in its predictions. Further, we observe that once a grid cell detects a pollution source, it canexactly pin-point the location of the source inside the grid with a relative MSE of 1.35e − 07. We also
demonstrate an accurate estimation of the emission strength for the pollution sources having an errorof 0.19. Figure 2 shows the ability of our proposed framework to accurately identify the pollution
sources as well as approximate their emission strengths on an example case. We also demonstratein Appendix Figures 5 and 4 that the Gaussian-Negative Loglikelihood formulation for estimating
uncertainties allows the estimate of the mean concentration field to be a smooth function, as the large
fluctuations near the pollution sources are taken care by the predicted variance, thereby preventing
oscillatory solutions due to the Delta-function-like behaviors around the sources.

5

Conclusion

We present a multi-task learning framework for identifying potential pollution sources and obtaining
reconstruction of spatio-temporal concentration fields from sparse sensor measurements. We also
propose a novel diffusive-masked convolution operations that employs the diffusion process inperforming masked-convolutions. The diffusive-masked convolution is realized by a spatial-Gaussianconvolution kernel, followed by diffusing the mask to nearby regions. Thus, by stacking multiple
such layers we are able to iteratively diffuse the information from a sparse sensor measurements(represented using the initial mask) in a principled manner until the learned representations spread overthe entire spatio-temporal domain. We also demonstrate precise reconstruction of the concentration
field along with accurate localization and emission strength estimation of the pollution sources on thetest simulations using our proposed framework.

6

Broader Impact

The problem of reconstructing the spatio-temporal fields of a complex-time evolving dynamicalsystem from sparse sensor observations is common in the field of science and engineering. Examples
of such cases include reconstructing the oceanic currents from sparse measurements from a bouy
network, or reconstructing the seismic waves from measurements taken at sparse station networks. In
this paper, we present a novel and principled approach of reconstructing the global fields from sparse
sensor measurements motivated by the process of diffusion. Our proposed framework is obtain highfidelity reconstruction of the target spatio-temporal field from any arbitrarily placed sensor network(with variable number of sensors), without the need to retrain the model. In addition, we also provide
a generic framework to solve inverse problems that require precise localization of sources and theircharacterizations. Again, a wide range of problems in the field of science and engineering fall under
this category such as locating the epicenter of seismic activities from sparse station observations.

5References[1] Mingkui Cao, Keith Gregson, and Stewart Marshall. Global methane emission from wetlands
and its sensitivity to climate change. Atmospheric environment, 32(19):3293–3299, 1998.
[2] Kai Fukami, Romit Maulik, Nesar Ramachandra, Koji Fukagata, and Kunihiko Taira. Global
field reconstruction from sparse sensors with voronoi tessellation-assisted deep learning. Nature
Machine Intelligence, 3(11):945–951, 2021.
[3] Robert W Howarth. A bridge to nowhere: methane emissions and the greenhouse gas footprintof natural gas. Energy Science & Engineering, 2(2):47–60, 2014.
[4] Robert W Howarth, Renee Santoro, and Anthony Ingraffea. Methane and the greenhouse-gas
footprint of natural gas from shale formations. Climatic change, 106(4):679–690, 2011.
[5] Youngdeok Hwang, Hang J Kim, Won Chang, Kyongmin Yeo, and Yongku Kim. Bayesianpollution source identification via an inverse physics model. Computational Statistics & Data
Analysis, 134:76–92, 2019.
[6] Guilin Liu, Fitsum A Reda, Kevin J Shih, Ting-Chun Wang, Andrew Tao, and Bryan Catanzaro.
Image inpainting for irregular holes using partial convolutions. In Proceedings of the European
conference on computer vision (ECCV), pages 85–100, 2018.
[7] Xiao Liu, Kyongmin Yeo, and Siyuan Lu. Statistical modeling for spatio-temporal data fromstochastic convection-diffusion processes. Journal of the American Statistical Association, 117(539):1482–1499, 2022.
[8] Fiona M O’Connor, O Boucher, N Gedney, CD Jones, GA Folberth, R Coppell, P Friedlingstein,
WJ Collins, J Chappellaz, J Ridley, et al. Possible role of wetlands, permafrost, and methane
hydrates in the methane cycle under future climate change: A review. Reviews of Geophysics,
48(4), 2010.
[9] Dave Reay and Pete Smith. Methane and climate change. Routledge, 2010.
[10] ED Schulze, S Luyssaert, P Ciais, A Freibauer, IA Janssens, Jean-François Soussana, P Smith,
John Grace, I Levin, B Thiruchittampalam, et al. Importance of methane and nitrous oxide for
europe’s terrestrial greenhouse-gas balance. Nature geoscience, 2(12):842–850, 2009.
[11] Trevor Standley, Amir Zamir, Dawn Chen, Leonidas Guibas, Jitendra Malik, and Silvio Savarese.
Which tasks should be learned together in multi-task learning? In International Conference on
Machine Learning, pages 9120–9132. PMLR, 2020.
[12] Yu Zhang and Qiang Yang. A survey on multi-task learning. IEEE Transactions on Knowledge
and Data Engineering, 2021.

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6

(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No](b) Did you specify all the training details (e.g., data splits, hyperparameters, how they
were chosen)? [Yes](c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No](d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)? [No] Appendix was strongly discouraged.
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A

Network Architecture for Proposed Multi-task Learning Framework

Our encoder-decoder architecture presented in the 3 is realized using a U-Net model with skip
connections, where each of the conventional convolutional layers (Conv3d in our case) are replacedby our proposed diffusive masked-convolutional layers. In our experiments, the encoder and decoder
are comprised of 4 layers each.

Figure 3: Neural Network Architecture of our proposed multi-task learning framework.

B

Additional Visualization of the Results

In this section, we provide some additional visualizations to qualitatively analyze the behavior of
our proposed framework. First, for a fixed value of the y-coordinate (y = 30 and y = 37) we plot
the variations in the concentration fields w.r.t. the x-axis at various times in Figure 4. We observethat the predictive intervals estimated by our model always engulf the ground truth concentration.
Additionally, we show that the Gaussian-Negative Loglikelihood (NLL) formulation for estimating
uncertainties allows the estimate of the mean concentration field to be a smooth function, as the large
fluctuations near the pollution sources are taken care by the predicted variance, thereby preventing anoscillations solution. This behavior can also be seen in Figure 5, where we see that the absolute error
without the NLL shows rectangular oscillatory patterns, which is nor observed for the model trainedwith NLL.
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Figure 4: Analyzing the empirical coverage of the 95% Prediction Interval (shown as the shaded
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Figure 5: Comparing the Absolute Errors of the model trained with and without the Gaussian Negative
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Figure 6: Demonstrating the reconstruction of the global field from sparse sensor measurements on arepresentative test example. The white crosses denote the position of the sensors. Top row: shows
the ground truth concentration fields at various time stamps, Middle row: the predicted mean of theconcentration fields, Bottom row: the predicted variance of the concentration fields

9