Physics-Informed CNNs for Super-Resolution of
Sparse Observations on Dynamical Systems

Daniel Kelshaw
Department of Aeronautics
Imperial College London
djk21@imperial.ac.uk

Georgios Rigas
Department of Aeronautics
Imperial College London
g.rigas@imperial.ac.uk

Luca Magri
Department of Aeronautics
Imperial College London,
Alan Turing Institutel.magri@imperial.ac.uk

Abstract
In the absence of high-resolution samples, super-resolution of sparse observations
on dynamical systems is a challenging problem with wide-reaching applications inexperimental settings. We showcase the application of physics-informed convolutional neural networks for super-resolution of sparse observations on grids. Results
are shown for the chaotic-turbulent Kolmogorov flow, demonstrating the potentialof this method for resolving finer scales of turbulence when compared with classic
interpolation methods, and thus effectively reconstructing missing physics.

Figure 1: Diagrammatic overview of the super-resolution task.

1

Introduction

The application of machine learning for dynamical systems is gaining traction, providing means to
extract physical information from data, and reducing the dependence on running computationallyexpensive simulations [1]. In many cases, access to only sparse or partial observations of a dynamicalsystem is a limiting factor, obscuring the underlying dynamics and providing a challenge for system
identification [2]. Super-resolution methods offer the means for high-fidelity state reconstruction
from limited observations, a problem of fundamental importance in the physical sciences.
Convolutional neural networks (CNNs) are prominent in the domain of image reconstruction due totheir inherent ability to exploit spatial correlations [3–5]. For the classic data-driven approach, there
is a dependency on access to samples of high-resolution data. In the absence of ground-truth labels,
as with observations on a dynamical system, a common approach is to impose prior knowledge of thephysics, regularising predictions with respect to known governing equations [6]. The introduction ofphysics-informed neural networks (PINNs) has provided new tools for physically-motivated problems,
exploiting the automatic-differentiation paradigm provided by neural networks to constrain gradientsof the physical system [7–9]. Applications of PINNs for super-resolution show promising results for
simple systems, using sparse observations for accurate reconstruction of high-resolution fields [10].
Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

Applications of physics-informed CNNs for super-resolution are less prevalent. While Liu et al. [11]demonstrate a CNN for super-resolution of observations on a dynamical system, there is a strict
dependence on high-resolution examples to train the model. Gao et al. [12] explore physics-informed
CNNs for the super-resolution of a steady flow field, producing a one-to-one mapping for stationary
solutions of the Navier-Stokes equations. Physics-based regularisation of the steady solution neglects
any temporal component, introducing further complexities when considering dynamical systems.
Our work extends on the current use of physics-informed CNNs for super-resolution, considering theapplication to dynamical systems where the temporal nature cannot be neglected. We showcase thesuper-resolution of sparse, spatial observations of the chaotic-turbulent Kolmogorov flow, highlighting
the ability to recover finer scales of turbulence through reconstruction of the missing physics.

2

Defining the Dynamical System

Our work considers super-resolution of sparse observations on dynamical systems of the form∂t u − N (u; λ) = 0

with u = u(x, t),x ∈ Ω ⊂ Rn ,t ∈ [0, T ] ⊂ R+ ,(1)

where u : Ω × [0, T ] → Rn , N is a sufficiently smooth differential operator, and λ are the physicalparameters of the system. We define the residual of the system as the left-hand side of equation (1)R(u; λ) ≜ ∂t u − N (u; λ),(2)such that R(u; λ) = 0 when u(x, t) is a solution to the partial differential equation (PDE).

3The Super-Resolution Task

Given sparse observations on a low-resolution grid, we aim to reconstruct the underlying solution to
the PDE on a high-resolution grid. Mathematically, we denote this process by the mappingfθ : u(ΩL , t) → u(ΩH , t),

(3)
n

n

where the domain Ω is discretised on uniform, structured grids ΩL ⊂ RN , ΩH ⊂ RM such that
ΩL ∩ ΩH = ΩL , and M = κN where κ ∈ N+ is the up-sampling factor. An overview is providedtin Figure 1. We further discretise the time-domain, providing T = {ti ∈ [0, T ]}N
i=0 for Nt samples.
Approximating the mapping fθ as a CNN, we optimise weights θ of the network to minimise the loss
Lθ = αLO + LR ,(4)

where α represents a fixed, empirical weighting parameter. We further define each loss term as
1 XΩL∥fθ (u(ΩL , t))− u(ΩL , t)∥2ΩL ,
|T |
t∈T
1 X
LR =
∥R(fθ (u(ΩL , t)); λ)∥2ΩH ,
|T |
LO =

(5)

t∈T

ΩL

where fθ ( · )denotes the corestriction of ΩH on ΩL , and ∥ · ∥Ω represents the ℓ2 -norm overthe given domain. The observation-based loss LO seeks to minimise the distance between known
observations and their corresponding predictions - not accounting for additional high-resolution
information unavailable to the system. We regularise network predictions with the residual-based
term LR , seeking to ensure that realisations of high-resolution fields satisfy the governing PDEs. As
a consequence of imposing prior knowledge of the governing equations through the residual-based
loss term, we are effectively able to condition the underlying physics on the observed data, allowing
us to recover the underlying solution on the grid ΩH .1
1

All code is available on GitHub: https://github.com/magrilab/PISR

2

4

Results

We consider super-resolution for the Kolmogorov flow, a solution of the incompressible NavierStokes equations. The system is evaluated on the domain Ω ∈ [0, 2π) ⊂ R2 with periodic boundary
conditions applied on ∂Ω, and a stationary, spatially-varying sinusoidal forcing term g(x). The
Kolmogorov flow is prominent in studies of turbulence, providing a suitable case study for thesuper-resolution task; this allows us to evaluate the quality of predictions with respect to the turbulent
energy cascade [13]. The standard continuity and momentum equations are∇ · u = 0,
∂t u + u · ∇u = −∇p + ν∆u + g(x),(6)

where p, ν denote the scalar pressure field and kinematic viscosity respectively. We take ν = 1/34 toensure chaotic-turbulent dynamics, and prescribe the standard forcing g(x) = [sin(4x2 ), 0]⊤ .
4.1

Differentiable Pseudospectral Discretisation

We utilise a differentiable pseudospectral spatial discretisation for the Kolmogorov flow, enabling
backpropagation for the residual-based loss, LR . By eliminating the pressure term, our discretisation
handles the continuity constraint implicitly, allowing us to neglect the term in the loss [14]. Datais generated by time-integration of the dynamical system with the forward-Euler scheme, taking
a time-step ∆t = 0.005 to ensure numerical stability according to the Courant–Friedrichs–Lewy
condition. We evaluate the residual-based loss LR in the Fourier domain, re-defining the loss
LR =

1 X
∥∂t fˆθ (u(ΩL , t)) − N̂ (fˆθ (u(ΩL , t)))∥2Ω̂ ,
k
|T |

(7)

t∈T

n
where Ω̂k ∈ ZK is the discretised Fourier wavespace grid, fˆθ = F ◦ fθ where F is the Fourier operator, and N̂ denotes the Fourier differential operator; represented by the differentiable discretisation.
We consider evaluation of the loss over discrete time-windows, each consisting of τ ≥ 2 consecutive
time-steps. This windowing procedure allows for localised computation of the residual in the time
domain, alleviating the requirement for the observations to be totally contiguous in time.

4.2

Numerical Experiments

We solve the flow in the Fourier domain on a grid Ω̂k ∈ Z61×61 . Data is generated on the highresolution grid ΩH ∈ R150×150 prior to extracting the low-resolution grid ΩL ∈ R10×10 of observations. A total of 2048 time-windows are used for training, with a further 256 for validation; taking
τ = 2 in each instance. We employ the VDSR architecture [15], a variation on the common VGG-net[16]. By prepending the convolutional layers with a bi-cubic upsampling layer, we reformulate theproblem in-terms of learning the residual between the interpolation, and the high-resolution field;
exploiting benefits associated with residual learning problems [17]. Periodic boundary conditions areembedded in the model through the use of periodic padding in the convolutional layers. For training
we use the Adam optimizer with a learning rate of 3 × 10−4 , weighting the loss with α = 1 × 105 .
Results are compared with bi-linear (BL) and bi-cubic (BC) interpolation, measuring performanceby the relative ℓ2 -error between network predictions fθ (ΩL , t), and the true high-resolution fields,u(ΩH , t). Figure 2 shows a snapshot of the results, with network predictions yielding qualitatively
superior results when compared with the interpolated alternatives. Quantitatively we observe a lower
error for the network predictions, achieving a relative ℓ2 -error of 0.0872, compared with 0.2091 for
bi-linear interpolation and 0.1717 for bi-cubic interpolation; averaging results over available data.2
The energy spectrum is characteristic of the direct energy cascade observed in turbulent flows, amulti-scale phenomenon which sees energy content decaying with increasing wavenumber. In Figure
2, we observe that the energy content of the low-resolution field diverges from that of the highresolution field; a consequence of spectral aliasing. Applying the 2/3 de-aliasing rule to the Nyquist
2

All experiments were run on a single NVIDIA Quadro RTX 8000.

3

u(ΩL , t)BL(u(ΩL , t))BC(u(ΩL , t))
1.0

0.5

0.0

−0.5

−1.0

fθ (u(ΩL , t))

Energy Spectrum
n
fΩ

L

fn
Ω̂k

104

E(|k|)u(ΩH , t)nNyquist fΩ
2/3f nΩu(ΩH , t)u(ΩL , t)BC(u(ΩL , t))BL(u(ΩL , t))fθ (u(ΩL , t))
10−15
100

101

102

|k|

Figure 2: Super-resolution of u(ΩL , t) ∈ R10×10 to u(ΩH , t) ∈ R150×150 , demonstrating: bi-linear
interpolation, BL(u(ΩL , t)); bi-cubic interpolation, BC(u(ΩL , t)); and predictions, fθ (u(ΩL , t)).
frequencies, we see that aliasing is introduced at |k| = 5, 28 for the low- and high-resolution fields
respectively. This correlates with the divergence in energy content observed in the spectrum. We
find that fields super-resolved by the network are capable of capturing finer scales of turbulencecompared to low-resolution and interpolation approaches, prior to diverging at |k| = 101 . This result
is indicative of the impact of the residual-based loss, signifying the ability of the network to act
beyond simple interpolation. The network is capable of de-aliasing, or inferring missing physics.
Despite the use of periodic padding in the convolutional layers, we observe artefacts on the boundariesof the predicted field. These artefacts are largely responsible for the divergence of the predictedenergy spectrum from the true spectrum and provide a consideration for future work.

5

Conclusion

In this work, we have demonstrated the application of physics-informed CNNs for the super-resolutionof the chaotic-turbulent Kolmogorov flow. Our results show improved performance when comparedwith standard interpolation methods. Evaluation of the turbulent energy spectrum highlights theability of the network to resolve turbulent structures on a finer scale than available in the sparse
observations. We note that this reconstruction of turbulence showcases the ability to resolve missing
physics in the sparse observations, a consequence of embedding prior knowledge of the physics in theloss term. With a view on the spectral representation, we find that the network effectively mitigates
the aliasing introduced by taking sparse observations. This work opens opportunities for the accurate
reconstruction of dynamical systems from sparse observations, as is often the case in experimental
settings for the physical sciences.

Acknowledgments and Disclosure of Funding
D. Kelshaw. and L. Magri. acknowledge support from the UK EPSRC. L. Magri gratefully acknowledges financial support from the ERC Starting Grant PhyCo 949388.
4

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5

Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
contributions and scope?
[Yes] We state the ability of the network to resolve finer scales of turbulence thanclassical interpolation methods as a result of the super-resolution; this is demonstrated in Figure 2 and discussed in the results section. We show quantitative
improvements in terms of relative ℓ2 -error, as well as justifying results with respect
to the turbulent energy cascade.
(b) Did you describe the limitations of your work?
[Yes] We discuss the presence of artefacts on the boundaries of the predicted
field in the final paragraph of the results, §4. This is largely responsible for thedivergence in the energy spectrum, as seen in Figure 2.
(c) Did you discuss any potential negative societal impacts of your work?
[No] We do not for-see any potential negative societal impacts.
(d) Have you read the ethics review guidelines and ensured that your paper conforms tothem? [Yes]2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [N/A](b) Did you include complete proofs of all theoretical results? [N/A]
3. If you ran experiments...
(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)?
[Yes] We include a link to the code in the footprint of §3.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they
were chosen)?
[Yes] We provide all necessary training details in §4, including: architecture,
hyperparameters, amount of data, physical parameters for simulation results.
Further details are available on the link to the repository.
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
[No] We do not believe the inclusion of error-bars is warranted for this study. Results in Figure 2 show snapshots of the super-resolution, and the energy spectrum
plot is not well-suited to error bars. Repeats were indeed run, but not reported.
(d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
[Yes] We provide a footnote in §4.2 which states the computational resource used.
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [N/A](b) Did you mention the license of the assets? [N/A](c) Did you include any new assets either in the supplemental material or as a URL?
[Yes] We provide a differentiable pseudospectral spatial discretisation for the
Kolmogorov flow, a link to the repository is available in §3 - currently redacted.
(d) Did you discuss whether and how consent was obtained from people whose data you’reusing/curating? [N/A](e) Did you discuss whether the data you are using/curating contains personally identifiable
information or offensive content? [N/A]
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if
applicable? [N/A](b) Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable? [N/A](c) Did you include the estimated hourly wage paid to participants and the total amountspent on participant compensation? [N/A]
6