[Paper Review] Physics-Informed CNNs for Super-Resolution of Sparse Observations on Dynamical Systems

Daniel Kelshaw, George Rigas+1 more|arXiv (Cornell University)|Oct 31, 2022

Advanced Image Processing Techniques4 citations

TL;DR

This paper proposes physics-informed convolutional neural networks (PINNs) for super-resolution of sparse observations in chaotic-turbulent dynamical systems, such as the Kolmogorov flow. By combining a residual learning architecture with a PDE residual loss that enforces the Navier-Stokes equations, the method reconstructs high-resolution fields with improved fidelity over interpolation, achieving a relative ℓ² error of 0.0872—significantly lower than bi-linear (0.2091) and bi-cubic (0.1717) interpolation.

ABSTRACT

In the absence of high-resolution samples, super-resolution of sparse observations on dynamical systems is a challenging problem with wide-reaching applications in experimental 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 potential of this method for resolving finer scales of turbulence when compared with classic interpolation methods, and thus effectively reconstructing missing physics.

Motivation & Objective

• To address the challenge of reconstructing high-resolution fields from sparse, partial observations in dynamical systems where high-fidelity data is unavailable.
• To improve super-resolution beyond standard interpolation by embedding physical laws into the learning process.
• To demonstrate that physics-informed CNNs can reconstruct missing turbulence physics in chaotic flows such as the Kolmogorov flow.
• To validate the method on a turbulent, chaotic system with complex multi-scale dynamics and spectral aliasing.
• To show that residual-based loss regularization enables de-aliasing and recovery of fine-scale structures not present in low-resolution data.

Proposed method

• A physics-informed CNN is trained using a hybrid loss combining observation fidelity and PDE residual minimization.
• The network maps low-resolution observations on Ωₗ to high-resolution predictions on Ωₕ using a VDSR-based architecture with bi-cubic upsampling.
• The residual loss ℒ_R enforces the Navier-Stokes equations by computing the time derivative and nonlinear operator of the predicted field.
• Time-windowing is applied to compute residuals over τ=2 consecutive time steps, enabling localized temporal regularization without requiring fully contiguous data.
• Periodic boundary conditions are enforced via periodic padding in convolutional layers to preserve spectral properties.
• The loss function is optimized using Adam with α=1×10⁵ to balance observation and residual terms.

Experimental results

Research questions

• RQ1 Can physics-informed CNNs reconstruct high-resolution fields from sparse observations in chaotic-turbulent dynamical systems?
• RQ2 Does incorporating the PDE residual into the loss function enable better super-resolution than standard interpolation?
• RQ3 To what extent can the model recover fine-scale turbulent structures not present in low-resolution data?
• RQ4 How does the method perform in mitigating spectral aliasing due to undersampling in the Kolmogorov flow?
• RQ5 Can the model generalize to reconstructing missing physics when ground-truth high-resolution data is unavailable?

Key findings

• The physics-informed CNN achieved a relative ℓ² error of 0.0872 on the high-resolution field, outperforming bi-linear (0.2091) and bi-cubic (0.1717) interpolation.
• The predicted energy spectrum captured finer-scale turbulence up to |k| ≈ 10¹, whereas interpolated fields diverged earlier due to aliasing.
• The model successfully mitigated spectral aliasing by learning to de-alias the low-resolution data through PDE regularization.
• The network reconstructed turbulent structures not visible in the low-resolution observations, indicating recovery of missing physics.
• Boundary artefacts were observed due to periodic padding, which caused divergence in the energy spectrum at high wavenumbers.
• The method demonstrated robustness to sparse, non-contiguous observations through time-windowed residual computation over τ=2 steps.