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Probabilistic Super-Resolution for High-Fidelity Physical System Simulations with Uncertainty Quantification

Pengyu Zhang, Connor Duffin, Alex Glyn-Davies, Arnaud Vadeboncoeur, Mark Girolami

TL;DR

This work tackles the high computational cost of obtaining high-fidelity PDE solutions by proposing ProbSR, a probabilistic super-resolution framework that uses statFEM-based priors and energy-based modeling to predict high-resolution fields from low-resolution data with uncertainty quantification. It replaces large HR training datasets with a Bayesian upscaling approach where a downscaling network learns residual corrections to bicubic upsampling, and employs Langevin dynamics to sample from posteriors for UQ. The key contributions are a data-efficient SR method for physical systems, a principled uncertainty quantification pipeline via posterior sampling, and demonstrated speedups over traditional high-resolution solvers on a 2D Poisson example. The approach enables reliable, efficient high-fidelity simulations suitable for engineering design and repeated evaluations, with future work aimed at extending to more complex physics and optimizing Langevin sampling with preconditioners.

Abstract

Super-resolution (SR) is a promising tool for generating high-fidelity simulations of physical systems from low-resolution data, enabling fast and accurate predictions in engineering applications. However, existing deep-learning based SR methods, require large labeled datasets and lack reliable uncertainty quantification (UQ), limiting their applicability in real-world scenarios. To overcome these challenges, we propose a probabilistic SR framework that leverages the Statistical Finite Element Method and energy-based generative modeling. Our method enables efficient high-resolution predictions with inherent UQ, while eliminating the need for extensive labeled datasets. The method is validated on a 2D Poisson example and compared with bicubic interpolation upscaling. Results demonstrate a computational speed-up over high-resolution numerical solvers while providing reliable uncertainty estimates.

Probabilistic Super-Resolution for High-Fidelity Physical System Simulations with Uncertainty Quantification

TL;DR

This work tackles the high computational cost of obtaining high-fidelity PDE solutions by proposing ProbSR, a probabilistic super-resolution framework that uses statFEM-based priors and energy-based modeling to predict high-resolution fields from low-resolution data with uncertainty quantification. It replaces large HR training datasets with a Bayesian upscaling approach where a downscaling network learns residual corrections to bicubic upsampling, and employs Langevin dynamics to sample from posteriors for UQ. The key contributions are a data-efficient SR method for physical systems, a principled uncertainty quantification pipeline via posterior sampling, and demonstrated speedups over traditional high-resolution solvers on a 2D Poisson example. The approach enables reliable, efficient high-fidelity simulations suitable for engineering design and repeated evaluations, with future work aimed at extending to more complex physics and optimizing Langevin sampling with preconditioners.

Abstract

Super-resolution (SR) is a promising tool for generating high-fidelity simulations of physical systems from low-resolution data, enabling fast and accurate predictions in engineering applications. However, existing deep-learning based SR methods, require large labeled datasets and lack reliable uncertainty quantification (UQ), limiting their applicability in real-world scenarios. To overcome these challenges, we propose a probabilistic SR framework that leverages the Statistical Finite Element Method and energy-based generative modeling. Our method enables efficient high-resolution predictions with inherent UQ, while eliminating the need for extensive labeled datasets. The method is validated on a 2D Poisson example and compared with bicubic interpolation upscaling. Results demonstrate a computational speed-up over high-resolution numerical solvers while providing reliable uncertainty estimates.

Paper Structure

This paper contains 15 sections, 17 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Statistical Finite Element Method (statFEM)
  4. Langevin Dynamics
  5. Proposed Methodology
  6. Model Structure
  7. Downscaling Network
  8. Training and Inference Procedure
  9. Training
  10. Inference
  11. Numerical Experiments
  12. Problem setup
  13. Results
  14. Conclusions and Future Work
  15. Acknowledgements

Figures (3)

  • Figure 1: The LR solution, and SR results of two test cases using direct bicubic upscaling and ProbSR.
  • Figure 2: Logarithm of the standard deviation field overlaid with black dots representing the original $40 \times 40$ grid.
  • Figure 3: Running time of two different methods for generating different resolutions of PDE solution.