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An Operator Learning Framework for Spatiotemporal Super-resolution of Scientific Simulations

Valentin Duruisseaux, Amit Chakraborty

TL;DR

The paper tackles the challenge of obtaining high-resolution solutions to parametric PDEs under limited resources by reframing super-resolution as operator learning. It introduces the Super Resolution Operator Network (SROpNet), which learns a continuous solution operator $\mathcal{S}$ that maps a low-resolution representation $u_{LR}$ to a high-resolution field $u$, allowing evaluation on arbitrary meshes via a 3-subnetwork (Branch, Sensor, Trunk) architecture that supports mesh-free predictions and nonuniform sensor layouts. A physics-informed variant is discussed, with a loss term $\mathcal{L}_{physics}$ to enforce PDE constraints when feasible, though data-driven training remains effective in many cases. Numerical experiments across 1D and 2D diffusion problems and a 2D Kolmogorov flow demonstrate strong generalization to unseen parameters and flexible sensor configurations, underscoring the practical impact for sensor-agnostic spatiotemporal super-resolution in scientific simulations.

Abstract

In numerous contexts, high-resolution solutions to partial differential equations are required to capture faithfully essential dynamics which occur at small spatiotemporal scales, but these solutions can be very difficult and slow to obtain using traditional methods due to limited computational resources. A recent direction to circumvent these computational limitations is to use machine learning techniques for super-resolution, to reconstruct high-resolution numerical solutions from low-resolution simulations which can be obtained more efficiently. The proposed approach, the Super Resolution Operator Network (SROpNet), frames super-resolution as an operator learning problem and draws inspiration from existing architectures to learn continuous representations of solutions to parametric differential equations from low-resolution approximations, which can then be evaluated at any desired location. In addition, no restrictions are imposed on the locations of (the fixed number of) spatiotemporal sensors at which the low-resolution approximations are provided, thereby enabling the consideration of a broader spectrum of problems arising in practice, for which many existing super-resolution approaches are not well-suited.

An Operator Learning Framework for Spatiotemporal Super-resolution of Scientific Simulations

TL;DR

The paper tackles the challenge of obtaining high-resolution solutions to parametric PDEs under limited resources by reframing super-resolution as operator learning. It introduces the Super Resolution Operator Network (SROpNet), which learns a continuous solution operator that maps a low-resolution representation to a high-resolution field , allowing evaluation on arbitrary meshes via a 3-subnetwork (Branch, Sensor, Trunk) architecture that supports mesh-free predictions and nonuniform sensor layouts. A physics-informed variant is discussed, with a loss term to enforce PDE constraints when feasible, though data-driven training remains effective in many cases. Numerical experiments across 1D and 2D diffusion problems and a 2D Kolmogorov flow demonstrate strong generalization to unseen parameters and flexible sensor configurations, underscoring the practical impact for sensor-agnostic spatiotemporal super-resolution in scientific simulations.

Abstract

In numerous contexts, high-resolution solutions to partial differential equations are required to capture faithfully essential dynamics which occur at small spatiotemporal scales, but these solutions can be very difficult and slow to obtain using traditional methods due to limited computational resources. A recent direction to circumvent these computational limitations is to use machine learning techniques for super-resolution, to reconstruct high-resolution numerical solutions from low-resolution simulations which can be obtained more efficiently. The proposed approach, the Super Resolution Operator Network (SROpNet), frames super-resolution as an operator learning problem and draws inspiration from existing architectures to learn continuous representations of solutions to parametric differential equations from low-resolution approximations, which can then be evaluated at any desired location. In addition, no restrictions are imposed on the locations of (the fixed number of) spatiotemporal sensors at which the low-resolution approximations are provided, thereby enabling the consideration of a broader spectrum of problems arising in practice, for which many existing super-resolution approaches are not well-suited.
Paper Structure (22 sections, 2 theorems, 15 equations, 12 figures, 1 table)

This paper contains 22 sections, 2 theorems, 15 equations, 12 figures, 1 table.

Table of Contents

  1. Preliminaries
  2. Operator Learning
  3. Motivation
  4. Operator Learning
  5. Spatiotemporal Super-Resolution of Scientific Simulations
  6. Super-resolution as an Operator Learning Problem
  7. Super Resolution Operator Networks (SROpNets)
  8. Physics-Informing
  9. SROpNet Architecture Examples
  10. Examples of SROpNets using simple combinations of MLPs, CNNs, and LSTMs
  11. Examples of SROpNets based on existing super-resolution algorithms
  12. Numerical Experiments
  13. 1D Forced Diffusion
  14. 1D Diffusion with Parametric Forcing
  15. Forced 1D Diffusion with Varied Initial Conditions
  16. ...and 7 more sections

Key Result

Theorem 1.1

Suppose $D_u \subset \mathbb{R}^{d_u}$ and $D_v \subset \mathbb{R}^{d_v}$ are compact sets. Let $\mathcal{G}$ be a nonlinear continuous operator mapping a subset of $C(D_u)$ into $C(D_v)$. Then, given $\varepsilon>0$, there exists a DeepONet $\mathcal{G}_\theta$ such that $|\mathcal{G}(u)(y) - \math

Figures (12)

  • Figure 1: Results using different SROpNets on 4 previously-unobserved samples from the dataset of solutions to the 1D Forced Diffusion equation \ref{['eq: Diffusion 1D Exp1']}.
  • Figure 2: Super-resolution results for 4 previously-unobserved samples from the dataset of solutions to the 1D Forced Diffusion equation \ref{['eq: Diffusion 1D Exp2']}.
  • Figure 3: Randomly-selected 244 sensor locations and 3600 prediction locations.
  • Figure 4: Results for 4 previously-unobserved samples from the dataset of solutions to the forced diffusion equation \ref{['eq: Diffusion 1D with Random Grids']} with random sensor and prediction locations.
  • Figure 5: Example of low-resolution simulation from the dataset of solutions to the 2D Diffusion equation \ref{['eq: 2D Diffusion']} with fixed diffusion constant.
  • ...and 7 more figures

Theorems & Definitions (2)

  • Theorem 1.1
  • Theorem 1.2