The False Promise of Zero-Shot Super-Resolution in Machine-Learned Operators

TL;DR

The paper investigates whether machine-learned operators can perform zero-shot super-resolution across discretizations, scrutinizing the Fourier Neural Operator on standard PDE benchmarks. It shows that zero-shot interpolation and extrapolation fail due to aliasing and out-of-distribution sensitivity to discretization, and that physics-informed constraints and band-limited learning do not resolve these issues. As a practical remedy, the authors propose multi-resolution training, demonstrating that incorporating data from multiple resolutions—preferably mostly low-resolution data with some high-resolution samples—greatly improves cross-resolution generalization at low additional cost. This work clarifies the limitations of zero-shot approaches in scientific ML and provides a scalable training paradigm to enable robust multi-resolution inference in surrogate PDE models, with potential broad impact on PDEBench-style workflows and mesh-invariant modeling.

Abstract

A core challenge in scientific machine learning, and scientific computing more generally, is modeling continuous phenomena which (in practice) are represented discretely. Machine-learned operators (MLOs) have been introduced as a means to achieve this modeling goal, as this class of architecture can perform inference at arbitrary resolution. In this work, we evaluate whether this architectural innovation is sufficient to perform "zero-shot super-resolution," namely to enable a model to serve inference on higher-resolution data than that on which it was originally trained. We comprehensively evaluate both zero-shot sub-resolution and super-resolution (i.e., multi-resolution) inference in MLOs. We decouple multi-resolution inference into two key behaviors: 1) extrapolation to varying frequency information; and 2) interpolating across varying resolutions. We empirically demonstrate that MLOs fail to do both of these tasks in a zero-shot manner. Consequently, we find MLOs are not able to perform accurate inference at resolutions different from those on which they were trained, and instead they are brittle and susceptible to aliasing. To address these failure modes, we propose a simple, computationally-efficient, and data-driven multi-resolution training protocol that overcomes aliasing and that provides robust multi-resolution generalization.

Figures (33)

• Figure 1: Aliasing in zero-shot super-resolution. Model trained on resolution 16 data, and evaluated at varying resolutions: 16, 32, 64, 128. Top Row: Sample prediction for Darcy flow; notice striation artifacts at resolution 128. Middle Row: Average test set 2D energy spectrum of label and model prediction. Bottom Row: Average residual spectrum normalized by label spectrum.
• Figure 2: Accurate multi-resolution inference requires both interpolation and extrapolation.Original: Signal is sampled at a rate greater than its Nyquist frequency. Interpolation: Adapting to new sampling rates of a given signal. Extrapolation: Adapting to new frequency information under constant sampling rate. Super-Resolution: Sampling a system at a higher rate, which enables the capture of higher frequency information (interpolation and extrapolation). Aliasing: High-frequency information is misrepresented as a low-frequency information due to insufficient sampling.
• Figure 3: Resolution Interpolation. Four FNOs are trained on Darcy data at resolutions {16, 32, 64, 128} from left to right with constant frequency information (low-pass limit of 8$f$). We test whether each model can generalize to data with varying resolution, visualizing the normalized spectra of the residuals averaged across test data. Notice that the residual spectra (error) increases substantially in the higher frequencies. (Lower residual energy at all frequencies is better.)
• Figure 4: Information Extrapolation. Four FNOs trained on Darcy data of resolution 128 (constant sampling rate) and low-pass filtered with limits {8, 16, 32, 64}$f$ (varying frequency information) from left to right. We test whether each model can generalize to data with varying frequency information, visualizing the normalized spectra of the residuals averaged across test data. Notice that the residual spectra (error) increases substantially in the higher frequencies. (Lower residual energy at all frequencies is better.)
• Figure 5: FNOs do not generalize to higher or lower resolutions. Model trained on Navier Stokes dataset at resolution 255 (indicated by *), evaluated resolutions 510, 255, 128. Top Row: Ground truth, prediction at resolutions 510 (super-resolution), resolution 255 (same as train resolution), and resolution 128 (sub-resolution). Bottom Row: Average energy spectra over test data.