Fill the gaps: continuous in time interpolation of fluid dynamical simulations

TL;DR

This work tackles continuous-in-time interpolation of fluid simulations by marrying Fourier Neural Operator (FNO) with a physics-informed token transformer (PITT), extended to multi-channel data. The PITT FNO framework learns a time-continuous interpolation operator that preserves mass and energy while delivering 6–10x data efficiency over linear interpolation, and it demonstrates strong performance on a 2D Euler-equations dataset. However, fine-scale features remain challenging due to temporal undersampling and high-frequency limitations not fully resolved by simply increasing the Fourier cutoff, indicating a practical bound set by data cadence. The approach offers significant potential for post-processing and runtime sub-stepping in fluid simulations, enabling substantial storage savings and faster interpolations while maintaining physical consistency.

Abstract

Flexible and accurate interpolation schemes using machine learning could be of great benefit for many use-cases in numerical simulations and post-processing, such as temporal upsampling or storage reduction. In this work, we adapt the physics-informed token transformer (PITT) network for multi-channel data and couple it with Fourier neural operator (FNO). The resulting PITT FNO network is trained for interpolation tasks on a dataset governed by the Euler equations. We compare the performance of our machine learning model with a linear interpolation baseline and show that it requires $\sim6-10$ times less data to achieve the same mean square error of the interpolated quantities. Additionally, PITT FNO has excellent mass and energy conservation as a result of its physics-informed nature. We further discuss the ability of the network to recover fine detail using a spectral analysis. Our results suggest that loss of fine details is related to the decreasing correlation time of the data with increasing Fourier mode which cannot be resolved by simply increasing Fourier mode truncation in FNO.

Figures (8)

• Figure 1: The adapted PITT network uses the tokenised interpolation time and equation to create an update to the neural operator output. The start ($t=0$) and end ($t=1$) frame are both independently subtracted from the neural operator output and concatenated before input in the attention block to create symmetry in the network predictions. The attention block explores cross-correlations in the multi-channel data, which is used together with the equation and time embedding to update the neural operator output to target time $T$.
• Figure 2: Example of PITT FNO prediction for the Euler dataset. From top to bottom: Density, energy, momentum in $x$, momentum in $y$. The start frame ($t=0$) and end frame ($t=1$) are input frames into the network and are separated by 20 data frames, with the target being at interpolation time $t=0.55$. The PITT FNO prediction is compared to the target frame and linear interpolation at the interpolation time. By learning the dynamics between snapshots through the PDE embedding, PITT FNO is able to predict non-trivial features such as crossing shock fronts observed in the energy prediction. For a movie equivalent of this Figure, go to this https://youtu.be/JbK104xy74Y?si=sANgVU_EAeiiMg3l.
• Figure 3: Mean square error as a function of separation between the input frames for the Euler equations. PITT FNO is able to outperform stand-alone FNO and linear interpolation across all levels of separation. Compression ratios between PITT FNO and linear interpolation range from 6 for low MSE (0.1) to 10 at higher MSE (0.2-0.3). All separations and compression ratios are given in Table \ref{['tab:MSE-Euler']}.
• Figure 4: Mean square error split up between variables during a single simulation. From top to bottom: Density, energy, momentum in $x$, momentum in $y$. The dashed black line shows the location of the input frames used to predict the fluid state between the inputs, with separations of 20 frames. Compared to linear interpolation, PITT FNO provides remarkably consistent predictive power across the entire domain.
• Figure 5: Relative error in mass, energy and momentum conservation during a single simulation. From top to bottom: Density, energy, momentum in $x$, momentum in $y$. The dashed black line shows the location of the input frames used to predict the fluid state between the inputs, with separations of 20 frames. Likely as a result of PDE embedding, errors in mass and energy conservation are low throughout the simulation.