Efficient probabilistic surrogate modeling techniques for partially-observed large-scale dynamical systems

TL;DR

The paper tackles forecasting of partially observed, large-scale dynamical systems governed by PDEs using probabilistic surrogate models. It centers on flow matching and a suite of extensions—direct distillation, progressive distillation, adversarial diffusion distillation, and rectified flows—to accelerate sampling while maintaining physical fidelity, and validates them on challenging Navier–Stokes and Rayleigh–Taylor instability benchmarks, including 2D slices for inflow generation. Key findings show adversarial diffusion distillation provides the fastest, most plausible samples, while progressive distillation offers a simple and efficient alternative; deterministic baselines tend to blur outcomes, and GAN-based approaches can be unstable on some data. The work demonstrates the viability of fast, probabilistic surrogates for partially observed dynamics, with practical impact for accelerating solvers and enabling inflow generation in large-scale simulations; code is publicly available.

Abstract

This paper is concerned with probabilistic techniques for forecasting dynamical systems described by partial differential equations (such as, for example, the Navier-Stokes equations). In particular, it is investigating and comparing various extensions to the flow matching paradigm that reduce the number of sampling steps. In this regard, it compares direct distillation, progressive distillation, adversarial diffusion distillation, Wasserstein GANs and rectified flows. Moreover, experiments are conducted on a set of challenging systems. In particular, we also address the challenge of directly predicting 2D slices of large-scale 3D simulations, paving the way for efficient inflow generation for solvers.

Figures (10)

• Figure 1: Autoregressively generated trajectories on the "sliced" Rayleigh-Taylor instability dataset, starting with the same initial condition at (simulation) time $\tau_0$.
• Figure 2: Results after a single prediction step. $^1$FM solved with 5 midpoint steps on the dNSE and dRTI datasets, and with 20 midpoint steps on the sRTI dataset.
• Figure 3: Samples and partial observations from the three datasets we are working with.
• Figure 4: Statistics (sharpness, energy and error to real trajectory) of model predictions over prediction horizon. Results averaged over all initial conditions.
• Figure 5: Spectra of the kinetic energy density on the dNSE dataset after multiple autoregressive prediction steps with equal initial conditions. The spectra are then averaged over all initial conditions. (See also \ref{['fig:kinetic-energy-density-ns-offset']} in the appendix for further comparison.)

Theorems & Definitions (1)

• remark 1