Data-driven multiscale modeling for correcting dynamical systems

TL;DR

The paper addresses the challenge of unstable, offline-trained subgrid closures in dynamical systems by introducing a data-driven, multiscale learning framework that splits the prediction into downscale and buildup steps across three scales: true, high-resolution ($\text{hr}$), and low-resolution ($\text{lr}$). It formalizes the workflow with coarsening operators $C$ and scale transforms $D$, $D^{+}$, training separate networks to predict $S_{\mathsf{x,lr}}$ from coarse states and then refine to $S_{\mathsf{x}}$, and validates the approach on two climate-like models, QG and KF, showing improved stability and accuracy, especially for smaller networks, and enabling online closures; in KF, stability required training-time noise tuning. The key contribution is demonstrating that exploiting cross-scale information through non-end-to-end training can yield robust, efficient closures suitable for online deployment, while still enabling end-to-end pipelines. The findings highlight the potential for multiscale, self-similarity-aware learning to enhance predictive performance and stability in multiscale simulations, with implications for stochastic parameterizations and broader scientific computing tasks.

Abstract

We propose a multiscale approach for predicting quantities in dynamical systems which is explicitly structured to extract information in both fine-to-coarse and coarse-to-fine directions. We envision this method being generally applicable to problems with significant self-similarity or in which the prediction task is challenging and where stability of a learned model's impact on the target dynamical system is important. We evaluate our approach on a climate subgrid parameterization task in which our multiscale networks correct chaotic underlying models to reflect the contributions of unresolved, fine-scale dynamics.

Figures (13)

• Figure 1: The several scales involved in the experiments described below. The true resolution is used only while rolling out a ground truth reference trajectory and then discarded. Samples are resampled to lower resolutions using operators $C$ and $D$ described in Appendix \ref{['sec:coarsedef']}. The values $S_{\mathsf{x}}$ are computed following Equation \ref{['eq:forcingdef']} using both $\mathsf{x}_{\text{true}}$ and $\mathsf{x}_{\text{hr}}$. The high and low resolution samples are used for training. During evaluation the simulation is run online at the high resolution---following the dashed lines.
• Figure 2: Downscale vs. across separated prediction tasks. The networks referenced in Equation \ref{['eq:datasks']} are combinations of an inner network $f_{\theta}$ with the fixed rescaling operators $D$, $D^{+}$. The overall prediction is indicated with a dashed line.
• Figure 3: Buildup vs. direct separated prediction. The networks in Equation \ref{['eq:bdtasks']} are combinations of the networks $f_{\theta}$ with the indicated fixed operations. In Figure \ref{['fig:buildup']}$f_{\theta}$ predicts the details which are combined with $S_{\text{lr}}$ from an oracle.
• Figure 4: Flow for the combined prediction experiments. For the full combined flow each network $f_{\theta}$ is trained sequentially, and earlier weights are frozen and used to train networks later in the pipeline. The overall flow no longer requires an oracle for any additional inputs. Note also that the networks in the baseline flow in Figure \ref{['fig:comb-base']} have the same residual prediction structure. The baseline network is trained end-to-end and differs only in that it misses the additional supervision from the multiscale training. The variable $\mathsf{x}$ represents the differing scalar fields for the QG and KF systems.
• Figure 5: Evaluation results from both of the separated experiments for the MSE metric. These are the same numbers which are reported as averages in Table \ref{['tab:higherres']} and Table \ref{['tab:buildup']}. The plot here shows the three samples---one from each trained network---used to compute the means.