Oliver R. A. Dunbar

Physics‐based models of the earthquake cycle could be used for time‐dependent hazard assessment. For such an application, their parameters must be calibrated so that simulated earthquake sequences reproduce the statistics of past earthquakes, including recurrence statistics and magnitudes. This is challenging because the dynamics are described by nonlinear partial differential equations, initial conditions are unknown, and records of past earthquakes are sparse, incomplete, and noisy. To address this challenge, we present a Bayesian inverse‐problem framework to infer heterogeneous parameters in an earthquake‐cycle model from observations of past large earthquakes. Rather than matching a time‐series of individual events, we exploit ergodicity (long‐term statistics independent of initial conditions) and invert for stationary, spatially distributed summary statistics of event timing and slip, including mean interevent times, coefficients of variation, burstiness, and average coseismic slip, which are quantities that can in principle be estimated from paleoseismic and historical records. Observational noise and event‐detection thresholds are explicitly accounted for in the construction of these statistics. The inverse problem is solved using a derivative‐free ensemble Kalman inversion method, and uncertainty quantification is enhanced by an emulate–sample strategy that leverages machine‐learning for surrogate modeling, in which a Gaussian‐process surrogate of the parameter‐to‐statistics map enables efficient MCMC sampling without further expensive forward simulations. Synthetic experiments with realistic data noise suggest that the model parameters can be inverted from statistical quantities observable from paleoseismology at a limited number of observation points. The framework provides a principled and computationally tractable method to calibrate earthquake‐cycle models, providing a foundation for forecasting.

Ensemble Kalman methods were initially developed to solve nonlinear data assimilation problems in oceanography but are now popular in applications far beyond their original use cases. Of particular interest is climate model calibration. As hybrid physics and machine‐learning models advance, the number of parameters and complexity of parameterizations in climate models will continue to grow. To fully realize these advances, we must move from laborious hand‐tuning to calibration‐driven model development in rapid iteration cycles. Thus, robust calibration of these parameters plays an increasingly important role. We focus on learning climate model parameters by minimizing the misfit between modeled and observed climate statistics in an idealized setting. Ensemble Kalman methods are a natural choice for this problem because they are derivative‐free, scalable to high dimensions, and robust to noise caused by statistical observations. Given the many variants of ensemble methods proposed, an important question is: Which ensemble Kalman method should be used for climate model calibration ? To answer this question, we perform systematic numerical experiments to explore the relative computational efficiencies of several ensemble Kalman methods. The numerical experiments involve statistical observations of Lorenz‐type models of increasing complexity, frequently used to represent simplified atmospheric systems, with some featuring neural network parameterizations. For each test problem, several ensemble Kalman methods and a derivative‐based method “race” to reach a specified accuracy, and we measure the computational cost required to achieve the desired accuracy. We investigate how prior information and the parameter or data dimensions define the computational costs of the various methods.

Physical Review Different approaches to using data-driven methods for subgrid-scale closure modeling of geophysical turbulence have emerged recently. Most of these approaches are data hungry and lack interpretability and out-of-distribution generalizability. Here, we use a hybrid approach that combines turbulence theory, physics-based modeling, and data-driven methods to overcome these challenges. Specifically, we address the parametric uncertainty of well-known physics-based large-eddy simulation (LES) closures: the Smagorinsky (Smag) and Leith eddy-viscosity models (one free parameter) and the Jansen-Held (JH) backscattering model (two free parameters). For various cases of two-dimensional turbulence, optimal parameters are first learned online from data via ensemble Kalman inversion (EKI), such that for each case, the LES energy spectrum matches that of direct numerical simulation (DNS). We quantify the uncertainties on these parameters using a modern machine-learning-accelerated Bayesian workflow, “, , ” Only a small training dataset is needed (to calculate the DNS spectra); i.e., the approach is data-efficient. We find the optimized parameter(s) and their associated uncertainty for each closure to be constant across broad flow regimes that differ in dominant length scales, eddy/jet structures, and dynamics, suggesting that these closures are generalizable. Next, we show that the online learned constants agree with the predictions of a recent semianalytical derivation, providing further interpretability. In both and tests that include examining the extreme events, LES with optimized closures, especially with JH, outperforms the baselines (LES with standard Smag, dynamic Smag, or Leith). This work shows the promise of combining advances in theory, physics-based modeling (e.g., JH), and data-driven modeling (e.g., online learning with EKI) to develop data-efficient frameworks for accurate, interpretable, and generalizable closures for geophysical turbulence, with ultimate applications in weather and climate prediction.

This work integrates machine learning into an atmospheric parameterization to target uncertain mixing processes while maintaining interpretable, predictive, and well‐established physical equations. We adopt an eddy‐diffusivity mass‐flux (EDMF) parameterization for the unified modeling of various convective and turbulent regimes. To avoid drift and instability that plague offline‐trained machine learning parameterizations that are subsequently coupled with climate models, we frame learning as an inverse problem: Data‐driven models are embedded within the EDMF parameterization and trained online in a one‐dimensional vertical global climate model (GCM) column. Training is performed against output from large‐eddy simulations (LES) forced with GCM‐simulated large‐scale conditions in the Pacific. Rather than optimizing subgrid‐scale tendencies, our framework directly targets climate variables of interest, such as the vertical profiles of entropy and liquid water path. Specifically, we use ensemble Kalman inversion to simultaneously calibrate both the EDMF parameters and the parameters governing data‐driven lateral mixing rates. The calibrated parameterization outperforms existing EDMF schemes, particularly in tropical and subtropical locations of the present climate, and maintains high fidelity in simulating shallow cumulus and stratocumulus regimes under increased sea surface temperatures from AMIP4K experiments. The results showcase the advantage of physically constraining data‐driven models and directly targeting relevant variables through online learning to build robust and stable machine learning parameterizations.

We synthesize knowledge from numerical weather prediction, inverse theory, and statistics to address the problem of estimating a high‐dimensional covariance matrix from a small number of samples. This problem is fundamental in statistics, machine learning/artificial intelligence, and in modern Earth science. We create several new adaptive methods for high‐dimensional covariance estimation, but one method, which we call Noise‐Informed Covariance Estimation (NICE), stands out because it has three important properties: (a) NICE is conceptually simple and computationally efficient; (b) NICE guarantees symmetric positive semi‐definite covariance estimates; and (c) NICE is largely tuning‐free. We illustrate the use of NICE on a large set of Earth science–inspired numerical examples, including cycling data assimilation, inversion of geophysical field data, and training of feed‐forward neural networks with time‐averaged data from a chaotic dynamical system. Our theory, heuristics and numerical tests suggest that NICE may indeed be a viable option for high‐dimensional covariance estimation in many Earth science problems.

This work integrates machine learning into an atmospheric parameterization to target uncertain mixing processes while maintaining interpretable, predictive, and well-established physical equations. We adopt an eddy-diffusivity mass-flux (EDMF) parameterization for the unified modeling of various convective and turbulent regimes. To avoid drift and instability that plague offline-trained machine learning parameterizations that are subsequently coupled with climate models, we frame learning as an inverse problem: Data-driven models are embedded within the EDMF parameterization and trained online using output from large-eddy simulations (LES) forced with GCM-simulated large-scale conditions in the Pacific. Rather than optimizing subgrid-scale tendencies, our framework directly targets climate variables of interest, such as the vertical profiles of entropy and liquid water path. Specifically, we use ensemble Kalman inversion to simultaneously calibrate both the EDMF parameters and the parameters governing data-driven lateral mixing rates. The calibrated parameterization outperforms existing EDMF schemes, particularly in tropical and subtropical locations of the present climate, and maintains high fidelity in simulating shallow cumulus and stratocumulus regimes under increased sea surface temperatures from AMIP4K experiments. The results showcase the advantage of physically-constraining data-driven models and directly targeting relevant variables through online learning to build robust and stable machine learning parameterizations.

A Julia language (Bezanson et al., 2017) package providing practical and modular implementation of "Calibrate, Emulate, Sample" (Cleary et

We synthesize knowledge from numerical weather prediction, inverse theory and statistics to address the problem of estimating a high-dimensional covariance matrix from a small number of samples. This problem is fundamental in statistics, machine learning/artificial intelligence, and in modern Earth science. We create several new adaptive methods for high-dimensional covariance estimation, but one method, which we call NICE (Noise-Informed Covariance Estimation), stands out because it has three important properties: (i) NICE is conceptually simple and computationally efficient; (ii) NICE guarantees symmetric positive semi-definite covariance estimates; and (iii) NICE is largely tuning-free. We illustrate the use of NICE on a large set of Earth-science-inspired numerical examples, including cycling data assimilation, geophysical inversion of field data, and training of feed-forward neural networks with time-averaged data from a chaotic dynamical system. Our theory, heuristics and numerical tests suggest that NICE may indeed be a viable option for high-dimensional covariance estimation in many Earth science problems.