Machine Learning Approaches for Data-Driven Analysis and Forecasting of High-Dimensional Chaotic Dynamical Systems

We consider problems in the forecasting of large, complex, spatiotemporal chaotic systems and the possibility that machine learning might be a useful tool for significant improvement of such forecasts. Focusing on weather forecasting as perhaps the most important example of such systems, we note that physics-based weather models have substantial error due to various factors including imperfect modeling of subgrid-scale dynamics and incomplete knowledge of physical processes. In this thesis, we ask if machine learning can potentially correct for such knowledge deficits. First, we demonstrate the effectiveness of using machine learning for model- free prediction of spatiotemporally chaotic systems of arbitrarily large spatial extent and attractor dimension purely from observations of the system’s past evolution. We present a parallel scheme with an example implementation based on the reservoir computing paradigm and demonstrate the scalability of our scheme using the Kuramoto-Sivashinsky equation as an example of a spatiotemporally chaotic system. We then demonstrate the use of machine learning for inferring fundamental properties of dynamical systems, namely the Lyapunov exponents, purely from observed data. We obtain results of unprecedented fidelity with our novel technique, making it possible to find the Lyapunov exponents of large systems where previously known techniques have failed. Next, we propose a general method that combines a physics-informed knowledge-based model and a machine learning technique to build a hybrid forecasting scheme. We further extend our hybrid forecasting approach to the difficult case where only partial measurements of the state of the dynamical system are available. For this purpose, we propose a novel technique that combines machine learning with a data assimilation method called an Ensemble Transform Kalman Filter (ETKF)

Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere

Fourier Neural Operators (FNOs) have proven to be an efficient and effective method for resolution-independent operator learning in a broad variety of application areas across scientific machine learning. A key reason for their success is their ability to accurately model long-range dependencies in spatio-temporal data by learning global convolutions in a computationally efficient manner. To this end, FNOs rely on the discrete Fourier transform (DFT), however, DFTs cause visual and spectral artifacts as well as pronounced dissipation when learning operators in spherical coordinates since they incorrectly assume a flat geometry. To overcome this limitation, we generalize FNOs on the sphere, introducing Spherical FNOs (SFNOs) for learning operators on spherical geometries. We apply SFNOs to forecasting atmospheric dynamics, and demonstrate stable auto\-regressive rollouts for a year of simulated time (1,460 steps), while retaining physically plausible dynamics. The SFNO has important implications for machine learning-based simulation of climate dynamics that could eventually help accelerate our response to climate change

ClimSim: A large multi-scale dataset for hybrid physics-ML climate emulation

Modern climate projections lack adequate spatial and temporal resolution due to computational constraints. A consequence is inaccurate and imprecise predictions of critical processes such as storms. Hybrid methods that combine physics with machine learning (ML) have introduced a new generation of higher fidelity climate simulators that can sidestep Moore's Law by outsourcing compute-hungry, short, high-resolution simulations to ML emulators. However, this hybrid ML-physics simulation approach requires domain-specific treatment and has been inaccessible to ML experts because of lack of training data and relevant, easy-to-use workflows. We present ClimSim, the largest-ever dataset designed for hybrid ML-physics research. It comprises multi-scale climate simulations, developed by a consortium of climate scientists and ML researchers. It consists of 5.7 billion pairs of multivariate input and output vectors that isolate the influence of locally-nested, high-resolution, high-fidelity physics on a host climate simulator's macro-scale physical state.The dataset is global in coverage, spans multiple years at high sampling frequency, and is designed such that resulting emulators are compatible with downstream coupling into operational climate simulators. We implement a range of deterministic and stochastic regression baselines to highlight the ML challenges and their scoring. The data (https://huggingface.co/datasets/LEAP/ClimSim_high-res) and code (https://leap-stc.github.io/ClimSim) are released openly to support the development of hybrid ML-physics and high-fidelity climate simulations for the benefit of science and society