Neural operators (NOs) have emerged as effective tools for modeling complex physical systems in scientific machine learning. In NOs, a central characteristic is to learn the governing physical laws directly from data. In contrast to other machine learning applications, partial knowledge is often known a priori about the physical system at hand whereby quantities such as mass, energy and momentum are exactly conserved. Currently, NOs have to learn these conservation laws from data and can only approximately satisfy them due to finite training data and random noise. In this work, we introduce conservation law-encoded neural operators (clawNOs), a suite of NOs that endow inference with automatic satisfaction of such conservation laws. ClawNOs are built with a divergence-free prediction of the solution field, with which the continuity equation is automatically guaranteed. As a consequence, clawNOs are compliant with the most fundamental and ubiquitous conservation laws essential for correct physical consistency. As demonstrations, we consider a wide variety of scientific applications ranging from constitutive modeling of material deformation, incompressible fluid dynamics, to atmospheric simulation. ClawNOs significantly outperform the state-of-the-art NOs in learning efficacy, especially in small-data regimes. Deep neural networks (DNN) have achieved tremendous progress in areas where the underlying laws are unknown. In computer vision applications convolutional neural networks have been very popular even though it is unclear how the lower-dimensional manifold of "valid" images is parameterized (He et al., 2016; Ren et al., 2015; Krizhevsky et al., 2012). That means, such a manifold has to be discovered in a purely data-driven way by feeding the network vast amounts of data.

General circulation models (GCMs) are the foundation of weather and climate prediction. GCMs are physics-based simulators which combine a numerical solver for large-scale dynamics with tuned representations for small-scale processes such as cloud formation. Recently, machine learning (ML) models trained on reanalysis data achieved comparable or better skill than GCMs for deterministic weather forecasting. However, these models have not demonstrated improved ensemble forecasts, or shown sufficient stability for long-term weather and climate simulations. Here we present the first GCM that combines a differentiable solver for atmospheric dynamics with ML components, and show that it can generate forecasts of deterministic weather, ensemble weather and climate on par with the best ML and physics-based methods. NeuralGCM is competitive with ML models for 1-10 day forecasts, and with the European Centre for Medium-Range Weather Forecasts ensemble prediction for 1-15 day forecasts. With prescribed sea surface temperature, NeuralGCM can accurately track climate metrics such as global mean temperature for multiple decades, and climate forecasts with 140 km resolution exhibit emergent phenomena such as realistic frequency and trajectories of tropical cyclones. For both weather and climate, our approach offers orders of magnitude computational savings over conventional GCMs. Our results show that end-to-end deep learning is compatible with tasks performed by conventional GCMs, and can enhance the large-scale physical simulations that are essential for understanding and predicting the Earth system.

Participants of the Berlin Summit on Earth Virtualization Engines (EVEs) discussed ideas and concepts to improve our ability to cope with climate change. EVEs aim to provide interactive and accessible climate simulations and data for a wide range of users. They combine high-resolution physics-based models with machine learning techniques to improve the fidelity, efficiency, and interpretability of climate projections. At their core, EVEs offer a federated data layer that enables simple and fast access to exabyte-sized climate data through simple interfaces. In this article, we summarize the technical challenges and opportunities for developing EVEs, and argue that they are essential for addressing the consequences of climate change. We are all witnessing the effects of climate change. Hotter summers, prolonged droughts, massive flooding, or ocean heat waves are examples of extreme weather and climate events that are growing in frequency and intensity. Many agree that addressing climate mitigation and adaptation is the biggest problem humanity faces today. A large group of scientists and practitioners from different climate-related domains, including some computer scientists, got together for a week in Berlin this July to discuss the concept of "Earth Virtualization Engines" (EVEs). The summit kicked off with the question: "If climate change is the most critical problem today, why are we not using the largest computers to help solve it?".