Statistical downscaling is a technique used in climate modeling to increase the resolution of climate simulations. High-resolution climate information is essential for various high-impact applications, including natural hazard risk assessment. However, simulating climate at high resolution is intractable. Thus, climate simulations are often conducted at a coarse scale and then downscaled to the desired resolution. Existing downscaling techniques are either simulation-based methods with high computational costs, or statistical approaches with limitations in accuracy or application specificity. We introduce Generative Bias Correction and Super-Resolution (GenBCSR), a two-stage probabilistic framework for statistical downscaling that overcomes the limitations of previous methods. GenBCSR employs two transformations to match high-dimensional distributions at different resolutions: (i) the first stage, bias correction, aligns the distributions at coarse scale, (ii) the second stage, statistical super-resolution, lifts the corrected coarse distribution by introducing fine-grained details. Each stage is instantiated by a state-of-the-art generative model, resulting in an efficient and effective computational pipeline for the well-studied distribution matching problem. By framing the downscaling problem as distribution matching, GenBCSR relaxes the constraints of supervised learning, which requires samples to be aligned. Despite not requiring such correspondence, we show that GenBCSR surpasses standard approaches in predictive accuracy of critical impact variables, particularly in predicting the tails (99% percentile) of composite indexes composed of interacting variables, achieving up to 4-5 folds of error reduction.

Regional high-resolution climate projections are crucial for many applications, such as agriculture, hydrology, and natural hazard risk assessment. Dynamical downscaling, the state-of-the-art method to produce localized future climate information, involves running a regional climate model (RCM) driven by an Earth System Model (ESM), but it is too computationally expensive to apply to large climate projection ensembles. We propose a novel approach combining dynamical downscaling with generative artificial intelligence to reduce the cost and improve the uncertainty estimates of downscaled climate projections. In our framework, an RCM dynamically downscales ESM output to an intermediate resolution, followed by a generative diffusion model that further refines the resolution to the target scale. This approach leverages the generalizability of physics-based models and the sampling efficiency of diffusion models, enabling the downscaling of large multi-model ensembles. We evaluate our method against dynamically-downscaled climate projections from the CMIP6 ensemble. Our results demonstrate its ability to provide more accurate uncertainty bounds on future regional climate than alternatives such as dynamical downscaling of smaller ensembles, or traditional empirical statistical downscaling methods. We also show that dynamical-generative downscaling results in significantly lower errors than bias correction and spatial disaggregation (BCSD), and captures more accurately the spectra and multivariate correlations of meteorological fields. These characteristics make the dynamical-generative framework a flexible, accurate, and efficient way to downscale large ensembles of climate projections, currently out of reach for pure dynamical downscaling.

We present a generative AI algorithm for addressing the challenging task of fast, accurate and robust statistical computation of three-dimensional turbulent fluid flows. Our algorithm, termed as GenCFD, is based on a conditional score-based diffusion model. Through extensive numerical experimentation with both incompressible and compressible fluid flows, we demonstrate that GenCFD provides very accurate approximation of statistical quantities of interest such as mean, variance, point pdfs, higher-order moments, while also generating high quality realistic samples of turbulent fluid flows and ensuring excellent spectral resolution. In contrast, ensembles of operator learning baselines which are trained to minimize mean (absolute) square errors regress to the mean flow. We present rigorous theoretical results uncovering the surprising mechanisms through which diffusion models accurately generate fluid flows. These mechanisms are illustrated with solvable toy models that exhibit the relevant features of turbulent fluid flows while being amenable to explicit analytical formulas.

Learning dynamics from dissipative chaotic systems is notoriously difficult due to their inherent instability, as formalized by their positive Lyapunov exponents, which exponentially amplify errors in the learned dynamics. However, many of these systems exhibit ergodicity and an attractor: a compact and highly complex manifold, to which trajectories converge in finite-time, that supports an invariant measure, i.e., a probability distribution that is invariant under the action of the dynamics, which dictates the long-term statistical behavior of the system. In this work, we leverage this structure to propose a new framework that targets learning the invariant measure as well as the dynamics, in contrast with typical methods that only target the misfit between trajectories, which often leads to divergence as the trajectories' length increases. We use our framework to propose a tractable and sample efficient objective that can be used with any existing learning objectives. Our Dynamics Stable Learning by Invariant Measures (DySLIM) objective enables model training that achieves better point-wise tracking and long-term statistical accuracy relative to other learning objectives. By targeting the distribution with a scalable regularization term, we hope that this approach can be extended to more complex systems exhibiting slowly-variant distributions, such as weather and climate models.

We introduce a two-stage probabilistic framework for statistical downscaling using unpaired data. Statistical downscaling seeks a probabilistic map to transform low-resolution data from a biased coarse-grained numerical scheme to high-resolution data that is consistent with a high-fidelity scheme. Our framework tackles the problem by composing two transformations: (i) a debiasing step via an optimal transport map, and (ii) an upsampling step achieved by a probabilistic diffusion model with a posteriori conditional sampling. This approach characterizes a conditional distribution without needing paired data, and faithfully recovers relevant physical statistics from biased samples. We demonstrate the utility of the proposed approach on one- and two-dimensional fluid flow problems, which are representative of the core difficulties present in numerical simulations of weather and climate. Our method produces realistic high-resolution outputs from low-resolution inputs, by upsampling resolutions of 8x and 16x. Moreover, our procedure correctly matches the statistics of physical quantities, even when the low-frequency content of the inputs and outputs do not match, a crucial but difficult-to-satisfy assumption needed by current state-of-the-art alternatives. Code for this work is available at: https://github.com/google-research/swirl-dynamics/tree/main/swirl_dynamics/projects/probabilistic_diffusion.

Despite their ubiquity throughout science and engineering, only a handful of partial differential equations (PDEs) have analytical, or closed-form solutions. This motivates a vast amount of classical work on numerical simulation of PDEs and more recently, a whirlwind of research into data-driven techniques leveraging machine learning (ML). A recent line of work indicates that a hybrid of classical numerical techniques and machine learning can offer significant improvements over either approach alone. In this work, we show that the choice of the numerical scheme is crucial when incorporating physics-based priors. We build upon Fourier-based spectral methods, which are known to be more efficient than other numerical schemes for simulating PDEs with smooth and periodic solutions. Specifically, we develop ML-augmented spectral solvers for three common PDEs of fluid dynamics. Our models are more accurate (2-4x) than standard spectral solvers at the same resolution but have longer overall runtimes (~2x), due to the additional runtime cost of the neural network component. We also demonstrate a handful of key design principles for combining machine learning and numerical methods for solving PDEs.

Diffusion models are a class of probabilistic generative models that have been widely used as a prior for image processing tasks like text conditional generation and inpainting. We demonstrate that these models can be adapted to make predictions and provide uncertainty quantification for chaotic dynamical systems. In these applications, diffusion models can implicitly represent knowledge about outliers and extreme events; however, querying that knowledge through conditional sampling or measuring probabilities is surprisingly difficult. Existing methods for conditional sampling at inference time seek mainly to enforce the constraints, which is insufficient to match the statistics of the distribution or compute the probability of the chosen events. To achieve these ends, optimally one would use the conditional score function, but its computation is typically intractable. In this work, we develop a probabilistic approximation scheme for the conditional score function which provably converges to the true distribution as the noise level decreases. With this scheme we are able to sample conditionally on nonlinear userdefined events at inference time, and matches data statistics even when sampling from the tails of the distribution.

We introduce a data-driven learning framework that assimilates two powerful ideas: ideal large eddy simulation (LES) from turbulence closure modeling and neural stochastic differential equations (SDE) for stochastic modeling. The ideal LES models the LES flow by treating each full-order trajectory as a random realization of the underlying dynamics, as such, the effect of small-scales is marginalized to obtain the deterministic evolution of the LES state. However, ideal LES is analytically intractable. In our work, we use a latent neural SDE to model the evolution of the stochastic process and an encoder-decoder pair for transforming between the latent space and the desired ideal flow field. This stands in sharp contrast to other types of neural parameterization of closure models where each trajectory is treated as a deterministic realization of the dynamics. We show the effectiveness of our approach (niLES - neural ideal LES) on a challenging chaotic dynamical system: Kolmogorov flow at a Reynolds number of 20,000. Compared to competing methods, our method can handle non-uniform geometries using unstructured meshes seamlessly. In particular, niLES leads to trajectories with more accurate statistics and enhances stability, particularly for long-horizon rollouts.

We present a data-driven, space-time continuous framework to learn surrogate models for complex physical systems described by advection-dominated partial differential equations. Those systems have slow-decaying Kolmogorov n-width that hinders standard methods, including reduced order modeling, from producing high-fidelity simulations at low cost. In this work, we construct hypernetworkbased latent dynamical models directly on the parameter space of a compact representation network. We leverage the expressive power of the network and a specially designed consistency-inducing regularization to obtain latent trajectories that are both low-dimensional and smooth. These properties render our surrogate models highly efficient at inference time. We show the efficacy of our framework by learning models that generate accurate multi-step rollout predictions at much faster inference speed compared to competitors, for several challenging examples. High-fidelity numerical simulation of physical systems modeled by time-dependent partial differential equations (PDEs) has been at the center of many technological advances in the last century. However, for engineering applications such as design, control, optimization, data assimilation, and uncertainty quantification, which require repeated model evaluation over a potentially large number of parameters, or initial conditions, high-fidelity simulations remain prohibitively expensive, even with state-of-art PDE solvers. The necessity of reducing the overall cost for such downstream applications has led to the development of surrogate models, which captures the core behavior of the target system but at a fraction of the cost. One of the most popular frameworks in the last decades (Aubry et al., 1988) to build such surrogates has been reduced order models (ROMs). In a nutshell, they construct lower-dimensional representations and their corresponding reduced dynamics that capture the system's behavior of interest. The computational gains then stem from the evolution of a lower-dimensional latent representation (see Benner et al. (2015) for a comprehensive review).

We propose an end-to-end deep learning framework that comprehensively solves the inverse wave scattering problem across all length scales. Our framework consists of the newly introduced wide-band butterfly network coupled with a simple training procedure that dynamically injects noise during training. While our trained network provides competitive results in classical imaging regimes, most notably it also succeeds in the super-resolution regime where other comparable methods fail. This encompasses both (i) reconstruction of scatterers with sub-wavelength geometric features, and (ii) accurate imaging when two or more scatterers are separated by less than the classical diffraction limit. We demonstrate these properties are retained even in the presence of strong noise and extend to scatterers not previously seen in the training set. In addition, our network is straightforward to train requiring no restarts and has an online runtime that is an order of magnitude faster than optimization-based algorithms. We perform experiments with a variety of wave scattering mediums and we demonstrate that our proposed framework outperforms both classical inversion and competing network architectures that specialize in oscillatory wave scattering data.