Tobias Bischoff

Katherine Deck, Tobias Bischoff

Generated images of score-based models can suffer from errors in their spatial means, an effect, referred to as a color shift, which grows for larger images. This paper investigates a previously-introduced approach to mitigate color shifts in score-based diffusion models. We quantify the performance of a nonlinear bypass connection in the score network, designed to process the spatial mean of the input and to predict the mean of the score function. We show that this network architecture substantially improves the resulting quality of the generated images, and that this improvement is approximately independent of the size of the generated images. As a result, this modified architecture offers a simple solution for the color shift problem across image sizes. We additionally discuss the origin of color shifts in an idealized setting in order to motivate the approach.

Ludovico Theo Giorgini, Katherine Deck, Tobias Bischoff et al.

We introduce an approach for analyzing the responses of dynamical systems to external perturbations that combines score-based generative modeling with the Generalized Fluctuation-Dissipation Theorem (GFDT). The methodology enables accurate estimation of system responses, including those with non-Gaussian statistics. We numerically validate our approach using time-series data from three different stochastic partial differential equations of increasing complexity: an Ornstein-Uhlenbeck process with spatially correlated noise, a modified stochastic Allen-Cahn equation, and the 2D Navier-Stokes equations. We demonstrate the improved accuracy of the methodology over conventional methods and discuss its potential as a versatile tool for predicting the statistical behavior of complex dynamical systems.

Bryan Riel, Tobias Bischoff

Recent developments in generative modeling have utilized score-based methods coupled with stochastic differential equations to sample from complex probability distributions. However, these and other performant sampling methods generally require gradients of the target probability distribution, which can be unavailable or computationally prohibitive in many scientific and engineering applications. Here, we introduce ensembles within score-based sampling methods to develop gradient-free approximate sampling techniques that leverage the collective dynamics of particle ensembles to compute approximate reverse diffusion drifts. We introduce the underlying methodology, emphasizing its relationship with generative diffusion models and the previously introduced Föllmer sampler. We demonstrate the efficacy of the ensemble strategies through various examples, ranging from low- to medium-dimensionality sampling problems, including multi-modal and highly non-Gaussian probability distributions, and provide comparisons to traditional methods like the No-U-Turn Sampler. Additionally, we showcase these strategies in the context of a high-dimensional Bayesian inversion problem within the geophysical sciences. Our findings highlight the potential of ensemble strategies for modeling complex probability distributions in situations where gradients are unavailable.

Tobias Bischoff, Katherine Deck

We present a method to downscale idealized geophysical fluid simulations using generative models based on diffusion maps. By analyzing the Fourier spectra of images drawn from different data distributions, we show how one can chain together two independent conditional diffusion models for use in domain translation. The resulting transformation is a diffusion bridge between a low resolution and a high resolution dataset and allows for new sample generation of high-resolution images given specific low resolution features. The ability to generate new samples allows for the computation of any statistic of interest, without any additional calibration or training. Our unsupervised setup is also designed to downscale images without access to paired training data; this flexibility allows for the combination of multiple source and target domains without additional training. We demonstrate that the method enhances resolution and corrects context-dependent biases in geophysical fluid simulations, including in extreme events. We anticipate that the same method can be used to downscale the output of climate simulations, including temperature and precipitation fields, without needing to train a new model for each application and providing a significant computational cost savings.

Krzysztof Chalupka, Tobias Bischoff, Pietro Perona et al.

We show that the climate phenomena of El Nino and La Nina arise naturally as states of macro-variables when our recent causal feature learning framework (Chalupka 2015, Chalupka 2016) is applied to micro-level measures of zonal wind (ZW) and sea surface temperatures (SST) taken over the equatorial band of the Pacific Ocean. The method identifies these unusual climate states on the basis of the relation between ZW and SST patterns without any input about past occurrences of El Nino or La Nina. The simpler alternatives of (i) clustering the SST fields while disregarding their relationship with ZW patterns, or (ii) clustering the joint ZW-SST patterns, do not discover El Nino. We discuss the degree to which our method supports a causal interpretation and use a low-dimensional toy example to explain its success over other clustering approaches. Finally, we propose a new robust and scalable alternative to our original algorithm (Chalupka 2016), which circumvents the need for high-dimensional density learning.