Conditioning diffusion and flow models have proven effective for super-resolving small-scale details in natural images. However, in physical sciences such as weather, super-resolving small-scale details poses significant challenges due to: (i) misalignment between input and output distributions (i.e., solutions to distinct partial differential equations (PDEs) follow different trajectories), (ii) multi-scale dynamics, deterministic dynamics at large scales vs. stochastic at small scales, and (iii) limited data, increasing the risk of overfitting. To address these challenges, we propose encoding the inputs to a latent base distribution that is closer to the target distribution, followed by flow matching to generate small-scale physics. The encoder captures the deterministic components, while flow matching adds stochastic small-scale details. To account for uncertainty in the deterministic part, we inject noise into the encoder's output using an adaptive noise scaling mechanism, which is dynamically adjusted based on maximum-likelihood estimates of the encoder's predictions. We conduct extensive experiments on both the realworld CWA weather dataset and the PDE-based Kolmogorov dataset, with the CWA task involving super-resolving the weather variables for the region of Taiwan from 25 km to 2 km scales. Our results show that the proposed stochastic flow matching (SFM) framework significantly outperforms existing methods such as conditional diffusion and flows. Resolving small-scale physics is crucial in many scientific applications (Wilby et al., 1998; Rampal et al., 2022; 2024). For instance, in the atmospheric sciences, accurately capturing small-scale dynamics is essential for local planning and disaster mitigation. The success of conditional diffusion models in super-resolving natural images and videos (Song et al., 2021; Batzolis et al., 2021; Hoogeboom et al., 2023) has recently been extended to super-resolving small-scale physics (Aich et al., 2024; Ling et al., 2024). However, this task faces significant challenges: (C1) Input and target data are often spatially misaligned due to differing PDE solutions operating at various resolutions, leading to divergent trajectories. Additionally, the input and target variables (channels) often represent different physical quantities, causing further misalignment. Few efforts have been made to directly address these challenges in generative learning. Prior work typically relies on residual learning approaches (Mardani et al., 2023; Zhao et al., 2021).

Recently, Zhang and Chen [25] have proposed the Diffusion Exponential Integrator Sampler (DEIS) for fast generation of samples from Diffusion Models. It leverages the semi-linear nature of the probability flow ordinary differential equation (ODE) in order to greatly reduce integration error and improve generation quality at low numbers of function evaluations (NFEs). Key to this approach is the score function reparameterisation, which reduces the integration error incurred from using a fixed score function estimate over each integration step. The original authors use the default parameterisation used by models trained for noise prediction - multiply the score by the standard deviation of the conditional forward noising distribution. We find that although the mean absolute value of this score parameterisation is close to constant for a large portion of the reverse sampling process, it changes rapidly at the end of sampling. As a simple fix, we propose to instead reparameterise the score (at inference) by dividing it by the average absolute value of previous score estimates at that time step collected from offline high NFE generations. We find that our score normalisation (DEIS-SN) consistently improves FID compared to vanilla DEIS, showing an improvement at 10 NFEs from 6.44 to 5.57 on CIFAR-10 and from 5.9 to 4.95 on LSUN-Church (64 64).

The field of image generation has made significant progress thanks to the introduction of Diffusion Models, which learn to progressively reverse a given image corruption. Recently, a few studies introduced alternative ways of corrupting images in Diffusion Models, with an emphasis on blurring. However, these studies are purely empirical and it remains unclear what is the optimal procedure for corrupting an image. In this work, we hypothesize that the optimal procedure minimizes the length of the path taken when corrupting an image towards a given final state. We propose the Fisher metric for the path length, measured in the space of probability distributions. We compute the shortest path according to this metric, and we show that it corresponds to a combination of image sharpening, rather than blurring, and noise deblurring. While the corruption was chosen arbitrarily in previous work, our Shortest Path Diffusion (SPD) determines uniquely the entire spatiotemporal structure of the corruption. We show that SPD improves on strong baselines without any hyperparameter tuning, and outperforms all previous Diffusion Models based on image blurring. Furthermore, any small deviation from the shortest path leads to worse performance, suggesting that SPD provides the optimal procedure to corrupt images. Our work sheds new light on observations made in recent works and provides a new approach to improve diffusion models on images and other types of data.

Deep networks have become increasingly of interest in dynamical system prediction, but generalization remains elusive. In this work, we consider the physical parameters of ODEs as factors of variation of the data generating process. By leveraging ideas from supervised disentanglement in VAEs, we aim to separate the ODE parameters from the dynamics in the latent space. Experiments show that supervised disentanglement allows VAEs to capture the variability in the dynamics and extrapolate better to ODE parameter spaces that were not present in the training data.

In this work, we evaluate adversarial robustness in the context of transfer learning from a source trained on CIFAR 100 to a target network trained on CIFAR 10. Specifically, we study the effects of using robust optimisation in the source and target networks. This allows us to identify transfer learning strategies under which adversarial defences are successfully retained, in addition to revealing potential vulnerabilities. We study the extent to which features learnt by a fast gradient sign method (FGSM) and its iterative alternative (PGD) can preserve their defence properties against black and white-box attacks under three different transfer learning strategies. We find that using PGD examples during training on the source task leads to more general robust features that are easier to transfer. Furthermore, under successful transfer, it achieves 5.2% more accuracy against white-box PGD attacks than suitable baselines. Overall, our empirical evaluations give insights on how well adversarial robustness under transfer learning can generalise.