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 Measure (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. Code to reproduce our experiments is available here: https://github.com/google-research/swirl-dynamics/tree/main/swirl_dynamics/projects/ergodic.

Automatic differentiation (autodiff) has revolutionized machine learning. Itallows to express complex computations by composing elementary ones in creativeways and removes the burden of computing their derivatives by hand. Morerecently, differentiation of optimization problem solutions has attractedwidespread attention with applications such as optimization layers, and inbi-level problems such as hyper-parameter optimization and meta-learning.However, so far, implicit differentiation remained difficult to use forpractitioners, as it often required case-by-case tedious mathematicalderivations and implementations. In this paper, we proposeautomatic implicit differentiation, an efficientand modular approach for implicit differentiation of optimization problems. Inour approach, the user defines directly in Python a function $F$ capturing theoptimality conditions of the problem to be differentiated. Once this is done, weleverage autodiff of $F$ and the implicit function theorem to automaticallydifferentiate the optimization problem. Our approach thus combines the benefitsof implicit differentiation and autodiff. It is efficient as it can be added ontop of any state-of-the-art solver and modular as the optimality conditionspecification is decoupled from the implicit differentiation mechanism. We showthat seemingly simple principles allow to recover many existing implicitdifferentiation methods and create new ones easily. We demonstrate the ease offormulating and solving bi-level optimization problems using our framework. Wealso showcase an application to the sensitivity analysis of molecular dynamics.

Variational data assimilation optimizes for an initial state of a dynamical system such that its evolution fits observational data. The physical model can subsequently be evolved into the future to make predictions. This principle is a cornerstone of large scale forecasting applications such as numerical weather prediction. As such, it is implemented in current operational systems of weather forecasting agencies across the globe. However, finding a good initial state poses a difficult optimization problem in part due to the non-invertible relationship between physical states and their corresponding observations. We learn a mapping from observational data to physical states and show how it can be used to improve optimizability. We employ this mapping in two ways: to better initialize the non-convex optimization problem, and to reformulate the objective function in better behaved physics space instead of observation space. Our experimental results for the Lorenz96 model and a two-dimensional turbulent fluid flow demonstrate that this procedure significantly improves forecast quality for chaotic systems.

BACKGROUND: Large image datasets acquired on automated microscopes typically have some fraction of low quality, out-of-focus images, despite the use of hardware autofocus systems. Identification of these images using automated image analysis with high accuracy is important for obtaining a clean, unbiased image dataset. Complicating this task is the fact that image focus quality is only well-defined in foreground regions of images, and as a result, most previous approaches only enable a computation of the relative difference in quality between two or more images, rather than an absolute measure of quality. RESULTS: We present a deep neural network model capable of predicting an absolute measure of image focus on a single image in isolation, without any user-specified parameters. The model operates at the image-patch level, and also outputs a measure of prediction certainty, enabling interpretable predictions. The model was trained on only 384 in-focus Hoechst (nuclei) stain images of U2OS cells, which were synthetically defocused to one of 11 absolute defocus levels during training. The trained model can generalize on previously unseen real Hoechst stain images, identifying the absolute image focus to within one defocus level (approximately 3 pixel blur diameter difference) with 95% accuracy. On a simpler binary in/out-of-focus classification task, the trained model outperforms previous approaches on both Hoechst and Phalloidin (actin) stain images (F-scores of 0.89 and 0.86, respectively over 0.84 and 0.83), despite only having been presented Hoechst stain images during training. Lastly, we observe qualitatively that the model generalizes to two additional stains, Hoechst and Tubulin, of an unseen cell type (Human MCF-7) acquired on a different instrument. CONCLUSIONS: Our deep neural network enables classification of out-of-focus microscope images with both higher accuracy and greater precision than previous approaches via interpretable patch-level focus and certainty predictions. The use of synthetically defocused images precludes the need for a manually annotated training dataset. The model also generalizes to different image and cell types. The framework for model training and image prediction is available as a free software library and the pre-trained model is available for immediate use in Fiji (ImageJ) and CellProfiler.