Anna Krause

Predicting the evolution of spatiotemporal physical systems from sparse and scattered observational data poses a significant challenge in various scientific domains. Traditional methods rely on dense grid-structured data, limiting their applicability in scenarios with sparse observations. To address this challenge, we introduce GrINd (Grid Interpolation Network for Scattered Observations), a novel network architecture that leverages the high-performance of grid-based models by mapping scattered observations onto a high-resolution grid using a Fourier Interpolation Layer. In the high-resolution space, a NeuralPDE-class model predicts the system's state at future timepoints using differentiable ODE solvers and fully convolutional neural networks parametrizing the system's dynamics. We empirically evaluate GrINd on the DynaBench benchmark dataset, comprising six different physical systems observed at scattered locations, demonstrating its state-of-the-art performance compared to existing models. GrINd offers a promising approach for forecasting physical systems from sparse, scattered observational data, extending the applicability of deep learning methods to real-world scenarios with limited data availability.

Previous work on learning physical systems from data has focused on high-resolution grid-structured measurements. However, real-world knowledge of such systems (e.g. weather data) relies on sparsely scattered measuring stations. In this paper, we introduce a novel simulated benchmark dataset, DynaBench, for learning dynamical systems directly from sparsely scattered data without prior knowledge of the equations. The dataset focuses on predicting the evolution of a dynamical system from low-resolution, unstructured measurements. We simulate six different partial differential equations covering a variety of physical systems commonly used in the literature and evaluate several machine learning models, including traditional graph neural networks and point cloud processing models, with the task of predicting the evolution of the system. The proposed benchmark dataset is expected to advance the state of art as an out-of-the-box easy-to-use tool for evaluating models in a setting where only unstructured low-resolution observations are available. The benchmark is available at https://anonymous.4open.science/r/code-2022-dynabench/.

Modeling data obtained from dynamical systems has gained attention in recent years as a challenging task for machine learning models. Previous approaches assume the measurements to be distributed on a grid. However, for real-world applications like weather prediction, the observations are taken from arbitrary locations within the spatial domain. In this paper, we propose TaylorPDENet - a novel machine learning method that is designed to overcome this challenge. Our algorithm uses the multidimensional Taylor expansion of a dynamical system at each observation point to estimate the spatial derivatives to perform predictions. TaylorPDENet is able to accomplish two objectives simultaneously: accurately forecast the evolution of a complex dynamical system and explicitly reconstruct the underlying differential equation describing the system. We evaluate our model on a variety of advection-diffusion equations with different parameters and show that it performs similarly to equivalent approaches on grid-structured data while being able to process unstructured data as well.

Climate models are the tool of choice for scientists researching climate change. Like all models they suffer from errors, particularly systematic and location-specific representation errors. One way to reduce these errors is model output statistics (MOS) where the model output is fitted to observational data with machine learning. In this work, we assess the use of convolutional Deep Learning climate MOS approaches and present the ConvMOS architecture which is specifically designed based on the observation that there are systematic and location-specific errors in the precipitation estimates of climate models. We apply ConvMOS models to the simulated precipitation of the regional climate model REMO, showing that a combination of per-location model parameters for reducing location-specific errors and global model parameters for reducing systematic errors is indeed beneficial for MOS performance. We find that ConvMOS models can reduce errors considerably and perform significantly better than three commonly used MOS approaches and plain ResNet and U-Net models in most cases. Our results show that non-linear MOS models underestimate the number of extreme precipitation events, which we alleviate by training models specialized towards extreme precipitation events with the imbalanced regression method DenseLoss. While we consider climate MOS, we argue that aspects of ConvMOS may also be beneficial in other domains with geospatial data, such as air pollution modeling or weather forecasts.