Towards Efficient Training of Graph Neural Networks: A Multiscale Approach

Graph Neural Networks (GNNs) have emerged as a powerful tool for learning and inferring from graph-structured data, and are widely used in a variety of applications, often considering large amounts of data and large graphs. However, training on such data requires large memory and extensive computations. In this paper, we introduce a novel framework for efficient multiscale training of GNNs, designed to integrate information across multiscale representations of a graph. Our approach leverages a hierarchical graph representation, taking advantage of coarse graph scales in the training process, where each coarse scale graph has fewer nodes and edges. Based on this approach, we propose a suite of GNN training methods: such as coarse-to-fine, sub-to-full, and multiscale gradient computation. We demonstrate the effectiveness of our methods on various datasets and learning tasks.

Probabilistic Forecasting for Dynamical Systems with Missing or Imperfect Data

The modeling of dynamical systems is essential in many fields, but applying machine learning techniques is often challenging due to incomplete or noisy data. This study introduces a variant of stochastic interpolation (SI) for probabilistic forecasting, estimating future states as distributions rather than single-point predictions. We explore its mathematical foundations and demonstrate its effectiveness on various dynamical systems, including the challenging WeatherBench dataset.

Iterative Flow Matching -- Path Correction and Gradual Refinement for Enhanced Generative Modeling

Generative models for image generation are now commonly used for a wide variety of applications, ranging from guided image generation for entertainment to solving inverse problems. Nonetheless, training a generator is a non-trivial feat that requires fine-tuning and can lead to so-called hallucinations, that is, the generation of images that are unrealistic. In this work, we explore image generation using flow matching. We explain and demonstrate why flow matching can generate hallucinations, and propose an iterative process to improve the generation process. Our iterative process can be integrated into virtually $\textit{any}$ generative modeling technique, thereby enhancing the performance and robustness of image synthesis systems.

Inversion of Magnetic Data using Learned Dictionaries and Scale Space

Magnetic data inversion is an important tool in geophysics, used to infer subsurface magnetic susceptibility distributions from surface magnetic field measurements. This inverse problem is inherently ill-posed, characterized by non-unique solutions, depth ambiguity, and sensitivity to noise. Traditional inversion approaches rely on predefined regularization techniques to stabilize solutions, limiting their adaptability to complex or diverse geological scenarios. In this study, we propose an approach that integrates variable dictionary learning and scale-space methods to address these challenges. Our method employs learned dictionaries, allowing for adaptive representation of complex subsurface features that are difficult to capture with predefined bases. Additionally, we extend classical variational inversion by incorporating multi-scale representations through a scale-space framework, enabling the progressive introduction of structural detail while mitigating overfitting. We implement both fixed and dynamic dictionary learning techniques, with the latter introducing iteration-dependent dictionaries for enhanced flexibility. Using a synthetic dataset to simulate geological scenarios, we demonstrate significant improvements in reconstruction accuracy and robustness compared to conventional variational and dictionary-based methods. Our results highlight the potential of learned dictionaries, especially when coupled with scale-space dynamics, to improve model recovery and noise handling. These findings underscore the promise of our data-driven approach for advance magnetic data inversion and its applications in geophysical exploration, environmental assessment, and mineral prospecting.

DAWN-SI: Data-Aware and Noise-Informed Stochastic Interpolation for Solving Inverse Problems

Inverse problems, which involve estimating parameters from incomplete or noisy observations, arise in various fields such as medical imaging, geophysics, and signal processing. These problems are often ill-posed, requiring regularization techniques to stabilize the solution. In this work, we employ $\textit{Stochastic Interpolation}$ (SI), a generative framework that integrates both deterministic and stochastic processes to map a simple reference distribution, such as a Gaussian, to the target distribution. Our method $\textbf{DAWN-SI}$: $\textbf{D}$ata-$\textbf{AW}$are and $\textbf{N}$oise-informed $\textbf{S}$tochastic $\textbf{I}$nterpolation incorporates data and noise embedding, allowing the model to access representations about the measured data explicitly and also account for noise in the observations, making it particularly robust in scenarios where data is noisy or incomplete. By learning a time-dependent velocity field, SI not only provides accurate solutions but also enables uncertainty quantification by generating multiple plausible outcomes. Unlike pre-trained diffusion models, which may struggle in highly ill-posed settings, our approach is trained specifically for each inverse problem and adapts to varying noise levels. We validate the effectiveness and robustness of our method through extensive numerical experiments on tasks such as image deblurring and tomography.

Learning Regularization for Graph Inverse Problems

In recent years, Graph Neural Networks (GNNs) have been utilized for various applications ranging from drug discovery to network design and social networks. In many applications, it is impossible to observe some properties of the graph directly; instead, noisy and indirect measurements of these properties are available. These scenarios are coined as Graph Inverse Problems (GRIP). In this work, we introduce a framework leveraging GNNs to solve GRIPs. The framework is based on a combination of likelihood and prior terms, which are used to find a solution that fits the data while adhering to learned prior information. Specifically, we propose to combine recent deep learning techniques that were developed for inverse problems, together with GNN architectures, to formulate and solve GRIP. We study our approach on a number of representative problems that demonstrate the effectiveness of the framework.

Fully invertible hyperbolic neural networks for segmenting large-scale surface and sub-surface data

The large spatial/temporal/frequency scale of geoscience and remote-sensing datasets causes memory issues when using convolutional neural networks for (sub-) surface data segmentation. Recently developed fully reversible or fully invertible networks can mostly avoid memory limitations by recomputing the states during the backward pass through the network. This results in a low and fixed memory requirement for storing network states, as opposed to the typical linear memory growth with network depth. This work focuses on a fully invertible network based on the telegraph equation. While reversibility saves the major amount of memory used in deep networks by the data, the convolutional kernels can take up most memory if fully invertible networks contain multiple invertible pooling/coarsening layers. We address the explosion of the number of convolutional kernels by combining fully invertible networks with layers that contain the convolutional kernels in a compressed form directly. A second challenge is that invertible networks output a tensor the same size as its input. This property prevents the straightforward application of invertible networks to applications that map between different input-output dimensions, need to map to outputs with more channels than present in the input data, or desire outputs that decrease/increase the resolution compared to the input data. However, we show that by employing invertible networks in a non-standard fashion, we can still use them for these tasks. Examples in hyperspectral land-use classification, airborne geophysical surveying, and seismic imaging illustrate that we can input large data volumes in one chunk and do not need to work on small patches, use dimensionality reduction, or employ methods that classify a patch to a single central pixel.

Advection Augmented Convolutional Neural Networks

Many problems in physical sciences are characterized by the prediction of space-time sequences. Such problems range from weather prediction to the analysis of disease propagation and video prediction. Modern techniques for the solution of these problems typically combine Convolution Neural Networks (CNN) architecture with a time prediction mechanism. However, oftentimes, such approaches underperform in the long-range propagation of information and lack explainability. In this work, we introduce a physically inspired architecture for the solution of such problems. Namely, we propose to augment CNNs with advection by designing a novel semi-Lagrangian push operator. We show that the proposed operator allows for the non-local transformation of information compared with standard convolutional kernels. We then complement it with Reaction and Diffusion neural components to form a network that mimics the Reaction-Advection-Diffusion equation, in high dimensions. We demonstrate the effectiveness of our network on a number of spatio-temporal datasets that show their merit.

Deep Optimal Experimental Design for Parameter Estimation Problems

Optimal experimental design is a well studied field in applied science and engineering. Techniques for estimating such a design are commonly used within the framework of parameter estimation. Nonetheless, in recent years parameter estimation techniques are changing rapidly with the introduction of deep learning techniques to replace traditional estimation methods. This in turn requires the adaptation of optimal experimental design that is associated with these new techniques. In this paper we investigate a new experimental design methodology that uses deep learning. We show that the training of a network as a Likelihood Free Estimator can be used to significantly simplify the design process and circumvent the need for the computationally expensive bi-level optimization problem that is inherent in optimal experimental design for non-linear systems. Furthermore, deep design improves the quality of the recovery process for parameter estimation problems. As proof of concept we apply our methodology to two different systems of Ordinary Differential Equations.

Graph Neural Reaction Diffusion Models

The integration of Graph Neural Networks (GNNs) and Neural Ordinary and Partial Differential Equations has been extensively studied in recent years. GNN architectures powered by neural differential equations allow us to reason about their behavior, and develop GNNs with desired properties such as controlled smoothing or energy conservation. In this paper we take inspiration from Turing instabilities in a Reaction Diffusion (RD) system of partial differential equations, and propose a novel family of GNNs based on neural RD systems. We \textcolor{black}{demonstrate} that our RDGNN is powerful for the modeling of various data types, from homophilic, to heterophilic, and spatio-temporal datasets. We discuss the theoretical properties of our RDGNN, its implementation, and show that it improves or offers competitive performance to state-of-the-art methods.