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Nutrient load simulators are large, deterministic, models that simulate the hydrodynamics and biogeochemical processes in aquatic ecosystems. They are central tools for planning cost efficient actions to fight eutrophication since they allow scenario predictions on impacts of nutrient load reductions to, e.g., harmful algal biomass growth. Due to being computationally heavy, the uncertainties related to these predictions are typically not rigorously assessed though. In this work, we developed a novel Bayesian computational approach for estimating the uncertainties in predictions of the Finnish coastal nutrient load model FICOS. First, we constructed a likelihood function for the multivariate spatiotemporal outputs of the FICOS model. Then, we used Bayes optimization to locate the posterior mode for the model parameters conditional on long term monitoring data. After that, we constructed a space filling design for FICOS model runs around the posterior mode and used it to train a Gaussian process emulator for the (log) posterior density of the model parameters. We then integrated over this (approximate) parameter posterior to produce probabilistic predictions for algal biomass and chlorophyll a concentration under alternative nutrient load reduction scenarios. Our computational algorithm allowed for fast posterior inference and the Gaussian process emulator had good predictive accuracy within the highest posterior probability mass region. The posterior predictive scenarios showed that the probability to reach the EUs Water Framework Directive objectives in the Finnish Archipelago Sea is generally low even under large load reductions.

Dense linear layers are the dominant computational bottleneck in large neural networks, presenting a critical need for more efficient alternatives. Previous efforts focused on a small number of hand-crafted structured matrices and neglected to investigate whether these structures can surpass dense layers in terms of compute-optimal scaling laws when both the model size and training examples are optimally allocated. In this work, we present a unifying framework that enables searching among all linear operators expressible via an Einstein summation. This framework encompasses many previously proposed structures, such as low-rank, Kronecker, Tensor-Train, Block Tensor-Train (BTT), and Monarch, along with many novel structures. To analyze the framework, we develop a taxonomy of all such operators based on their computational and algebraic properties and show that differences in the compute-optimal scaling laws are mostly governed by a small number of variables that we introduce. Namely, a small $\omega$ (which measures parameter sharing) and large $\psi$ (which measures the rank) reliably led to better scaling laws. Guided by the insight that full-rank structures that maximize parameters per unit of compute perform the best, we propose BTT-MoE, a novel Mixture-of-Experts (MoE) architecture obtained by sparsifying computation in the BTT structure. In contrast to the standard sparse MoE for each entire feed-forward network, BTT-MoE learns an MoE in every single linear layer of the model, including the projection matrices in the attention blocks. We find BTT-MoE provides a substantial compute-efficiency gain over dense layers and standard MoE.

This paper explores alternative formulations of the Kolmogorov Superposition Theorem (KST) as a foundation for neural network design. The original KST formulation, while mathematically elegant, presents practical challenges due to its limited insight into the structure of inner and outer functions and the large number of unknown variables it introduces. Kolmogorov-Arnold Networks (KANs) leverage KST for function approximation, but they have faced scrutiny due to mixed results compared to traditional multilayer perceptrons (MLPs) and practical limitations imposed by the original KST formulation. To address these issues, we introduce ActNet, a scalable deep learning model that builds on the KST and overcomes many of the drawbacks of Kolmogorov's original formulation. We evaluate ActNet in the context of Physics-Informed Neural Networks (PINNs), a framework well-suited for leveraging KST's strengths in low-dimensional function approximation, particularly for simulating partial differential equations (PDEs). In this challenging setting, where models must learn latent functions without direct measurements, ActNet consistently outperforms KANs across multiple benchmarks and is competitive against the current best MLP-based approaches. These results present ActNet as a promising new direction for KST-based deep learning applications, particularly in scientific computing and PDE simulation tasks.

To effectively study complex causal systems, it is often useful to construct representations that simplify parts of the system by discarding irrelevant details while preserving key features. The Information Bottleneck (IB) method is a widely used approach in representation learning that compresses random variables while retaining information about a target variable. Traditional methods like IB are purely statistical and ignore underlying causal structures, making them ill-suited for causal tasks. We propose the Causal Information Bottleneck (CIB), a causal extension of the IB, which compresses a set of chosen variables while maintaining causal control over a target variable. This method produces representations which are causally interpretable, and which can be used when reasoning about interventions. We present experimental results demonstrating that the learned representations accurately capture causality as intended.

The effect of relative entropy asymmetry is analyzed in the context of empirical risk minimization (ERM) with relative entropy regularization (ERM-RER). Two regularizations are considered: $(a)$ the relative entropy of the measure to be optimized with respect to a reference measure (Type-I ERM-RER); or $(b)$ the relative entropy of the reference measure with respect to the measure to be optimized (Type-II ERM-RER). The main result is the characterization of the solution to the Type-II ERM-RER problem and its key properties. By comparing the well-understood Type-I ERM-RER with Type-II ERM-RER, the effects of entropy asymmetry are highlighted. The analysis shows that in both cases, regularization by relative entropy forces the solution's support to collapse into the support of the reference measure, introducing a strong inductive bias that can overshadow the evidence provided by the training data. Finally, it is shown that Type-II regularization is equivalent to Type-I regularization with an appropriate transformation of the empirical risk function.

Common practice in modern machine learning involves fitting a large number of parameters relative to the number of observations. These overparameterized models can exhibit surprising generalization behavior, e.g., ``double descent'' in the prediction error curve when plotted against the raw number of model parameters, or another simplistic notion of complexity. In this paper, we revisit model complexity from first principles, by first reinterpreting and then extending the classical statistical concept of (effective) degrees of freedom. Whereas the classical definition is connected to fixed-X prediction error (in which prediction error is defined by averaging over the same, nonrandom covariate points as those used during training), our extension of degrees of freedom is connected to random-X prediction error (in which prediction error is averaged over a new, random sample from the covariate distribution). The random-X setting more naturally embodies modern machine learning problems, where highly complex models, even those complex enough to interpolate the training data, can still lead to desirable generalization performance under appropriate conditions. We demonstrate the utility of our proposed complexity measures through a mix of conceptual arguments, theory, and experiments, and illustrate how they can be used to interpret and compare arbitrary prediction models.

The execution of graph algorithms using neural networks has recently attracted significant interest due to promising empirical progress. This motivates further understanding of how neural networks can replicate reasoning steps with relational data. In this work, we study the ability of transformer networks to simulate algorithms on graphs from a theoretical perspective. The architecture we use is a looped transformer with extra attention heads that interact with the graph. We prove by construction that this architecture can simulate individual algorithms such as Dijkstra's shortest path, Breadth- and Depth-First Search, and Kosaraju's strongly connected components, as well as multiple algorithms simultaneously. The number of parameters in the networks does not increase with the input graph size, which implies that the networks can simulate the above algorithms for any graph. Despite this property, we show a limit to simulation in our solution due to finite precision. Finally, we show a Turing Completeness result with constant width when the extra attention heads are utilized.

In Influence Maximization (IM), the objective is to -- given a budget -- select the optimal set of entities in a network to target with a treatment so as to maximize the total effect. For instance, in marketing, the objective is to target the set of customers that maximizes the total response rate, resulting from both direct treatment effects on targeted customers and indirect, spillover, effects that follow from targeting these customers. Recently, new methods to estimate treatment effects in the presence of network interference have been proposed. However, the issue of how to leverage these models to make better treatment allocation decisions has been largely overlooked. Traditionally, in Uplift Modeling (UM), entities are ranked according to estimated treatment effect, and the top entities are allocated treatment. Since, in a network context, entities influence each other, the UM ranking approach will be suboptimal. The problem of finding the optimal treatment allocation in a network setting is combinatorial and generally has to be solved heuristically. To fill the gap between IM and UM, we propose OTAPI: Optimizing Treatment Allocation in the Presence of Interference to find solutions to the IM problem using treatment effect estimates. OTAPI consists of two steps. First, a causal estimator is trained to predict treatment effects in a network setting. Second, this estimator is leveraged to identify an optimal treatment allocation by integrating it into classic IM algorithms. We demonstrate that this novel method outperforms classic IM and UM approaches on both synthetic and semi-synthetic datasets.

Deep neural networks have revolutionized many machine learning tasks in power systems, ranging from pattern recognition to signal processing. The data in these tasks is typically represented in Euclidean domains. Nevertheless, there is an increasing number of applications in power systems, where data are collected from non-Euclidean domains and represented as the graph-structured data with high dimensional features and interdependency among nodes. The complexity of graph-structured data has brought significant challenges to the existing deep neural networks defined in Euclidean domains. Recently, many studies on extending deep neural networks for graph-structured data in power systems have emerged. In this paper, a comprehensive overview of graph neural networks (GNNs) in power systems is proposed. Specifically, several classical paradigms of GNNs structures (e.g., graph convolutional networks, graph recurrent neural networks, graph attention networks, graph generative networks, spatial-temporal graph convolutional networks, and hybrid forms of GNNs) are summarized, and key applications in power systems such as fault diagnosis, power prediction, power flow calculation, and data generation are reviewed in detail. Furthermore, main issues and some research trends about the applications of GNNs in power systems are discussed.

We introduce a multi-task setup of identifying and classifying entities, relations, and coreference clusters in scientific articles. We create SciERC, a dataset that includes annotations for all three tasks and develop a unified framework called Scientific Information Extractor (SciIE) for with shared span representations. The multi-task setup reduces cascading errors between tasks and leverages cross-sentence relations through coreference links. Experiments show that our multi-task model outperforms previous models in scientific information extraction without using any domain-specific features. We further show that the framework supports construction of a scientific knowledge graph, which we use to analyze information in scientific literature.