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A neural network has an activation bottleneck if one of its hidden layers has a bounded image. We show that networks with an activation bottleneck cannot forecast unbounded sequences such as straight lines, random walks, or any sequence with a trend: The difference between prediction and ground truth becomes arbitrary large, regardless of the training procedure. Widely-used neural network architectures such as LSTM and GRU suffer from this limitation. In our analysis, we characterize activation bottlenecks and explain why they prevent sigmoidal networks from learning unbounded sequences. We experimentally validate our findings and discuss modifications to network architectures which mitigate the effects of activation bottlenecks.

Understanding the functional architecture of complex systems is crucial to illuminate their inner workings and enable effective methods for their prediction and control. Recent advances have introduced tools to characterise emergent macroscopic levels; however, while these approaches are successful in identifying when emergence takes place, they are limited in the extent they can determine how it does. Here we address this limitation by developing a computational approach to emergence, which characterises macroscopic processes in terms of their computational capabilities. Concretely, we articulate a view on emergence based on how software works, which is rooted on a mathematical formalism that articulates how macroscopic processes can express self-contained informational, interventional, and computational properties. This framework establishes a hierarchy of nested self-contained processes that determines what computations take place at what level, which in turn delineates the functional architecture of a complex system. This approach is illustrated on paradigmatic models from the statistical physics and computational neuroscience literature, which are shown to exhibit macroscopic processes that are akin to software in human-engineered systems. Overall, this framework enables a deeper understanding of the multi-level structure of complex systems, revealing specific ways in which they can be efficiently simulated, predicted, and controlled., Comment: 33 pages, 13 figures

This paper employs physics-informed neural networks (PINNs) to solve Fisher's equation, a fundamental representation of a reaction-diffusion system with both simplicity and significance. The focus lies specifically in investigating Fisher's equation under conditions of large reaction rate coefficients, wherein solutions manifest as traveling waves, posing a challenge for numerical methods due to the occurring steepness of the wave front. To address optimization challenges associated with the standard PINN approach, a residual weighting scheme is introduced. This scheme is designed to enhance the tracking of propagating wave fronts by considering the reaction term in the reaction-diffusion equation. Furthermore, a specific network architecture is studied which is tailored for solutions in the form of traveling waves. Lastly, the capacity of PINNs to approximate an entire family of solutions is assessed by incorporating the reaction rate coefficient as an additional input to the network architecture. This modification enables the approximation of the solution across a broad and continuous range of reaction rate coefficients, thus solving a class of reaction-diffusion systems using a single PINN instance., Comment: 14 pages, 7 figures

Flamelet models are widely used in computational fluid dynamics to simulate thermochemical processes in turbulent combustion. These models typically employ memory-expensive lookup tables that are predetermined and represent the combustion process to be simulated. Artificial neural networks (ANNs) offer a deep learning approach that can store this tabular data using a small number of network weights, potentially reducing the memory demands of complex simulations by orders of magnitude. However, ANNs with standard training losses often struggle with underrepresented targets in multivariate regression tasks, e.g., when learning minor species mass fractions as part of lookup tables. This paper seeks to improve the accuracy of an ANN when learning multiple species mass fractions of a hydrogen (\ce{H2}) combustion lookup table. We assess a simple, yet effective loss weight adjustment that outperforms the standard mean-squared error optimization and enables accurate learning of all species mass fractions, even of minor species where the standard optimization completely fails. Furthermore, we find that the loss weight adjustment leads to more balanced gradients in the network training, which explains its effectiveness., Comment: 8 pages. Part of Scientific Computing 2023 Conference Proceedings (ISBN e-Book: 978-3-903318-20-5)

The turbulent jet ignition concept using prechambers is a promising solution to achieve stable combustion at lean conditions in large gas engines, leading to high efficiency at low emission levels. Due to the wide range of design and operating parameters for large gas engine prechambers, the preferred method for evaluating different designs is computational fluid dynamics (CFD), as testing in test bed measurement campaigns is time-consuming and expensive. However, the significant computational time required for detailed CFD simulations due to the complexity of solving the underlying physics also limits its applicability. In optimization settings similar to the present case, i.e., where the evaluation of the objective function(s) is computationally costly, Bayesian optimization has largely replaced classical design-of-experiment. Thus, the present study deals with the computationally efficient Bayesian optimization of large gas engine prechambers design using CFD simulation. Reynolds-averaged-Navier-Stokes simulations are used to determine the target values as a function of the selected prechamber design parameters. The results indicate that the chosen strategy is effective to find a prechamber design that achieves the desired target values., Comment: 9 pages. Part of Scientific Computing 2023 Conference Proceedings (ISBN e-Book: 978-3-903318-20-5)

The information-theoretic framework promises to explain the predictive power of neural networks. In particular, the information plane analysis, which measures mutual information (MI) between input and representation as well as representation and output, should give rich insights into the training process. This approach, however, was shown to strongly depend on the choice of estimator of the MI. The problem is amplified for deterministic networks if the MI between input and representation is infinite. Thus, the estimated values are defined by the different approaches for estimation, but do not adequately represent the training process from an information-theoretic perspective. In this work, we show that dropout with continuously distributed noise ensures that MI is finite. We demonstrate in a range of experiments that this enables a meaningful information plane analysis for a class of dropout neural networks that is widely used in practice., Comment: Published as a conference paper at ICLR2023

Rate-distortion theory-based outlier detection builds upon the rationale that a good data compression will encode outliers with unique symbols. Based on this rationale, we propose Cluster Purging, which is an extension of clustering-based outlier detection. This extension allows one to assess the representivity of clusterings, and to find data that are best represented by individual unique clusters. We propose two efficient algorithms for performing Cluster Purging, one being parameter-free, while the other algorithm has a parameter that controls representivity estimations, allowing it to be tuned in supervised setups. In an experimental evaluation, we show that Cluster Purging improves upon outliers detected from raw clusterings, and that Cluster Purging competes strongly against state-of-the-art alternatives.

We consider the problem of finding an input to a stochastic black box function such that the scalar output of the black box function is as close as possible to a target value in the sense of the expected squared error. While the optimization of stochastic black boxes is classic in (robust) Bayesian optimization, the current approaches based on Gaussian processes predominantly focus either on i) maximization/minimization rather than target value optimization or ii) on the expectation, but not the variance of the output, ignoring output variations due to stochasticity in uncontrollable environmental variables. In this work, we fill this gap and derive acquisition functions for common criteria such as the expected improvement, the probability of improvement, and the lower confidence bound, assuming that aleatoric effects are Gaussian with known variance. Our experiments illustrate that this setting is compatible with certain extensions of Gaussian processes, and show that the thus derived acquisition functions can outperform classical Bayesian optimization even if the latter assumptions are violated. An industrial use case in billet forging is presented., Comment: 24 pages; submitted to Computers and Industrial Engineering

Learning invariant representations that remain useful for a downstream task is still a key challenge in machine learning. We investigate a set of related information funnels and bottleneck problems that claim to learn invariant representations from the data. We also propose a new element to this family of information-theoretic objectives: The Conditional Privacy Funnel with Side Information, which we investigate in fully and semi-supervised settings. Given the generally intractable objectives, we derive tractable approximations using amortized variational inference parameterized by neural networks and study the intrinsic trade-offs of these objectives. We describe empirically the proposed approach and show that with a few labels it is possible to learn fair classifiers and generate useful representations approximately invariant to unwanted sources of variation. Furthermore, we provide insights about the applicability of these methods in real-world scenarios with ordinary tabular datasets when the data is scarce., Comment: 28 pages

In this paper, we frame homogeneous-feature multi-task learning (MTL) as a hierarchical representation learning problem, with one task-agnostic and multiple task-specific latent representations. Drawing inspiration from the information bottleneck principle and assuming an additive independent noise model between the task-agnostic and task-specific latent representations, we limit the information contained in each task-specific representation. It is shown that our resulting representations yield competitive performance for several MTL benchmarks. Furthermore, for certain setups, we show that the trained parameters of the additive noise model are closely related to the similarity of different tasks. This indicates that our approach yields a task-agnostic representation that is disentangled in the sense that its individual dimensions may be interpretable from a task-specific perspective., Comment: Accepted by the 2022 International Joint Conference on Neural Networks (IJCNN 2022)

Online social networks are a dominant medium in everyday life to stay in contact with friends and to share information. In Twitter, users can connect with other users by following them, who in turn can follow back. In recent years, researchers studied several properties of social networks and designed random graph models to describe them. Many of these approaches either focus on the generation of undirected graphs or on the creation of directed graphs without modeling the dependencies between reciprocal (i.e., two directed edges of opposite direction between two nodes) and directed edges. We propose an approach to generate directed social network graphs that creates reciprocal and directed edges and considers the correlation between the respective degree sequences. Our model relies on crawled directed graphs in Twitter, on which information w.r.t. a topic is exchanged or disseminated. While these graphs exhibit a high clustering coefficient and small average distances between random node pairs (which is typical in real-world networks), their degree sequences seem to follow a $\chi^2$-distribution rather than power law. To achieve high clustering coefficients, we apply an edge rewiring procedure that preserves the node degrees. We compare the crawled and the created graphs, and simulate certain algorithms for information dissemination and epidemic spreading on them. The results show that the created graphs exhibit very similar topological and algorithmic properties as the real-world graphs, providing evidence that they can be used as surrogates in social network analysis. Furthermore, our model is highly scalable, which enables us to create graphs of arbitrary size with almost the same properties as the corresponding real-world networks., Comment: 11 pages, 7 figures; published at ACM Web Conference 2022

We survey information-theoretic approaches to the reduction of Markov chains. Our survey is structured in two parts: The first part considers Markov chain coarse graining, which focuses on projecting the Markov chain to a process on a smaller state space that is informative}about certain quantities of interest. The second part considers Markov chain model reduction, which focuses on replacing the original Markov model by a simplified one that yields similar behavior as the original Markov model. We discuss the practical relevance of both approaches in the field of knowledge discovery and data mining by formulating problems of unsupervised machine learning as reduction problems of Markov chains. Finally, we briefly discuss the concept of lumpability, the phenomenon when a coarse graining yields a reduced Markov model., Comment: 16 pages, 3 figures; survey paper

This paper empirically studies commonly observed training difficulties of Physics-Informed Neural Networks (PINNs) on dynamical systems. Our results indicate that fixed points which are inherent to these systems play a key role in the optimization of the in PINNs embedded physics loss function. We observe that the loss landscape exhibits local optima that are shaped by the presence of fixed points. We find that these local optima contribute to the complexity of the physics loss optimization which can explain common training difficulties and resulting nonphysical predictions. Under certain settings, e.g., initial conditions close to fixed points or long simulations times, we show that those optima can even become better than that of the desired solution., Comment: 22 pages

This paper introduces a method for the detection of knock occurrences in an internal combustion engine (ICE) using a 1D convolutional neural network trained on in-cylinder pressure data. The model architecture was based on considerations regarding the expected frequency characteristics of knocking combustion. To aid the feature extraction, all cycles were reduced to 60{\deg} CA long windows, with no further processing applied to the pressure traces. The neural networks were trained exclusively on in-cylinder pressure traces from multiple conditions and labels provided by human experts. The best-performing model architecture achieves an accuracy of above 92% on all test sets in a tenfold cross-validation when distinguishing between knocking and non-knocking cycles. In a multi-class problem where each cycle was labeled by the number of experts who rated it as knocking, 78% of cycles were labeled perfectly, while 90% of cycles were classified at most one class from ground truth. They thus considerably outperform the broadly applied MAPO (Maximum Amplitude of Pressure Oscillation) detection method, as well as other references reconstructed from previous works. Our analysis indicates that the neural network learned physically meaningful features connected to engine-characteristic resonance frequencies, thus verifying the intended theory-guided data science approach. Deeper performance investigation further shows remarkable generalization ability to unseen operating points. In addition, the model proved to classify knocking cycles in unseen engines with increased accuracy of 89% after adapting to their features via training on a small number of exclusively non-knocking cycles. The algorithm takes below 1 ms (on CPU) to classify individual cycles, effectively making it suitable for real-time engine control., Comment: accepted for publication in IEEE/ASME Transactions on Mechatronics. (c) IEEE 2022

We connect the problem of semi-supervised clustering to constrained Markov aggregation, i.e., the task of partitioning the state space of a Markov chain. We achieve this connection by considering every data point in the dataset as an element of the Markov chain's state space, by defining the transition probabilities between states via similarities between corresponding data points, and by incorporating semi-supervision information as hard constraints in a Hartigan-style algorithm. The introduced Constrained Markov Clustering (CoMaC) is an extension of a recent information-theoretic framework for (unsupervised) Markov aggregation to the semi-supervised case. Instantiating CoMaC for certain parameter settings further generalizes two previous information-theoretic objectives for unsupervised clustering. Our results indicate that CoMaC is competitive with the state-of-the-art., Comment: 13 pages, 6 figures; this is an extended version of a short paper accepted at ACM SAC 2022 (minor changes to the text; error in source code corrected)