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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.

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)

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)

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)

Physics-informed neural networks (PINNs) have emerged as a promising deep learning method, capable of solving forward and inverse problems governed by differential equations. Despite their recent advance, it is widely acknowledged that PINNs are difficult to train and often require a careful tuning of loss weights when data and physics loss functions are combined by scalarization of a multi-objective (MO) problem. In this paper, we aim to understand how parameters of the physical system, such as characteristic length and time scales, the computational domain, and coefficients of differential equations affect MO optimization and the optimal choice of loss weights. Through a theoretical examination of where these system parameters appear in PINN training, we find that they effectively and individually scale the loss residuals, causing imbalances in MO optimization with certain choices of system parameters. The immediate effects of this are reflected in the apparent Pareto front, which we define as the set of loss values achievable with gradient-based training and visualize accordingly. We empirically verify that loss weights can be used successfully to compensate for the scaling of system parameters, and enable the selection of an optimal solution on the apparent Pareto front that aligns well with the physically valid solution. We further demonstrate that by altering the system parameterization, the apparent Pareto front can shift and exhibit locally convex parts, resulting in a wider range of loss weights for which gradient-based training becomes successful. This work explains the effects of system parameters on MO optimization in PINNs, and highlights the utility of proposed loss weighting schemes., Comment: 11 pages

Drive towards improved performance of machine learning models has led to the creation of complex features representing a database of condensed matter systems. The complex features, however, do not offer an intuitive explanation on which physical attributes do improve the performance. The effect of the database on the performance of the trained model is often neglected. In this work we seek to understand in depth the effect that the choice of features and the properties of the database have on a machine learning application. In our experiments, we consider the complex phase space of carbon as a test case, for which we use a set of simple, human understandable and cheaply computable features for the aim of predicting the total energy of the crystal structure. Our study shows that (i) the performance of the machine learning model varies depending on the set of features and the database, (ii) is not transferable to every structure in the phase space and (iii) depends on how well structures are represented in the database., Comment: 18 pages, 11 figures

Complex systems, abstractly represented as networks, are ubiquitous in everyday life. Analyzing and understanding these systems requires, among others, tools for community detection. As no single best community detection algorithm can exist, robustness across a wide variety of problem settings is desirable. In this work, we present Synwalk, a random walk-based community detection method. Synwalk builds upon a solid theoretical basis and detects communities by synthesizing the random walk induced by the given network from a class of candidate random walks. We thoroughly validate the effectiveness of our approach on synthetic and empirical networks, respectively, and compare Synwalk's performance with the performance of Infomap and Walktrap. Our results indicate that Synwalk performs robustly on networks with varying mixing parameters and degree distributions. We outperform Infomap on networks with high mixing parameter, and Infomap and Walktrap on networks with many small communities and low average degree. Our work has a potential to inspire further development of community detection via synthesis of random walks and we provide concrete ideas for future research., Comment: 31 pages, 13 figures

Distance-based classification is among the most competitive classification methods for time series data. The most critical component of distance-based classification is the selected distance function. Past research has proposed various different distance metrics or measures dedicated to particular aspects of real-world time series data, yet there is an important aspect that has not been considered so far: Robustness against arbitrary data contamination. In this work, we propose a novel distance metric that is robust against arbitrarily "bad" contamination and has a worst-case computational complexity of $\mathcal{O}(n\log n)$. We formally argue why our proposed metric is robust, and demonstrate in an empirical evaluation that the metric yields competitive classification accuracy when applied in k-Nearest Neighbor time series classification., Comment: MileTS Workshop at KDD'19

We review the current literature concerned with information plane analyses of neural network classifiers. While the underlying information bottleneck theory and the claim that information-theoretic compression is causally linked to generalization are plausible, empirical evidence was found to be both supporting and conflicting. We review this evidence together with a detailed analysis of how the respective information quantities were estimated. Our survey suggests that compression visualized in information planes is not necessarily information-theoretic, but is rather often compatible with geometric compression of the latent representations. This insight gives the information plane a renewed justification. Aside from this, we shed light on the problem of estimating mutual information in deterministic neural networks and its consequences. Specifically, we argue that even in feed-forward neural networks the data processing inequality need not hold for estimates of mutual information. Similarly, while a fitting phase, in which the mutual information between the latent representation and the target increases, is necessary (but not sufficient) for good classification performance, depending on the specifics of mutual information estimation such a fitting phase need not be visible in the information plane., Comment: 12 pages, 3 figures; accepted for publication in IEEE Transactions on Neural Networks and Learning Systems. (c) 2021 IEEE

We propose a semi-supervised generative model, SeGMA, which learns a joint probability distribution of data and their classes and which is implemented in a typical Wasserstein auto-encoder framework. We choose a mixture of Gaussians as a target distribution in latent space, which provides a natural splitting of data into clusters. To connect Gaussian components with correct classes, we use a small amount of labeled data and a Gaussian classifier induced by the target distribution. SeGMA is optimized efficiently due to the use of Cramer-Wold distance as a maximum mean discrepancy penalty, which yields a closed-form expression for a mixture of spherical Gaussian components and thus obviates the need of sampling. While SeGMA preserves all properties of its semi-supervised predecessors and achieves at least as good generative performance on standard benchmark data sets, it presents additional features: (a) interpolation between any pair of points in the latent space produces realistically-looking samples; (b) combining the interpolation property with disentangled class and style variables, SeGMA is able to perform a continuous style transfer from one class to another; (c) it is possible to change the intensity of class characteristics in a data point by moving the latent representation of the data point away from specific Gaussian components.

In this draft, which reports on work in progress, we 1) adapt the information bottleneck functional by replacing the compression term by class-conditional compression, 2) relax this functional using a variational bound related to class-conditional disentanglement, 3) consider this functional as a training objective for stochastic neural networks, and 4) show that the latent representations are learned such that they can be used in a naive Bayes classifier. We continue by suggesting a series of experiments along the lines of Nonlinear In-formation Bottleneck [Kolchinsky et al., 2018], Deep Variational Information Bottleneck [Alemi et al., 2017], and Information Dropout [Achille and Soatto, 2018]. We furthermore suggest a neural network where the decoder architecture is a parameterized naive Bayes decoder., Comment: draft; work in progress

In this work, we investigate the use of three information-theoretic quantities -- entropy, mutual information with the class variable, and a class selectivity measure based on Kullback-Leibler divergence -- to understand and study the behavior of already trained fully-connected feed-forward neural networks. We analyze the connection between these information-theoretic quantities and classification performance on the test set by cumulatively ablating neurons in networks trained on MNIST, FashionMNIST, and CIFAR-10. Our results parallel those recently published by Morcos et al., indicating that class selectivity is not a good indicator for classification performance. However, looking at individual layers separately, both mutual information and class selectivity are positively correlated with classification performance, at least for networks with ReLU activation functions. We provide explanations for this phenomenon and conclude that it is ill-advised to compare the proposed information-theoretic quantities across layers. Furthermore, we show that cumulative ablation of neurons with ascending or descending information-theoretic quantities can be used to formulate hypotheses regarding the joint behavior of multiple neurons, such as redundancy and synergy, with comparably low computational cost. We also draw connections to the information bottleneck theory for neural networks., Comment: 12 pages; accepted for publication in IEEE Transactions on Neural Networks and Learning Systems

In this theory paper, we investigate training deep neural networks (DNNs) for classification via minimizing the information bottleneck (IB) functional. We show that the resulting optimization problem suffers from two severe issues: First, for deterministic DNNs, either the IB functional is infinite for almost all values of network parameters, making the optimization problem ill-posed, or it is piecewise constant, hence not admitting gradient-based optimization methods. Second, the invariance of the IB functional under bijections prevents it from capturing properties of the learned representation that are desirable for classification, such as robustness and simplicity. We argue that these issues are partly resolved for stochastic DNNs, DNNs that include a (hard or soft) decision rule, or by replacing the IB functional with related, but more well-behaved cost functions. We conclude that recent successes reported about training DNNs using the IB framework must be attributed to such solutions. As a side effect, our results indicate limitations of the IB framework for the analysis of DNNs. We also note that rather than trying to repair the inherent problems in the IB functional, a better approach may be to design regularizers on latent representation enforcing the desired properties directly., Comment: 16 pages, to appear in IEEE Trans. Pattern Analysis and Machine Intelligence