arXiv subjects

Lucas Böttcher

Publications and source records attributed to Lucas Böttcher.

At least 37 records · Page 2Linked to original sources

arXiv2023

Gradient-free training of neural ODEs for system identification and control using ensemble Kalman inversion

Ensemble Kalman inversion (EKI) is a sequential Monte Carlo method used to solve inverse problems within a Bayesian framework. Unlike backpropagation, EKI is a gradient-free optimization method that only necessitates the evaluation of artificial neural networks in forward passes. In this study, we examine the effectiveness of EKI in training neural ordinary differential equations (neural ODEs) for system identification and control tasks. To apply EKI to optimal control problems, we formulate inverse problems that incorporate a Tikhonov-type regularization term. Our numerical results demonstrate that EKI is an efficient method for training neural ODEs in system identification and optimal control problems, with runtime and quality of solutions that are competitive with commonly used gradient-based optimizers.

cs.LG↗

arXiv2023

Control of Dual-Sourcing Inventory Systems using Recurrent Neural Networks

A key challenge in inventory management is to identify policies that optimally replenish inventory from multiple suppliers. To solve such optimization problems, inventory managers need to decide what quantities to order from each supplier, given the net inventory and outstanding orders, so that the expected backlogging, holding, and sourcing costs are jointly minimized. Inventory management problems have been studied extensively for over 60 years, and yet even basic dual-sourcing problems, in which orders from an expensive supplier arrive faster than orders from a regular supplier, remain intractable in their general form. In addition, there is an emerging need to develop proactive, scalable optimization algorithms that can adjust their recommendations to dynamic demand shifts in a timely fashion. In this work, we approach dual sourcing from a neural network--based optimization lens and incorporate information on inventory dynamics and its replenishment (i.e., control) policies into the design of recurrent neural networks. We show that the proposed neural network controllers (NNCs) are able to learn near-optimal policies of commonly used instances within a few minutes of CPU time on a regular personal computer. To demonstrate the versatility of NNCs, we also show that they can control inventory dynamics with empirical, non-stationary demand distributions that are challenging to tackle effectively using alternative, state-of-the-art approaches. Our work shows that high-quality solutions of complex inventory management problems with non-stationary demand can be obtained with deep neural-network optimization approaches that directly account for inventory dynamics in their optimization process. As such, our research opens up new ways of efficiently managing complex, high-dimensional inventory dynamics.

cs.LG↗

arXiv2023

Modeling and forecasting age-specific drug overdose mortality in the United States

Drug overdose deaths continue to increase in the United States for all major drug categories. Over the past two decades the total number of overdose fatalities has increased more than five-fold; since 2013 the surge in overdose rates is primarily driven by fentanyl and methamphetamines. Different drug categories and factors such as age, gender, and ethnicity are associated with different overdose mortality characteristics that may also change in time. For example, the average age at death from a drug overdose has decreased from 1940 to 1990 while the overall mortality rate has steadily increased. To provide insight into the population-level dynamics of drug-overdose mortality, we develop an age-structured model for drug addiction. Using an augmented ensemble Kalman filter (EnKF), we show through a simple example how our model can be combined with synthetic observation data to estimate mortality rate and an age-distribution parameter. Finally, we use an EnKF to combine our model with observation data on overdose fatalities in the United States from 1999 to 2020 to forecast the evolution of overdose trends and estimate model parameters.

stat.AP↗

arXiv2023

Spectrally Adapted Physics-Informed Neural Networks for Solving Unbounded Domain Problems

Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined on an unbounded domain arises in numerous physical applications. Accurately solving unbounded domain PDEs requires efficient numerical methods that can resolve the dependence of the PDE on the unbounded variable over at least several orders of magnitude. We propose a solution to such problems by combining two classes of numerical methods: (i) adaptive spectral methods and (ii) physics-informed neural networks (PINNs). The numerical approach that we develop takes advantage of the ability of physics-informed neural networks to easily implement high-order numerical schemes to efficiently solve PDEs and extrapolate numerical solutions at any point in space and time. We then show how recently introduced adaptive techniques for spectral methods can be integrated into PINN-based PDE solvers to obtain numerical solutions of unbounded domain problems that cannot be efficiently approximated by standard PINNs. Through a number of examples, we demonstrate the advantages of the proposed spectrally adapted PINNs in solving PDEs and estimating model parameters from noisy observations in unbounded domains.

cs.LG↗

arXiv2023

Impact of random and targeted disruptions on information diffusion during outbreaks

Outbreaks are complex multi-scale processes that are impacted not only by cellular dynamics and the ability of pathogens to effectively reproduce and spread, but also by population-level dynamics and the effectiveness of mitigation measures. A timely exchange of information related to the spread of novel pathogens, stay-at-home orders, and other containment measures can be effective at containing an infectious disease, particularly during in the early stages when testing infrastructure, vaccines, and other medical interventions may not be available at scale. Using a multiplex epidemic model that consists of an information layer (modeling information exchange between individuals) and a spatially embedded epidemic layer (representing a human contact network), we study how random and targeted disruptions in the information layer (\eg, errors and intentional attacks on communication infrastructure) impact outbreak dynamics. We calibrate our model to the early outbreak stages of the SARS-CoV-2 pandemic in 2020. Mitigation campaign can still be effective under random disruptions, such as failure of information channels between a few individuals. However, targeted disruptions or sabotage of hub nodes that exchange information with a large number of individuals can abruptly change outbreak characteristics such as the time to reach the peak infection. Our results emphasize the importance of using a robust communication infrastructure that can withstand both random and targeted disruptions.

physics.soc-ph↗

arXiv2022

Near-optimal control of dynamical systems with neural ordinary differential equations

Optimal control problems naturally arise in many scientific applications where one wishes to steer a dynamical system from a certain initial state $\mathbf{x}_0$ to a desired target state $\mathbf{x}^*$ in finite time $T$. Recent advances in deep learning and neural network-based optimization have contributed to the development of methods that can help solve control problems involving high-dimensional dynamical systems. In particular, the framework of neural ordinary differential equations (neural ODEs) provides an efficient means to iteratively approximate continuous time control functions associated with analytically intractable and computationally demanding control tasks. Although neural ODE controllers have shown great potential in solving complex control problems, the understanding of the effects of hyperparameters such as network structure and optimizers on learning performance is still very limited. Our work aims at addressing some of these knowledge gaps to conduct efficient hyperparameter optimization. To this end, we first analyze how truncated and non-truncated backpropagation through time affect runtime performance and the ability of neural networks to learn optimal control functions. Using analytical and numerical methods, we then study the role of parameter initializations, optimizers, and neural-network architecture. Finally, we connect our results to the ability of neural ODE controllers to implicitly regularize control energy.

cs.LG↗

arXiv2022

Epidemic Management and Control Through Risk-Dependent Individual Contact Interventions

Testing, contact tracing, and isolation (TTI) is an epidemic management and control approach that is difficult to implement at scale because it relies on manual tracing of contacts. Exposure notification apps have been developed to digitally scale up TTI by harnessing contact data obtained from mobile devices; however, exposure notification apps provide users only with limited binary information when they have been directly exposed to a known infection source. Here we demonstrate a scalable improvement to TTI and exposure notification apps that uses data assimilation (DA) on a contact network. Network DA exploits diverse sources of health data together with the proximity data from mobile devices that exposure notification apps rely upon. It provides users with continuously assessed individual risks of exposure and infection, which can form the basis for targeting individual contact interventions. Simulations of the early COVID-19 epidemic in New York City prove the concepts. In the simulations, network DA identifies up to a factor 2 more infections than contact tracing when both harness the same contact data and diagnostic test data. This remains true even when only a relatively small fraction of the population uses network DA. When a sufficiently large fraction of the population ($\gtrsim 75\%$) uses network DA and complies with individual contact interventions, targeting contact interventions with network DA reduces deaths by up to a factor 4 relative to TTI. Network DA can be implemented by expanding the computational backend of existing exposure notification apps, thus greatly enhancing their capabilities. Implemented at scale, it has the potential to precisely and effectively control future epidemics while minimizing economic disruption.

stat.AP↗

arXiv2021

Neural Ordinary Differential Equation Control of Dynamics on Graphs

We study the ability of neural networks to calculate feedback control signals that steer trajectories of continuous time non-linear dynamical systems on graphs, which we represent with neural ordinary differential equations (neural ODEs). To do so, we present a neural-ODE control (NODEC) framework and find that it can learn feedback control signals that drive graph dynamical systems into desired target states. While we use loss functions that do not constrain the control energy, our results show, in accordance with related work, that NODEC produces low energy control signals. Finally, we evaluate the performance and versatility of NODEC against well-known feedback controllers and deep reinforcement learning. We use NODEC to generate feedback controls for systems of more than one thousand coupled, non-linear ODEs that represent epidemic processes and coupled oscillators.

cs.LG↗

arXiv2021

Tradeoffs in Hierarchical Voting Systems

Condorcet's jury theorem states that the correct outcome is reached in direct majority voting systems with sufficiently large electorates as long as each voter's independent probability of voting for that outcome is greater than 0.5. Yet, in situations where direct voting systems are infeasible, such as due to high implementation and infrastructure costs, hierarchical voting systems provide a reasonable alternative. We study differences in outcome precision between hierarchical and direct voting systems for varying group sizes, abstention rates, and voter competencies. Using asymptotic expansions of the derivative of the reliability function (or Banzhaf number), we first prove that indirect systems differ most from their direct counterparts when group size and number are equal to each other, and therefore to $\sqrt{N_{\rm d}}$, where $N_{\rm d}$ is the total number of voters in the direct system. In multitier systems, we prove that this difference is maximized when group size equals $\sqrt[n]{N_{\rm d}}$, where $n$ is the number of hierarchical levels. Second, we show that while direct majority rule always outperforms hierarchical voting for homogeneous electorates that vote with certainty, as group numbers and size increase, hierarchical majority voting gains in its ability to represent all eligible voters. Furthermore, when voter abstention and competency are correlated within groups, hierarchical systems often outperform direct voting, which we show by using a generating function approach that is able to analytically characterize heterogeneous voting systems.

math.CO↗

arXiv2021

Controlling epidemics through optimal allocation of test kits and vaccine doses across networks

Efficient testing and vaccination protocols are critical aspects of epidemic management. To study the optimal allocation of limited testing and vaccination resources in a heterogeneous contact network of interacting susceptible, recovered, and infected individuals, we present a degree-based testing and vaccination model for which we use control-theoretic methods to derive optimal testing and vaccination policies. Within our framework, we find that optimal intervention policies first target high-degree nodes before shifting to lower-degree nodes in a time-dependent manner. Using such optimal policies, it is possible to delay outbreaks and reduce incidence rates to a greater extent than uniform and reinforcement-learning-based interventions, particularly on certain scale-free networks.

q-bio.PE↗

arXiv2021

On the accuracy of short-term COVID-19 fatality forecasts

Forecasting new cases, hospitalizations, and disease-induced deaths is an important part of infectious disease surveillance and helps guide health officials in implementing effective countermeasures. For disease surveillance in the U.S., the Centers for Disease Control and Prevention (CDC) combine more than 65 individual forecasts of these numbers in an ensemble forecast at national and state levels. We collected data on CDC ensemble forecasts of COVID-19 fatalities in the United States, and compare them with easily interpretable ``Euler'' forecasts serving as a model-free benchmark that is only based on the local rate of change of the incidence curve. The term ``Euler method'' is motivated by the eponymous numerical integration scheme that calculates the value of a function at a future time step based on the current rate of change. Our results show that CDC ensemble forecasts are not more accurate than ``Euler'' forecasts on short-term forecasting horizons of one week. However, CDC ensemble forecasts show a better performance on longer forecasting horizons. Using the current rate of change in incidences as estimates of future incidence changes is useful for epidemic forecasting on short time horizons. An advantage of the proposed method over other forecasting approaches is that it can be implemented with a very limited amount of work and without relying on additional data (e.g., human mobility and contact patterns) and high-performance computing systems.

physics.soc-ph↗

arXiv2021

A statistical model of COVID-19 testing in populations: effects of sampling bias and testing errors

We develop a statistical model for the testing of disease prevalence in a population. The model assumes a binary test result, positive or negative, but allows for biases in sample selection and both type I (false positive) and type II (false negative) testing errors. Our model also incorporates multiple test types and is able to distinguish between retesting and exclusion after testing. Our quantitative framework allows us to directly interpret testing results as a function of errors and biases. By applying our testing model to COVID-19 testing data and actual case data from specific jurisdictions, we are able to estimate and provide uncertainty quantification of indices that are crucial in a pandemic, such as disease prevalence and fatality ratios.

stat.AP↗

arXiv2021

Classical and quantum random-walk centrality measures in multilayer networks

Multilayer network analysis is a useful approach for studying the structural properties of entities with diverse, multitudinous relations. Classifying the importance of nodes and node-layer tuples is an important aspect of the study of multilayer networks. To do this, it is common to calculate various centrality measures, which allow one to rank nodes and node-layers according to a variety of structural features. In this paper, we formulate occupation, PageRank, betweenness, and closeness centralities in terms of node-occupation properties of different types of continuous-time classical and quantum random walks on multilayer networks. We apply our framework to a variety of synthetic and real-world multilayer networks, and we identify marked differences between classical and quantum centrality measures. Our computations also give insights into the correlations between certain random-walk-based and geodesic-path-based centralities.

physics.soc-ph↗

arXiv2021

Implicit energy regularization of neural ordinary-differential-equation control

Although optimal control problems of dynamical systems can be formulated within the framework of variational calculus, their solution for complex systems is often analytically and computationally intractable. In this Letter we present a versatile neural ordinary-differential-equation control (NODEC) framework with implicit energy regularization and use it to obtain neural-network-generated control signals that can steer dynamical systems towards a desired target state within a predefined amount of time. We demonstrate the ability of NODEC to learn control signals that closely resemble those found by corresponding optimal control frameworks in terms of control energy and deviation from the desired target state. Our results suggest that NODEC is capable to solve a wide range of control and optimization problems, including those that are analytically intractable.

cs.LG↗

arXiv2021

Using excess deaths and testing statistics to improve estimates of COVID-19 mortalities

Factors such as non-uniform definitions of mortality, uncertainty in disease prevalence, and biased sampling complicate the quantification of fatality during an epidemic. Regardless of the employed fatality measure, the infected population and the number of infection-caused deaths need to be consistently estimated for comparing mortality across regions. We combine historical and current mortality data, a statistical testing model, and an SIR epidemic model, to improve estimation of mortality. We find that the average excess death across the entire US is 13$\%$ higher than the number of reported COVID-19 deaths. In some areas, such as New York City, the number of weekly deaths is about eight times higher than in previous years. Other countries such as Peru, Ecuador, Mexico, and Spain exhibit excess deaths significantly higher than their reported COVID-19 deaths. Conversely, we find negligible or negative excess deaths for part and all of 2020 for Denmark, Germany, and Norway.

q-bio.QM↗

arXiv2020

Aging and equilibration in bistable contagion dynamics

We analyze the late-time relaxation dynamics for a general contagion model. In this model, nodes are either active or failed. Active nodes can fail either "spontaneously" at any time or "externally" if their neighborhoods are sufficiently damaged. Failed nodes may always recover spontaneously. At late times, the breaking of time-translation-invariance is a necessary condition for physical aging. We observe that time-translational invariance is lost for initial conditions that lie between the basins of attraction of the model's two stable stationary states. Based on corresponding mean-field predictions, we characterize the observed model behavior in terms of a phase diagram spanned by the fractions of spontaneously and externally failed nodes. For the square lattice, the phases in which the dynamics approaches one of the two stable stationary states are not linearly separable due to spatial correlation effects. Our results provide new insights into aging and relaxation phenomena that are observable in a model of social contagion processes.

physics.soc-ph↗

arXiv2020

From classical to quantum walks with stochastic resetting on networks

Random walks are fundamental models of stochastic processes with applications in various fields including physics, biology, and computer science. We study classical and quantum random walks under the influence of stochastic resetting on arbitrary networks. Based on the mathematical formalism of quantum stochastic walks, we provide a framework of classical and quantum walks whose evolution is determined by graph Laplacians. We study the influence of quantum effects on the stationary and long-time average probability distribution by interpolating between the classical and quantum regime. We compare our analytical results on stationary and long-time average probability distributions with numerical simulations on different networks, revealing differences in the way resets affect the sampling properties of classical and quantum walks.

cond-mat.stat-mech↗

arXiv2020

A Random-Line-Graph Approach to Overlapping Line Segments

We study graphs that are formed by independently-positioned needles (i.e., line segments) in the unit square. To mathematically characterize the graph structure, we derive the probability that two line segments intersect and determine related quantities such as the distribution of intersections, given a certain number of line segments $N$. We interpret intersections between line segments as nodes and connections between them as edges in a spatial network that we refer to as random-line graph (RLG). Using methods from the study of random-geometric graphs, we show that the probability of RLGs to be connected undergoes a sharp transition if the number of lines exceeds a threshold $N^*$.

cond-mat.soft↗