Felix Dietrich

Technische Universität München | TUM · School of Computation Information and Technology

We introduce a probability distribution, combined with an efficient sampling algorithm, for weights and biases of fully-connected neural networks. In a supervised learning context, no iterative optimization or gradient computations of internal network parameters are needed to obtain a trained network. The sampling is based on the idea of random fea...

Quantum process tomography conventionally uses a multitude of initial quantum states and then performs state tomography on the process output. Here we propose and study an alternative approach which requires only a single (or few) known initial states together with time-delayed measurements for reconstructing the unitary map and corresponding Hamil...

We identify effective stochastic differential equations (SDEs) for coarse observables of fine-grained particle- or agent-based simulations; these SDEs then provide useful coarse surrogate models of the fine scale dynamics. We approximate the drift and diffusivity functions in these effective SDEs through neural networks, which can be thought of as...

Neural networks have recently gained attention in the context of solving inverse problems. Physics-Informed Neural Networks (PINNs) are a prominent methodology for the task of solving both forward and inverse problems. In the paper at hand, full waveform inversion is the inverse problem under consideration. The performance of PINNs is compared agai...

Even the best scientific equipment can only partially observe reality. Recorded data is often lower-dimensional, e.g., two-dimensional pictures of the three-dimensional world. Combining data from multiple experiments then results in a marginal density. This work shows how to transport such lower-dimensional marginal densities into a more informativ...

We propose a machine-learning approach to model long-term out-of-sample dynamics of brain activity from task-dependent fMRI data. Our approach is a three stage one. First, we exploit Diffusion maps (DMs) to discover a set of variables that parametrize the low-dimensional manifold on which the emergent high-dimensional fMRI time series evolve. Then,...

Neural networks have recently gained attention in solving inverse problems. One prominent methodology are Physics-Informed Neural Networks (PINNs) which can solve both forward and inverse problems. In the paper at hand, full waveform inversion is the considered inverse problem. The performance of PINNs is compared against classical adjoint optimiza...

Safe Policy Improvement (SPI) is an important technique for offline reinforcement learning in safety critical applications as it improves the behavior policy with a high probability. We classify various SPI approaches from the literature into two groups, based on how they utilize the uncertainty of state-action pairs. Focusing on the Soft-SPIBB (Sa...

We construct a reduced, data-driven, parameter dependent effective Stochastic Differential Equation (eSDE) for electric-field mediated colloidal crystallization using data obtained from Brownian dynamics simulations.

Meta-learning of numerical algorithms for a given task consist of the data-driven identification and adaptation of an algorithmic structure and the associated hyperparameters. To limit the complexity of the meta-learning problem, neural architectures with a certain inductive bias towards favorable algorithmic structures can, and should, be used. We...

Wheeled vehicles are the most convenient and widespread locomotion machines for the majority of research, industrial or private tasks. A perceptible share of wheeled vehicles is used on soft soil. Modelling wheel locomotion in these situations is challenging, because of the non-proportional relation between applied shear stress and the soil’s defor...

We present a data-driven approach to characterizing nonidentifiability of a model’s parameters and illustrate it through dynamic as well as steady kinetic models. By employing Diffusion Maps and their extensions, we discover the minimal combinations of parameters required to characterize the output behavior of a chemical system: a set of effective...

Safe Policy Improvement (SPI) is an important technique for offline reinforcement learning in safety critical applications as it improves the behavior policy with a high probability. We classify various SPI approaches from the literature into two groups, based on how they utilize the uncertainty of state-action pairs. Focusing on the Soft-SPIBB (Sa...

We propose an approach to learn effective evolution equations for large systems of interacting agents. This is demonstrated on two examples, a well-studied system of coupled normal form oscillators and a biologically motivated example of coupled Hodgkin-Huxley-like neurons. For such types of systems there is no obvious space coordinate in which to...

We construct a reduced, data-driven, parameter dependent effective Stochastic Differential Equation (eSDE) for electric-field mediated colloidal crystallization using data obtained from Brownian Dynamics Simulations. We use Diffusion Maps (a manifold learning algorithm) to identify a set of useful latent observables. In this latent space we identif...

We introduce a data-driven approach to building reduced dynamical models through manifold learning; the reduced latent space is discovered using Diffusion Maps (a manifold learning technique) on time series data. A second round of Diffusion Maps on those latent coordinates allows the approximation of the reduced dynamical models. This second round...

In this work, we propose a method to learn multivariate probability distributions using sample path data from stochastic differential equations. Specifically, we consider temporally evolving probability distributions (e.g., those produced by integrating local or nonlocal Fokker-Planck equations). We analyze this evolution through machine learning a...

Safe Policy Improvement (SPI) aims at provable guarantees that a learned policy is at least approximately as good as a given baseline policy. Building on SPI with Soft Baseline Bootstrapping (Soft-SPIBB) by Nadjahi et al., we identify theoretical issues in their approach, provide a corrected theory, and derive a new algorithm that is provably safe...

Quantum process tomography conventionally uses a multitude of initial quantum states and then performs state tomography on the process output. Here we propose and study an alternative approach which requires only a single (or few) known initial states together with time-delayed measurements for reconstructing the unitary map and corresponding Hamil...

Describing and forecasting city traffic is challenging, given the array of factors influencing the movement of pedestrians and vehicles. Faced with this complexity, research has focused on machine learning as a way to capture spatio-temporal traffic patterns, based on past sensor data. While the methods can accurately forecast high-dimensional obse...

We present a data-driven approach to characterizing nonidentifiability of a model's parameters and illustrate it through dynamic kinetic models. By employing Diffusion Maps and their extensions, we discover the minimal combinations of parameters required to characterize the dynamic output behavior: a set of effective parameters for the model. Furth...

We discuss the correspondence between Gaussian process regression and Geometric Harmonics, two similar kernel-based methods that are typically used in different contexts. Research communities surrounding the two concepts often pursue different goals. Results from both camps can be successfully combined, providing alternative interpretations of unce...

In this work, we propose a method to learn probability distributions using sample path data from stochastic differential equations. Specifically, we consider temporally evolving probability distributions (e.g., those produced by integrating local or nonlocal Fokker-Planck equations). We analyze this evolution through machine learning assisted const...

We identify effective stochastic differential equations (SDE) for coarse observables of fine-grained particle- or agent-based simulations; these SDE then provide coarse surrogate models of the fine scale dynamics. We approximate the drift and diffusivity functions in these effective SDE through neural networks, which can be thought of as effective...

We study the meta-learning of numerical algorithms for scientific computing, which combines the mathematically driven, handcrafted design of general algorithm structure with a data-driven adaptation to specific classes of tasks. This represents a departure from the classical approaches in numerical analysis, which typically do not feature such lear...

We extract data-driven, intrinsic spatial coordinates from observations of the dynamics of large systems of coupled heterogeneous agents. These coordinates then serve as an emergent space in which to learn predictive models in the form of partial differential equations (PDEs) for the collective description of the coupled-agent system. They play the...

Significance A fundamental issue in empirical science is the ability to calibrate between different types of measurements/observations of the same phenomenon. This naturally suggests the selection of canonical variables, in the spirit of principal components, to enable matching/calibration among different observation modalities/instruments. We deve...

We apply the Koopman operator framework to pedestrian dynamics. In an example scenario, we generate crowd density time series data with a microscopic pedestrian simulator. We then approximate the Koopman operator in matrix form through Extended Dynamic Mode Decomposition, using Geometric Harmonics on the data as a dictionary. The Koopman matrix is...

It is difficult to provide live simulation systems for decision support. Time is limited and uncertainty quantification requires many simulation runs. We combine a surrogate model with the stochastic collocation method to overcome time and storage restrictions and show a proof of concept for a de-boarding scenario of a train.

We propose to test, and when possible establish, an equivalence between two different artificial neural networks by attempting to construct a data-driven transformation between them, using manifold-learning techniques. In particular, we employ diffusion maps with a Mahalanobis-like metric. If the construction succeeds, the two networks can be thoug...

We propose a deep-learning based method for obtaining standardized data coordinates from scientific measurements.Data observations are modeled as samples from an unknown, non-linear deformation of an underlying Riemannian manifold, which is parametrized by a few normalized latent variables. By leveraging a repeated measurement sampling strategy, we...

In this paper, we propose a spectral method for deriving functions that are jointly smooth on multiple observed manifolds. Our method is unsupervised and primarily consists of two steps. First, using kernels, we obtain a subspace spanning smooth functions on each manifold. Then, we apply a spectral method to the obtained subspaces and discover func...

Data mining is routinely used to organize ensembles of short temporal observations so as to reconstruct useful, low-dimensional realizations of an underlying dynamical system. In this paper, we use manifold learning to organize unstructured ensembles of observations (“trials”) of a system’s response surface. We have no control over where every tria...

Model predictive control (MPC) is a de facto standard control algorithm across the process industries. There remain, however, applications where MPC is impractical because an optimization problem is solved at each time step. We present a link between explicit MPC formulations and manifold learning to enable facilitated prediction of the MPC policy....

Concise, accurate descriptions of physical systems through their conserved quantities abound in the natural sciences. In data science, however, current research often focuses on regression problems, without routinely incorporating additional assumptions about the system that generated the data. Here, we propose to explore a particular type of under...

Concise, accurate descriptions of physical systems through their conserved quantities abound in the natural sciences. In data science, however, current research often focuses on regression problems, without routinely incorporating additional assumptions about the system that generated the data. Here, we propose to explore a particular type of under...

A systematic mathematical framework for the study of numerical algorithms would allow comparisons , facilitate conjugacy arguments, as well as enable the discovery of improved, accelerated, data-driven algorithms. Over the course of the last century, the Koopman operator has provided a mathematical framework for the study of dynamical systems, whic...

Different observations of a relation between inputs ("sources") and outputs ("targets") are often reported in terms of histograms (discretizations of the source and the target densities). Transporting these densities to each other provides insight regarding the underlying relation. In (forward) uncertainty quantification, one typically studies how...

The problem of domain adaptation has become central in many applications from a broad range of fields. Recently, it was proposed to use Optimal Transport (OT) to solve it. In this paper, we model the difference between the two domains by a diffeomorphism and use the polar factorization theorem to claim that OT is indeed optimal for domain adaptatio...

In statistical modeling with Gaussian Process regression, it has been shown that combining (few) high-fidelity data with (many) low-fidelity data can enhance prediction accuracy, compared to prediction based on the few high-fidelity data only. Such information fusion techniques for multifidelity data commonly approach the high-fidelity model $f_h(t...

Model predictive control (MPC) is among the most successful approaches for process control and has become a de facto standard across the process industries. There remain, however, applications for which MPC becomes difficult or impractical due to the demand that an optimization problem is solved at each time step. In this work, we present a link be...

Manifold-learning techniques are routinely used in mining complex spatiotemporal data to extract useful, parsimonious data representations/parametrizations; these are, in turn, useful in nonlinear model identification tasks. We focus here on the case of time series data that can ultimately be modelled as a spatially distributed system (e.g. a parti...

Data mining is routinely used to organize ensembles of short temporal observations so as to reconstruct useful, low-dimensional realizations of the underlying dynamical systems. By analogy, we use data mining to organize ensembles of a different type of short observations to reconstruct useful realizations of bifurcation diagrams. Here the observat...

To assess a computer model’s descriptive and predictive power, the model’s response to uncertainties in the input must be quantified. However, simulations of complex systems typically need a lot of computational resources, and thus prohibit exhaustive sweeps of high-dimensional spaces. Moreover, the time available to compute a result for decision s...

Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of nonlinear dynamic behavior (e.g. through normal forms). In this paper we will argue that the use of the Koopman operator and its spectrum is particularly well suited for th...

We discuss the interplay between manifold-learning techniques, which can extract intrinsic order from observations of complex dynamics, and systems modeling considerations. Tuning the scale of the data-mining kernels can guide the construction of dynamic models at different levels of coarse-graining. In particular, we focus on the observability of...

Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD) and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclus...

There are many well validated models of pedestrian movement on a flat surface. This is not the case for movement on stairs. Experiments show that pedestrians slow down when climbing or descending stairs. Hence, it is tempting to model movement on stairs by simply slowing down by a factor. But this would imply that, other than being slower, motion o...

Stop and go waves in granular flow can often be described mathematically by a dynamical system with a Hopf bifurcation. We show that a certain class of microscopic, ordinary differential equation-based models in crowd dynamics fulfil certain conditions of Hopf bifurcations. The class is based on the Gradient Navigation Model. An interesting phenome...

Models using a superposition of scalar fields for navigation are prevalent in microscopic pedestrian stream simulations. However, classifications, differences, and similarities of models are not clear at the conceptual level of navigation mechanisms. In this paper, we describe the superposition of scalar fields as an approach to microscopic crowd m...

Pedestrians adjust both speed and stride length when they navigate difficult situations such as tight corners or dense crowds. They do this with foresight reacting instantly when they encounter the difficulty. This has an impact on the movement of the whole crowd especially at bottlenecks where slower movement and smaller steps can be observed. Sta...

Pedestrian flow simulations are a modern method for computationally predicting pedestrian behavior. In contemporary research, the development of new and sophisticated hybrid pedestrian behavior models is ongoing. These models couple other pedestrian behavior models into a single concept. However, it was shown that a coupling of different models not...

Systems incorporating many particles such as agents, molecules, stars or birds are gaining popularity as research topics. When modeled, the state space of the model is often very high-dimensional, using a separate dimension for each variable of each particle. However, the interesting properties of such a system can often be described by very few va...

The document serves as a reference for researchers trying to capture a large portion of a mass event on video for several hours, while using a very limited budget.

The natural biomechanical motion process of many animals is stepwise. This feature of human movement and other bipeds is largely ignored in simulation models of pedestrians and crowds. We present a concise movement model for pedestrians based on stepwise movement. A series of controlled experiments was conducted to calibrate the model based on indi...

by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility Abstract Pedestrian stepping behaviour has been widely ignored in crowd simulation models. Yet, the continuous motion of pedestrian torsos is the result of decisions about discrete steps...

Cellular automata (CA) and ordinary differential equation (ODE) models compete for dominance in microscopic pedestrian dynamics. There are two major differences: movement in a CA is restricted to a grid and navigation is achieved by moving directly in the desired direction. Force based ODE models operate in continuous space and navigation is comput...

We present a microscopic ordinary differential equation (ODE)-based model for pedestrian dynamics: the gradient navigation model. The model uses a superposition of gradients of distance functions to directly change the direction of the velocity vector. The velocity is then integrated to obtain the location. The approach differs fundamentally from f...

This paper describes Monte-Carlo simulation techniques that calculate how effective the risk of a blockage in case of concurrent processes is. We start describing a common problem in current computer science, the deadlock. This is followed by a mathematical abstraction of the problem. Three solution models are presented for it, two of them designed...