Johannes Müller

Johannes Müller, Tobias Meuser, Ralf Steinmetz et al.

Cooperative information shared among a multi-agent system (MAS) can be useful to agents to efficiently fulfill their missions. Relying on wrong information, however, can have severe consequences. While classical approaches only consider measurement uncertainty, reliability information on the incoming data can be useful for decision making. In this work, a subjective logic based mechanism is proposed that amends reliability information to the data shared among the MAS. If multiple agents report the same event, their information is fused. In order to maintain high reliability, the mechanism detects and isolates misbehaving agents. Therefore, an attacker model is specified that includes faulty as well as malicious agents. The mechanism is applied to Intelligent Transportation Systems (ITS) and it is shown in simulation that the approach scales well with the size of the MAS and that it is able to efficiently detected and isolated misbehaving agents. Keywords: Multi-agent systems, Fault Detection, Sensor/data fusion, Control Applications

Jingtong Sun, Julius Berner, Lorenz Richter et al.

The task of sampling from a probability density can be approached as transporting a tractable density function to the target, known as dynamical measure transport. In this work, we tackle it through a principled unified framework using deterministic or stochastic evolutions described by partial differential equations (PDEs). This framework incorporates prior trajectory-based sampling methods, such as diffusion models or Schrödinger bridges, without relying on the concept of time-reversals. Moreover, it allows us to propose novel numerical methods for solving the transport task and thus sampling from complicated targets without the need for the normalization constant or data samples. We employ physics-informed neural networks (PINNs) to approximate the respective PDE solutions, implying both conceptional and computational advantages. In particular, PINNs allow for simulation- and discretization-free optimization and can be trained very efficiently, leading to significantly better mode coverage in the sampling task compared to alternative methods. Moreover, they can readily be fine-tuned with Gauss-Newton methods to achieve high accuracy in sampling.

Johannes Müller, Guido Montúfar

We study the convergence of several natural policy gradient (NPG) methods in infinite-horizon discounted Markov decision processes with regular policy parametrizations. For a variety of NPGs and reward functions we show that the trajectories in state-action space are solutions of gradient flows with respect to Hessian geometries, based on which we obtain global convergence guarantees and convergence rates. In particular, we show linear convergence for unregularized and regularized NPG flows with the metrics proposed by Kakade and Morimura and co-authors by observing that these arise from the Hessian geometries of conditional entropy and entropy respectively. Further, we obtain sublinear convergence rates for Hessian geometries arising from other convex functions like log-barriers. Finally, we interpret the discrete-time NPG methods with regularized rewards as inexact Newton methods if the NPG is defined with respect to the Hessian geometry of the regularizer. This yields local quadratic convergence rates of these methods for step size equal to the penalization strength.

Alexander Tsaregorodtsev, Johannes Müller, Jan Strohbeck et al.

Monocular camera sensors are vital to intelligent vehicle operation and automated driving assistance and are also heavily employed in traffic control infrastructure. Calibrating the monocular camera, though, is time-consuming and often requires significant manual intervention. In this work, we present an extrinsic camera calibration approach that automatizes the parameter estimation by utilizing semantic segmentation information from images and point clouds. Our approach relies on a coarse initial measurement of the camera pose and builds on lidar sensors mounted on a vehicle with high-precision localization to capture a point cloud of the camera environment. Afterward, a mapping between the camera and world coordinate spaces is obtained by performing a lidar-to-camera registration of the semantically segmented sensor data. We evaluate our method on simulated and real-world data to demonstrate low error measurements in the calibration results. Our approach is suitable for infrastructure sensors as well as vehicle sensors, while it does not require motion of the camera platform.

Jesse van Oostrum, Johannes Müller, Nihat Ay

The natural gradient field is a vector field that lives on a model equipped with a distinguished Riemannian metric, e.g. the Fisher-Rao metric, and represents the direction of steepest ascent of an objective function on the model with respect to this metric. In practice, one tries to obtain the corresponding direction on the parameter space by multiplying the ordinary gradient by the inverse of the Gram matrix associated with the metric. We refer to this vector on the parameter space as the natural parameter gradient. In this paper we study when the pushforward of the natural parameter gradient is equal to the natural gradient. Furthermore we investigate the invariance properties of the natural parameter gradient. Both questions are addressed in an overparametrised setting.

Johannes Müller, Michael Gabb, Michael Buchholz

Use of cooperative information, distributed by road-side units, offers large potential for intelligent vehicles (IVs). As vehicle automation progresses and cooperative perception is used to fill the blind spots of onboard sensors, the question of reliability of the data becomes increasingly important in safety considerations (SOTIF, Safety of the Intended Functionality). This paper addresses the problem to estimate the reliability of cooperative information for in-vehicle use. We propose a novel method to infer the reliability of received data based on the theory of Subjective Logic (SL). Using SL, we fuse multiple information sources, which individually only provide mild cues of the reliability, into a holistic estimate, which is statistically sound through an end-to-end modeling within the theory of SL. Using the proposed scheme for probabilistic SL-based fusion, IVs are able to separate faulty from correct data samples with a large margin of safety. Real world experiments show the applicability and effectiveness of our approach.

Johannes Müller, Guido Montúfar

Reward optimization in fully observable Markov decision processes is equivalent to a linear program over the polytope of state-action frequencies. Taking a similar perspective in the case of partially observable Markov decision processes with memoryless stochastic policies, the problem was recently formulated as the optimization of a linear objective subject to polynomial constraints. Based on this we present an approach for Reward Optimization in State-Action space (ROSA). We test this approach experimentally in maze navigation tasks. We find that ROSA is computationally efficient and can yield stability improvements over other existing methods.

Dominic Kohler, Johannes Müller, Utz Wever

We propose a novel density based numerical method for uncertainty propagation under certain partial differential equation dynamics. The main idea is to translate them into objects that we call cellular probabilistic automata and to evolve the latter. The translation is achieved by state discretization as in set oriented numerics and the use of the locality concept from cellular automata theory. We develop the method at the example of initial value uncertainties under deterministic dynamics and prove a consistency result. As an application we discuss arsenate transportation and adsorption in drinking water pipes and compare our results to Monte Carlo computations.

Johannes Müller, Marius Zeinhofer

We propose energy natural gradient descent, a natural gradient method with respect to a Hessian-induced Riemannian metric as an optimization algorithm for physics-informed neural networks (PINNs) and the deep Ritz method. As a main motivation we show that the update direction in function space resulting from the energy natural gradient corresponds to the Newton direction modulo an orthogonal projection onto the model's tangent space. We demonstrate experimentally that energy natural gradient descent yields highly accurate solutions with errors several orders of magnitude smaller than what is obtained when training PINNs with standard optimizers like gradient descent or Adam, even when those are allowed significantly more computation time.

Marcel Schloz, Johannes Müller, Thomas C. Pekin et al.

We present a method that lowers the dose required for a ptychographic reconstruction by adaptively scanning the specimen, thereby providing the required spatial information redundancy in the regions of highest importance. The proposed method is built upon a deep learning model that is trained by reinforcement learning (RL), using prior knowledge of the specimen structure from training data sets. We show that equivalent low-dose experiments using adaptive scanning outperform conventional ptychography experiments in terms of reconstruction resolution.

Felix Dangel, Johannes Müller, Marius Zeinhofer

Physics-informed neural networks (PINNs) are infamous for being hard to train. Recently, second-order methods based on natural gradient and Gauss-Newton methods have shown promising performance, improving the accuracy achieved by first-order methods by several orders of magnitude. While promising, the proposed methods only scale to networks with a few thousand parameters due to the high computational cost to evaluate, store, and invert the curvature matrix. We propose Kronecker-factored approximate curvature (KFAC) for PINN losses that greatly reduces the computational cost and allows scaling to much larger networks. Our approach goes beyond the established KFAC for traditional deep learning problems as it captures contributions from a PDE's differential operator that are crucial for optimization. To establish KFAC for such losses, we use Taylor-mode automatic differentiation to describe the differential operator's computation graph as a forward network with shared weights. This allows us to apply KFAC thanks to a recently-developed general formulation for networks with weight sharing. Empirically, we find that our KFAC-based optimizers are competitive with expensive second-order methods on small problems, scale more favorably to higher-dimensional neural networks and PDEs, and consistently outperform first-order methods and LBFGS.

Nikola Milosevic, Johannes Müller, Nico Scherf

Reinforcement Learning (RL) agents can solve diverse tasks but often exhibit unsafe behavior. Constrained Markov Decision Processes (CMDPs) address this by enforcing safety constraints, yet existing methods either sacrifice reward maximization or allow unsafe training. We introduce Constrained Trust Region Policy Optimization (C-TRPO), which reshapes the policy space geometry to ensure trust regions contain only safe policies, guaranteeing constraint satisfaction throughout training. We analyze its theoretical properties and connections to TRPO, Natural Policy Gradient (NPG), and Constrained Policy Optimization (CPO). Experiments show that C-TRPO reduces constraint violations while maintaining competitive returns.

Nikola Milosevic, Johannes Müller, Nico Scherf

In constrained Markov decision processes, enforcing constraints during training is often thought of as decreasing the final return. Recently, it was shown that constraints can be incorporated directly into the policy geometry, yielding an optimization trajectory close to the central path of a barrier method, which does not compromise final return. Building on this idea, we introduce Central Path Proximal Policy Optimization (C3PO), a simple modification of the PPO loss that produces policy iterates, that stay close to the central path of the constrained optimization problem. Compared to existing on-policy methods, C3PO delivers improved performance with tighter constraint enforcement, suggesting that central path-guided updates offer a promising direction for constrained policy optimization.

Johannes Müller, Semih Çaycı, Guido Montúfar

Kakade's natural policy gradient method has been studied extensively in recent years, showing linear convergence with and without regularization. We study another natural gradient method based on the Fisher information matrix of the state-action distributions which has received little attention from the theoretical side. Here, the state-action distributions follow the Fisher-Rao gradient flow inside the state-action polytope with respect to a linear potential. Therefore, we study Fisher-Rao gradient flows of linear programs more generally and show linear convergence with a rate that depends on the geometry of the linear program. Equivalently, this yields an estimate on the error induced by entropic regularization of the linear program which improves existing results. We extend these results and show sublinear convergence for perturbed Fisher-Rao gradient flows and natural gradient flows up to an approximation error. In particular, these general results cover the case of state-action natural policy gradients.

Victor Armegioiu, Juan Carrasquilla, Siddhartha Mishra et al.

We propose a framework for the design and analysis of optimization algorithms in variational quantum Monte Carlo, drawing on geometric insights into the corresponding function space. The framework translates infinite-dimensional optimization dynamics into tractable parameter-space algorithms through a Galerkin projection onto the tangent space of the variational ansatz. This perspective unifies existing methods such as stochastic reconfiguration and Rayleigh-Gauss-Newton, provides connections to classic function-space algorithms, and motivates the derivation of novel algorithms with geometrically principled hyperparameter choices. We validate our framework with numerical experiments demonstrating its practical relevance through the accurate estimation of ground-state energies for several prototypical models in condensed matter physics modeled with neural network wavefunctions.

Johannes Müller, Semih Cayci

We study the error introduced by entropy regularization in infinite-horizon discrete discounted Markov decision processes. We show that this error decreases exponentially in the inverse regularization strength, both in a weighted KL-divergence and in value with a problem-specific exponent. This is in contrast to previously known estimates, of the order $O(τ)$, where $τ$ is the regularization strength. We provide a lower bound that matches our upper bound up to a polynomial term, thereby characterizing the exponential convergence rate for entropy regularization. Our proof relies on the observation that the solutions of entropy-regularized Markov decision processes solve a gradient flow of the unregularized reward with respect to a Riemannian metric common in natural policy gradient methods. This correspondence allows us to identify the limit of this gradient flow as the generalized maximum entropy optimal policy, thereby characterizing the implicit bias of this gradient flow, which corresponds to a time-continuous version of the natural policy gradient method. We use our improved error estimates to show that for entropy-regularized natural policy gradient methods, the overall error decays exponentially in the square root of the number of iterations, improving over existing sublinear guarantees. Finally, we extend our analysis to settings beyond the entropy. In particular, we characterize the implicit bias regarding general convex potentials and their resulting generalized natural policy gradients.

Matti Henning, Johannes Müller, Fabian Gies et al.

Within the field of automated driving, a clear trend in environment perception tends towards more sensors, higher redundancy, and overall increase in computational power. This is mainly driven by the paradigm to perceive the entire environment as best as possible at all times. However, due to the ongoing rise in functional complexity, compromises have to be considered to ensure real-time capabilities of the perception system. In this work, we introduce a concept for situation-aware environment perception to control the resource allocation towards processing relevant areas within the data as well as towards employing only a subset of functional modules for environment perception, if sufficient for the current driving task. Specifically, we propose to evaluate the context of an automated vehicle to derive a multi-layer attention map (MLAM) that defines relevant areas. Using this MLAM, the optimum of active functional modules is dynamically configured and intra-module processing of only relevant data is enforced. We outline the feasibility of application of our concept using real-world data in a straight-forward implementation for our system at hand. While retaining overall functionality, we achieve a reduction of accumulated processing time of 59%.

Johannes Müller, Jan Strohbeck, Martin Herrmann et al.

Motion planning at urban intersections that accounts for the situation context, handles occlusions, and deals with measurement and prediction uncertainty is a major challenge on the way to urban automated driving. In this work, we address this challenge with a sampling-based optimization approach. For this, we formulate an optimal control problem that optimizes for low risk and high passenger comfort. The risk is calculated on the basis of the perception information and the respective uncertainty using a risk model. The risk model combines set-based methods and probabilistic approaches. Thus, the approach provides safety guarantees in a probabilistic sense, while for a vanishing risk, the formal safety guarantees of the set-based methods are inherited. By exploring all available behavior options, our approach solves decision making and longitudinal trajectory planning in one step. The available behavior options are provided by a formal representation of the situation context, which is also used to reduce calculation efforts. Occlusions are resolved using the external perception of infrastructure-mounted sensors. Yet, instead of merging external and ego perception with track-to-track fusion, the information is used in parallel. The motion planning scheme is validated through real-world experiments.

Johannes Müller, Guido Montúfar

We consider the problem of finding the best memoryless stochastic policy for an infinite-horizon partially observable Markov decision process (POMDP) with finite state and action spaces with respect to either the discounted or mean reward criterion. We show that the (discounted) state-action frequencies and the expected cumulative reward are rational functions of the policy, whereby the degree is determined by the degree of partial observability. We then describe the optimization problem as a linear optimization problem in the space of feasible state-action frequencies subject to polynomial constraints that we characterize explicitly. This allows us to address the combinatorial and geometric complexity of the optimization problem using recent tools from polynomial optimization. In particular, we estimate the number of critical points and use the polynomial programming description of reward maximization to solve a navigation problem in a grid world.

Johannes Müller, Marius Zeinhofer

We estimate the error of the Deep Ritz Method for linear elliptic equations. For Dirichlet boundary conditions, we estimate the error when the boundary values are imposed through the boundary penalty method. Our results apply to arbitrary sets of ansatz functions and estimate the error in dependence of the optimization accuracy, the approximation capabilities of the ansatz class and -- in the case of Dirichlet boundary values -- the penalization strength $λ$. To the best of our knowledge, our results are presently the only ones in the literature that treat the case of Dirichlet boundary conditions in full generality, i.e., without a lower order term that leads to coercivity on all of $H^1(Ω)$. Further, we discuss the implications of our results for ansatz classes which are given through ReLU networks and the relation to existing estimates for finite element functions. For high dimensional problems our results show that the favourable approximation capabilities of neural networks for smooth functions are inherited by the Deep Ritz Method.

Martin Herrmann, Aldi Piroli, Jan Strohbeck et al.

Autonomous vehicles need precise knowledge on dynamic objects in their surroundings. Especially in urban areas with many objects and possible occlusions, an infrastructure system based on a multi-sensor setup can provide the required environment model for the vehicles. Previously, we have published a concept of object reference points (e.g. the corners of an object), which allows for generic sensor "plug and play" interfaces and relatively cheap sensors. This paper describes a novel method to additionally incorporate multiple hypotheses for fusing the measurements of the object reference points using an extension to the previously presented Labeled Multi-Bernoulli (LMB) filter. In contrast to the previous work, this approach improves the tracking quality in the cases where the correct association of the measurement and the object reference point is unknown. Furthermore, this paper identifies options based on physical models to sort out inconsistent and unfeasible associations at an early stage in order to keep the method computationally tractable for real-time applications. The method is evaluated on simulations as well as on real scenarios. In comparison to comparable methods, the proposed approach shows a considerable performance increase, especially the number of non-continuous tracks is decreased significantly.

Thomas Griebel, Johannes Müller, Michael Buchholz et al.

Self-assessment is a key to safety and robustness in automated driving. In order to design safer and more robust automated driving functions, the goal is to self-assess the performance of each module in a whole automated driving system. One crucial component in automated driving systems is the tracking of surrounding objects, where the Kalman filter is the most fundamental tracking algorithm. For Kalman filters, some classical online consistency measures exist for self-assessment, which are based on classical probability theory. However, these classical approaches lack the ability to measure the explicit statistical uncertainty within the self-assessment, which is an important quality measure, particularly, if only a small number of samples is available for the self-assessment. In this work, we propose a novel online self-assessment method using subjective logic, which is a modern extension of probabilistic logic that explicitly models the statistical uncertainty. Thus, by embedding classical Kalman filtering into subjective logic, our method additionally features an explicit measure for statistical uncertainty in the self-assessment.

Johannes Müller, Marius Zeinhofer

Recently, progress has been made in the application of neural networks to the numerical analysis of partial differential equations (PDEs). In the latter the variational formulation of the Poisson problem is used in order to obtain an objective function - a regularised Dirichlet energy - that was used for the optimisation of some neural networks. In this notes we use the notion of $Γ$-convergence to show that ReLU networks of growing architecture that are trained with respect to suitably regularised Dirichlet energies converge to the true solution of the Poisson problem. We discuss how this approach generalises to arbitrary variational problems under certain universality assumptions of neural networks and see that this covers some nonlinear stationary PDEs like the $p$-Laplace.

Johannes Müller, Martin Herrmann, Jan Strohbeck et al.

Sensor calibration usually is a time consuming yet important task. While classical approaches are sensor-specific and often need calibration targets as well as a widely overlapping field of view (FOV), within this work, a cooperative intelligent vehicle is used as callibration target. The vehicleis detected in the sensor frame and then matched with the information received from the cooperative awareness messagessend by the coperative intelligent vehicle. The presented algorithm is fully automated as well as sensor-independent, relying only on a very common set of assumptions. Due to the direct registration on the world frame, no overlapping FOV is necessary. The algorithm is evaluated through experiment for four laserscanners as well as one pair of stereo cameras showing a repetition error within the measurement uncertainty of the sensors. A plausibility check rules out systematic errors that might not have been covered by evaluating the repetition error.

Johannes Müller

Residual networks (ResNets) are a deep learning architecture that substantially improved the state of the art performance in certain supervised learning tasks. Since then, they have received continuously growing attention. ResNets have a recursive structure $x_{k+1} = x_k + R_k(x_k)$ where $R_k$ is a neural network called a residual block. This structure can be seen as the Euler discretisation of an associated ordinary differential equation (ODE) which is called a neural ODE. Recently, ResNets were proposed as the space-time approximation of ODEs which are not of this neural type. To elaborate this connection we show that by increasing the number of residual blocks as well as their expressivity the solution of an arbitrary ODE can be approximated in space and time simultaneously by deep ReLU ResNets. Further, we derive estimates on the complexity of the residual blocks required to obtain a prescribed accuracy under certain regularity assumptions.

Noreen Jamil, Johannes Müller, Christof Lutteroth et al.

Many computer programs have graphical user interfaces (GUIs), which need good layout to make efficient use of the available screen real estate. Most GUIs do not have a fixed layout, but are resizable and able to adapt themselves. Constraints are a powerful tool for specifying adaptable GUI layouts: they are used to specify a layout in a general form, and a constraint solver is used to find a satisfying concrete layout, e.g.\ for a specific GUI size. The constraint solver has to calculate a new layout every time a GUI is resized or changed, so it needs to be efficient to ensure a good user experience. One approach for constraint solvers is based on the Gauss-Seidel algorithm and successive over-relaxation (SOR). Our observation is that a solution after resizing or changing is similar in structure to a previous solution. Thus, our hypothesis is that we can increase the computational performance of an SOR-based constraint solver if we reuse the solution of a previous layout to warm-start the solving of a new layout. In this paper we report on experiments to test this hypothesis experimentally for three common use cases: big-step resizing, small-step resizing and constraint change. In our experiments, we measured the solving time for randomly generated GUI layout specifications of various sizes. For all three cases we found that the performance is improved if an existing solution is used as a starting solution for a new layout.