Albert Chern

Zhuoyuan Wang, Albert Chern, Yorie Nakahira · cmu

Accurate estimation of long-term risk is essential for the design and analysis of stochastic dynamical systems. Existing risk quantification methods typically rely on extensive datasets involving risk events observed over extended time horizons, which can be prohibitively expensive to acquire. Motivated by this gap, we propose an efficient method for learning long-term risk probabilities using short-term samples with limited occurrence of risk events. Specifically, we establish that four distinct classes of long-term risk probabilities are characterized by specific partial differential equations (PDEs). Using this characterization, we introduce a physics-informed learning framework that combines empirical data with physics information to infer risk probabilities. We then analyze the theoretical properties of this framework in terms of generalization and convergence. Through numerical experiments, we demonstrate that our framework not only generalizes effectively beyond the sampled states and time horizons but also offers additional benefits such as improved sample efficiency, rapid online inference capabilities under changing system dynamics, and stable computation of probability gradients. These results highlight how embedding PDE constraints, which contain explicit gradient terms and inform how risk probabilities depend on state, time horizon, and system parameters, improves interpolation and generalization between/beyond the available data.

Albert Chern

Perfectly Matched Layer (PML) is a widely adopted non-reflecting boundary treatment for wave simulations. Reducing numerical reflections from a discretized PML has been a long lasting challenge. This paper presents a new discrete PML for the multi-dimensional scalar wave equation which produces no numerical reflection at all. The reflectionless discrete PML is discovered through a straightforward derivation using Discrete Complex Analysis. The resulting PML takes an easily-implementable finite difference form with compact stencil. In practice, the discrete waves are damped exponentially in the PML, and the error due to domain truncation is maintained at machine zero by a moderately thick PML. The numerical stability of the proposed PML is also demonstrated.

Zhuoyuan Wang, Haoming Jing, Christian Kurniawan et al. · cmu

This paper addresses the design of safety certificates for stochastic systems, with a focus on ensuring long-term safety through fast real-time control. In stochastic environments, set invariance-based methods that restrict the probability of risk events in infinitesimal time intervals may exhibit significant long-term risks due to cumulative uncertainties/risks. On the other hand, reachability-based approaches that account for the long-term future may require prohibitive computation in real-time decision making. To overcome this challenge involving stringent long-term safety vs. computation tradeoffs, we first introduce a novel technique termed `probabilistic invariance'. This technique characterizes the invariance conditions of the probability of interest. When the target probability is defined using long-term trajectories, this technique can be used to design myopic conditions/controllers with assured long-term safe probability. Then, we integrate this technique into safe control and learning. The proposed control methods efficiently assure long-term safety using neural networks or model predictive controllers with short outlook horizons. The proposed learning methods can be used to guarantee long-term safety during and after training. Finally, we demonstrate the performance of the proposed techniques in numerical simulations.

David Palmer, Dmitriy Smirnov, Stephanie Wang et al.

Recent techniques have been successful in reconstructing surfaces as level sets of learned functions (such as signed distance fields) parameterized by deep neural networks. Many of these methods, however, learn only closed surfaces and are unable to reconstruct shapes with boundary curves. We propose a hybrid shape representation that combines explicit boundary curves with implicit learned interiors. Using machinery from geometric measure theory, we parameterize currents using deep networks and use stochastic gradient descent to solve a minimal surface problem. By modifying the metric according to target geometry coming, e.g., from a mesh or point cloud, we can use this approach to represent arbitrary surfaces, learning implicitly defined shapes with explicitly defined boundary curves. We further demonstrate learning families of shapes jointly parameterized by boundary curves and latent codes.

Xi Wang, Albert Chern, Marc Alexa

Video-based glint-free eye tracking commonly estimates gaze direction based on the pupil center. The boundary of the pupil is fitted with an ellipse and the euclidean center of the ellipse in the image is taken as the center of the pupil. However, the center of the pupil is generally not mapped to the center of the ellipse by the projective camera transformation. This error resulting from using a point that is not the true center of the pupil directly affects eye tracking accuracy. We investigate the underlying geometric problem of determining the center of a circular object based on its projective image. The main idea is to exploit two concentric circles -- in the application scenario these are the pupil and the iris. We show that it is possible to computed the center and the ratio of the radii from the mapped concentric circles with a direct method that is fast and robust in practice. We evaluate our method on synthetically generated data and find that it improves systematically over using the center of the fitted ellipse. Apart from applications of eye tracking we estimate that our approach will be useful in other tracking applications.