As an example, consider the problem of tracking an autonomous robot with an unknown starting position in a d d grid (Figure 1). Suppose the agent's policy is known, and an observer sees that the agent moved a step without colliding into a wall. This information indicates how the observer should update their beliefs about the agent's position. Tracking these belief states can be challenging when they are either continuous or too large to enumerate (Solinas et al., 2023)--even when the agent's policy and the environment dynamics are known. A common approach frames belief state modeling as a Bayesian filtering problem in which a posterior is maintained and updated with each new observation. Classical Bayesian filters, such as the Kalman Filter (Kalman, 1960) and its nonlinear variants (e.g., Extended and Unscented Kalman Filters (Sorenson, 1985; Julier & Uhlmann, 2004)), assume that the underlying distributions are unimodal and approximately Gaussian. While computationally efficient, this limits their applicability in settings that do not satisfy these assumptions.

This paper presents a novel stochastic barrier function (SBF) framework for safety analysis of stochastic systems based on piecewise (PW) functions. We first outline a general formulation of PW-SBFs. Then, we focus on PW-Constant (PWC) SBFs and show how their simplicity yields computational advantages for general stochastic systems. Specifically, we prove that synthesis of PWC-SBFs reduces to a minimax optimization problem. Then, we introduce three efficient algorithms to solve this problem, each offering distinct advantages and disadvantages. The first algorithm is based on dual linear programming (LP), which provides an exact solution to the minimax optimization problem. The second is a more scalable algorithm based on iterative counter-example guided synthesis, which involves solving two smaller LPs. The third algorithm solves the minimax problem using gradient descent, which admits even better scalability. We provide an extensive evaluation of these methods on various case studies, including neural network dynamic models, nonlinear switched systems, and high-dimensional linear systems. Our benchmarks demonstrate that PWC-SBFs outperform state-of-the-art methods, namely sum-of-squares and neural barrier functions, and can scale to eight dimensional systems.

We consider the problem of using scientific machine learning (SciML) to predict solutions of high Mach fluid flows over irregular geometries. In this setting, data is limited, and so it is desirable for models to perform well in the low-data setting. We show that the neural basis function (NBF), which learns a basis of behavior modes from the data and then uses this basis to make predictions, is more effective than a basis-unaware baseline model. In addition, we identify continuing challenges in the space of predicting solutions for this type of problem.

The Neurobiofeedback machine gained popularity for its non-invasive and quantitative approach to behavior regulation, but its legitimacy remains in question by pediatricians, therapists, and other professionals. In academic-sounding terms, this machine (which I'll be abbreviating as NBF from now on) is built on the concept of feedback therapy, which exploits our ability to exert and/or regain control over physiological aspects in our body. NBF is a type of Brain-Computer Interface (BCI) machine that senses your brain wave activity in different ways (usually involving hardware-software interaction) and rewards you with an auditory or visual stimulus when your brain wave's frequency matches the desired frequency. This comes from the scientific notion that brain rhythms correspond to certain cognitive states. By "mind games", the'auditory or visual stimulus' I mentioned last paragraph usually comes in the form of a game.