Medical cyber-physical systems (MCPS) require computational physiology models that are both expressive and amenable to formal analysis. This paper presents a discrete, executable, and verifiable framework for physiology modeling tailored to MCPS. The framework models physiological systems as interacting components with finite state spaces and explicitly separates state, action, and impact to represent causal interactions. System evolution follows deterministic, staged execution semantics, yielding a direct interpretation as a finite-state transition system. Pathophysiological conditions are represented as localized modifications of a healthy baseline model, enabling systematic structural reuse. We evaluate the framework using two case studies. For glucose homeostasis, the model captures key physiological dynamics and generalizes from a small set of calibration scenarios to diverse multi-meal conditions; Type 1 and Type 2 Diabetes behaviors are derived via local component modifications. For respiratory gas exchange, a qualitative evaluation shows that the framework tracks characteristic oxygen and carbon dioxide cycle behavior. Overall, the results indicate that the proposed approach is a promising step toward interpretable, modular, and formally analyzable physiology modeling for MCPS.

Deep neural networks (DNNs) have achieved remarkable empirical success, yet the absence of a principled theoretical foundation continues to hinder their systematic development. In this survey, we present differential equations as a theoretical foundation for understanding, analyzing, and improving DNNs. We organize the discussion around three guiding questions: i) how differential equations offer a principled understanding of DNN architectures, ii) how tools from differential equations can be used to improve DNN performance in a principled way, and iii) what real-world applications benefit from grounding DNNs in differential equations. We adopt a two-fold perspective spanning the model level, which interprets the whole DNN as a differential equation, and the layer level, which models individual DNN components as differential equations. From these two perspectives, we review how this framework connects model design, theoretical analysis, and performance improvement. We further discuss real-world applications, as well as key challenges and opportunities for future research.

End-to-end deep neural networks have achieved remarkable success across various domains but are often criticized for their lack of interpretability. While post hoc explanation methods attempt to address this issue, they often fail to accurately represent these black-box models, resulting in misleading or incomplete explanations. To overcome these challenges, we propose an inherently transparent model architecture called Neural Probabilistic Circuits (NPCs), which enable compositional and interpretable predictions through logical reasoning. In particular, an NPC consists of two modules: an attribute recognition model, which predicts probabilities for various attributes, and a task predictor built on a probabilistic circuit, which enables logical reasoning over recognized attributes to make class predictions. To train NPCs, we introduce a three-stage training algorithm comprising attribute recognition, circuit construction, and joint optimization. Moreover, we theoretically demonstrate that an NPC's error is upper-bounded by a linear combination of the errors from its modules. To further demonstrate the interpretability of NPC, we provide both the most probable explanations and the counterfactual explanations. Empirical results on four benchmark datasets show that NPCs strike a balance between interpretability and performance, achieving results competitive even with those of end-to-end black-box models while providing enhanced interpretability.

Safe online reinforcement learning requires policies to respect safety constraints while maintaining smooth optimization dynamics. Existing approaches typically rely on either strict safety enforcement via action interventions, which introduce discontinuities in system interaction and learning, or soft safety constraint formulations, which preserve smooth learning but provide limited safety assurance. We propose AutoSafe, a safety-aware policy architecture that integrates structured safety monitoring and intervention directly into the action generation process. This design enables smooth, risk-dependent transitions between performance-driven and safety-preserving behaviors, resulting in continuous online interaction and learning dynamics. Empirical results across a suite of continuous-control benchmarks demonstrate strong safety enforcement without sacrificing learning smoothness. We further validate AutoSafe on a physical cart-pole system, highlighting its practical effectiveness for safe online learning in the real world.

Autonomous systems increasingly rely on machine-learning (ML) components for safety-critical tasks such as perception and control in autonomous vehicles (AVs). While ML enables essential capabilities, it inevitably exhibits long-tail faults that make it unsuitable for safety-critical tasks. Runtime assurance (RTA) mitigates this issue by pairing ML components with verifiable safety monitors, e.g., Control Simplex and Perception Simplex architectures. However, the limited performance of safety monitors remains a major bottleneck. The Synergistic Simplex (SS) architecture improves system performance by enabling bidirectional integration between ML components and safety monitors while preserving formal safety guarantees. The key innovation here is allowing safety monitors to use ML outputs, which is typically prohibited in RTA systems. We formally derive conditions under which this integration preserves safety and demonstrate the performance benefits. We present the design, analysis, and evaluation of SS for AV obstacle detection.

This paper presents a runtime learning framework for quadruped robots, enabling them to learn and adapt safely in dynamic wild environments. The framework integrates sensing, navigation, and control, forming a closed-loop system for the robot. The core novelty of this framework lies in two interactive and complementary components within the control module: the high-performance (HP)-Student and the high-assurance (HA)-Teacher. HP-Student is a deep reinforcement learning (DRL) agent that engages in self-learning and teaching-to-learn to develop a safe and high-performance action policy. HA-Teacher is a simplified yet verifiable physics-model-based controller, with the role of teaching HP-Student about safety while providing a backup for the robot's safe locomotion. HA-Teacher is innovative due to its real-time physics model, real-time action policy, and real-time control goals, all tailored to respond effectively to real-time wild environments, ensuring safety. The framework also includes a coordinator who effectively manages the interaction between HP-Student and HA-Teacher. Experiments involving a Unitree Go2 robot in Nvidia Isaac Gym and comparisons with state-of-the-art safe DRLs demonstrate the effectiveness of the proposed runtime learning framework.

We present VISAT, a novel open dataset and benchmarking suite for evaluating model robustness in the task of traffic sign recognition with the presence of visual attributes. Built upon the Mapillary Traffic Sign Dataset (MTSD), our dataset introduces two benchmarks that respectively emphasize robustness against adversarial attacks and distribution shifts. For our adversarial attack benchmark, we employ the state-of-the-art Projected Gradient Descent (PGD) method to generate adversarial inputs and evaluate their impact on popular models. Additionally, we investigate the effect of adversarial attacks on attribute-specific multi-task learning (MTL) networks, revealing spurious correlations among MTL tasks. The MTL networks leverage visual attributes (color, shape, symbol, and text) that we have created for each traffic sign in our dataset. For our distribution shift benchmark, we utilize ImageNet-C's realistic data corruption and natural variation techniques to perform evaluations on the robustness of both base and MTL models. Moreover, we further explore spurious correlations among MTL tasks through synthetic alterations of traffic sign colors using color quantization techniques. Our experiments focus on two major backbones, ResNet-152 and ViT-B/32, and compare the performance between base and MTL models. The VISAT dataset and benchmarking framework contribute to the understanding of model robustness for traffic sign recognition, shedding light on the challenges posed by adversarial attacks and distribution shifts. We believe this work will facilitate advancements in developing more robust models for real-world applications in autonomous driving and cyber-physical systems.

This paper introduces the Real-DRL framework for safety-critical autonomous systems, enabling runtime learning of a deep reinforcement learning (DRL) agent to develop safe and high-performance action policies in real plants while prioritizing safety. The Real-DRL consists of three interactive components: a DRL-Student, a PHY-Teacher, and a Trigger. The DRL-Student is a DRL agent that innovates in the dual self-learning and teaching-to-learn paradigm and the safety-status-dependent batch sampling. On the other hand, PHY-Teacher is a physics-model-based design of action policies that focuses solely on safety-critical functions. PHY-Teacher is novel in its real-time patch for two key missions: i) fostering the teaching-to-learn paradigm for DRL-Student and ii) backing up the safety of real plants. The Trigger manages the interaction between the DRL-Student and the PHY-Teacher. Powered by the three interactive components, the Real-DRL can effectively address safety challenges that arise from the unknown unknowns and the Sim2Real gap. Additionally, Real-DRL notably features i) assured safety, ii) automatic hierarchy learning (i.e., safety-first learning and then high-performance learning), and iii) safety-informed batch sampling to address the experience imbalance caused by corner cases. Experiments with a real quadruped robot, a quadruped robot in Nvidia Isaac Gym, and a cart-pole system, along with comparisons and ablation studies, demonstrate the Real-DRL's effectiveness and unique features.

The rapid advancements in air mobility vehicles is paving the way for air taxis to become a viable mode of public transportation. The next technological frontier for air taxis is fully autonomous operation. Developing safe and efficient autonomous control for air taxis presents greater challenges than for ground vehicles due to the inherent instability of aerial vehicles. Therefore, simulation solutions for autonomous air taxis will play a crucial role in accelerating their development and eventual safe deployment. This paper introduces AirTaxiSim, an end to end simulation framework for autonomous air taxis. AirTaxiSim is designed to model and analyze the complexities of autonomous air taxi operations in dynamic and cluttered urban environments. AirTaxiSim integrates highfidelity physical models of vertical take-off and landing air vehicles in photo-realistic urban environments. The primary purpose of AirTaxiSim is to evaluate the safety, performance, and efficiency of autonomous air taxi services, across a variety of scenarios, including dangerous edge cases. AirTaxiSim also provides methods for generating datasets and establishing benchmarks for autonomous air taxis. This paper describes the simulator's construction, functionalities, and some of the use cases, providing critical information to facilitate its use in advancing autonomy in aerial vehicles. https://github.com/CPS-IL/airtaxisim

This paper proposes the runtime learning machine for safety-critical learning-enabled cyber-physical systems (CPS). The learning machine has three interactive components: a high-performance (HP)-Student, a high-assurance (HA)-Teacher, and a Coordinator. The HP-Student is a high-performance but not fully verified Phy-DRL (physics-regulated deep reinforcement learning) agent that performs runtime learning in real CPS, using real-time sensor data from real-time physical environments. On the other hand, HA-Teacher is a verified but simplified design, focusing on safety-critical functions only. As a complementary, HA-Teacher's novelty lies in real-time patch for two missions: i) correcting unsafe learning of HP-Student, and ii) backing up safety. The Coordinator manages the interaction between HP-Student and HA-Teacher. Powered by the three interactive components, the runtime learning machine notably features i) assuring lifetime safety (i.e., safety guarantee in any runtime learning stage), ii) tolerating unknown unknowns, iii) addressing Sim2Real gap, and iv) automatic hierarchy learning (i.e., safety-first learning, and then high-performance learning). Experiments involving a cart-pole system and two quadruped robots, as well as comparisons with state-of-the-art safe DRL, fault-tolerant DRL, and approaches for addressing Sim2Real gap, demonstrate the learning machine's effectiveness and unique features.

Purely data-driven deep neural networks (DNNs) applied to physical engineering systems can infer relations that violate physics laws, thus leading to unexpected consequences. To address this challenge, we propose a physics-knowledge-enhanced DNN framework called Phy-Taylor, accelerating learning-compliant representations with physics knowledge. The Phy-Taylor framework makes two key contributions; it introduces a new architectural physics-compatible neural network (PhN) and features a novel compliance mechanism, which we call physics-guided neural network (NN) editing. The PhN aims to directly capture nonlinear physical quantities, such as kinetic energy, electrical power, and aerodynamic drag force. To do so, the PhN augments NN layers with two key components: 1) monomials of the Taylor series for capturing physical quantities and 2) a suppressor for mitigating the influence of noise. The NN editing mechanism further modifies network links and activation functions consistently with physics knowledge. As an extension, we also propose a self-correcting Phy-Taylor framework for safety-critical control of autonomous systems, which introduces two additional capabilities: 1) safety relationship learning and 2) automatic output correction when safety violations occur. Through experiments, we show that Phy-Taylor features considerably fewer parameters and a remarkably accelerated training process while offering enhanced model robustness and accuracy.

This paper proposes the SeC-Learning Machine: Simplex-enabled safe continual learning for safety-critical autonomous systems. The SeC-learning machine is built on Simplex logic (that is, “using simplicity to control complexity”) and physics-regulated deep reinforcement learning (Phy-DRL). The SeC-learning machine thus constitutes HP (high performance)-Student, HA (high assurance)-Teacher, and Coordinator. Specifically, the HP-Student is a pre-trained high-performance but not fully verified Phy-DRL, continuing to learn in a real plant to tune the action policy to be safe. In contrast, the HA-Teacher is a mission-reduced, physics-model-based, and verified design. As a complementary, HA-Teacher has two missions: backing up safety and correcting unsafe learning. The Coordinator triggers the interaction and the switch between HP-Student and HA-Teacher. Powered by the three interactive components, the SeC-learning machine can i) assure lifetime safety (i.e., safety guarantee in any continual-learning stage, regardless of HP-Student's success or convergence), ii) address the Sim2Real gap, and iii) learn to tolerate unknown unknowns in real plants. The experiments on a cart-pole system and a real quadruped robot demonstrate the distinguished features of the SeC-learning machine, compared with continual learning built on state-of-the-art safe DRL frameworks with approaches to addressing the Sim2Real gap.

This paper proposes the Phy-DRL: a physics-regulated deep reinforcement learning (DRL) framework for safety-critical autonomous systems. The Phy-DRL has three distinguished invariant-embedding designs: i) residual action policy (i.e., integrating data-driven-DRL action policy and physics-model-based action policy), ii) automatically constructed safety-embedded reward, and iii) physics-model-guided neural network (NN) editing, including link editing and activation editing. Theoretically, the Phy-DRL exhibits 1) a mathematically provable safety guarantee and 2) strict compliance of critic and actor networks with physics knowledge about the action-value function and action policy. Finally, we evaluate the Phy-DRL on a cart-pole system and a quadruped robot. The experiments validate our theoretical results and demonstrate that Phy-DRL features guaranteed safety compared to purely data-driven DRL and solely model-based design while offering remarkably fewer learning parameters and fast training towards safety guarantee.

This paper proposes and evaluates a new safety concept called backup plan safety for path planning of autonomous vehicles under mission uncertainty using model predictive control (MPC). Backup plan safety is defined as the ability to complete an alternative mission when the primary mission is aborted. To include this new safety concept in control problems, we formulate a feasibility maximization problem aiming to maximize the feasibility of the primary and alternative missions. The feasibility maximization problem is based on multi-objective MPC, where each objective (cost function) is associated with a different mission and balanced by a weight vector. Furthermore, the feasibility maximization problem incorporates additional control input horizons toward the alternative missions on top of the control input horizon toward the primary mission, denoted as multihorizon inputs, to evaluate the cost for each mission. We develop the backup plan constrained MPC algorithm, which designs the weight vector that ensures asymptotic stability of the closed-loop system, and generates the optimal control input by solving the feasibility maximization problem with computational efficiency. The performance of the proposed algorithm is validated through simulations of an unmanned aerial vehicle path planning problem.

Advances in deep learning have revolutionized cyber-physical applications, including the development of autonomous vehicles. However, real-world collisions involving autonomous control of vehicles have raised significant safety concerns regarding the use of deep neural networks (DNNs) in safety-critical tasks, particularly perception. The inherent unverifiability of DNNs poses a key challenge in ensuring their safe and reliable operation. In this work, we propose perception simplex (PS$$ \mathcal{PS} $$), a fault-tolerant application architecture designed for obstacle detection and collision avoidance. We analyse an existing LiDAR-based classical obstacle detection algorithm to establish strict bounds on its capabilities and limitations. Such analysis and verification have not been possible for deep learning-based perception systems yet. By employing verifiable obstacle detection algorithms, PS$$ \mathcal{PS} $$ identifies obstacle existence detection faults in the output of unverifiable DNN-based object detectors. When faults with potential collision risks are detected, appropriate corrective actions are initiated. Through extensive analysis and software-in-the-loop simulations, we demonstrate that PS$$ \mathcal{PS} $$ provides deterministic fault tolerance against obstacle existence detection faults, establishing a robust safety guarantee. The traditional autonomous driving system (mission layer) is monitored at runtime by the safety layer, which includes verifiable algorithms only. Utilizing the detectability model for such perception algorithms, safety layer provides deterministic fault detection and collision avoidance properties. This framework presents a promising approach towards the end goal of verifiable end-to-end safety in autonomous vehicles. Detectability model provides deterministic translation between safety policies, sensors and algorithm parameters and safety guarantees. Such a model can only be devised for fully analysable and verifiable solutions, a requirement not met by deep neural networks. image

Perception, Planning, and Control form the essential components of autonomy in advanced air mobility. This work advances the holistic integration of these components to enhance the performance and robustness of the complete cyber-physical system. We adapt Perception Simplex, a system for verifiable collision avoidance amidst obstacle detection faults, to the vertical landing maneuver for autonomous air mobility vehicles. We improve upon this system by replacing static assumptions of control capabilities with dynamic confirmation, i.e., real-time confirmation of control limitations of the system, ensuring reliable fulfillment of safety maneuvers and overrides, without dependence on overly pessimistic assumptions. Parameters defining control system capabilities and limitations, e.g., maximum deceleration, are continuously tracked within the system and used to make safety-critical decisions. We apply these techniques to propose a verifiable collision avoidance solution for autonomous aerial mobility vehicles operating in cluttered and potentially unsafe environments.

The paper discusses algorithmic priority inversion in mission-critical machine inference pipelines used in modern neural-network-based perception subsystems and describes a solution to mitigate its effect. In general, priority inversion occurs in computing systems when computations that are "less important" are performed together with or ahead of those that are "more important." Significant priority inversion occurs in existing machine inference pipelines when they do not differentiate between critical and less critical data. We describe a framework to resolve this problem and demonstrate that it improves a perception system's ability to react to critical inputs, while at the same time reducing platform cost.

Learning-based solutions have enabled incredible capabilities for autonomous systems. Autonomous vehicles, both aerial and ground, rely on DNN for various integral tasks, including perception. The efficacy of supervised learning solutions hinges on the quality of the training data. Discrepancies between training data and operating conditions result in faults that can lead to catastrophic incidents. However, collecting vast amounts of context-sensitive data, with broad coverage of possible operating environments, is prohibitively difficult. Synthetic data generation techniques for DNN allow for the easy exploration of diverse scenarios. However, synthetic data generation solutions for aerial vehicles are still lacking. This work presents a data augmentation framework for aerial vehicle's perception training, leveraging photorealistic simulation integrated with high-fidelity vehicle dynamics. Safe landing is a crucial challenge in the development of autonomous air taxis, therefore, landing maneuver is chosen as the focus of this work. With repeated simulations of landing in varying scenarios we assess the landing performance of the VTOL type UAV and gather valuable data. The landing performance is used as the objective function to optimize the DNN through retraining. Given the high computational cost of DNN retraining, we incorporated Bayesian Optimization in our framework that systematically explores the data augmentation parameter space to retrain the best-performing models. The framework allowed us to identify high-performing data augmentation parameters that are consistently effective across different landing scenarios. Utilizing the capabilities of this data augmentation framework, we obtained a robust perception model. The model consistently improved the perception-based landing success rate by at least 20 different lighting and weather conditions.

Autonomous air taxis are poised to revolutionize urban mass transportation, however, ensuring their safety and reliability remains an open challenge. Validating autonomy solutions on air taxis in the real world presents complexities, risks, and costs that further convolute this challenge. Verification and Validation (V V) frameworks play a crucial role in the design and development of highly reliable systems by formally verifying safety properties and validating algorithm behavior across diverse operational scenarios. Advancements in high-fidelity simulators have significantly enhanced their capability to emulate real-world conditions, encouraging their use for validating autonomous air taxi solutions, especially during early development stages. This evolution underscores the growing importance of simulation environments, not only as complementary tools to real-world testing but as essential platforms for evaluating algorithms in a controlled, reproducible, and scalable manner. This work presents a V V framework for a vision-based landing system for air taxis with vertical take-off and landing (VTOL) capabilities. Specifically, we use Verse, a tool for formal verification, to model and verify the safety of the system by obtaining and analyzing the reachable sets. To conduct this analysis, we utilize a photorealistic simulation environment. The simulation environment, built on Unreal Engine, provides realistic terrain, weather, and sensor characteristics to emulate real-world conditions with high fidelity. To validate the safety analysis results, we conduct extensive scenario-based testing to assess the reachability set and robustness of the landing algorithm in various conditions. This approach showcases the representativeness of high-fidelity simulators, offering an effective means to analyze and refine algorithms before real-world deployment.