# A Physics-Informed Machine Learning Framework for Safe and Optimal Control of Autonomous Systems

arXiv: https://arxiv.org/abs/2502.11057 · HTML: https://mlsanity.com/paper/2502.11057/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 40 references that are in the map.
- First posted on arXiv 16 Feb 2025.

## Builds on (most cited first, 22 of 22)

- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Constrained Policy Optimization](https://mlsanity.com/paper/1705.10528/) (2017)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [DeepReach: A Deep Learning Approach to High-Dimensional Reachability](https://mlsanity.com/paper/2011.02082/) (2020)
- [Information Theoretic Model Predictive Control: Theory and Applications to Autonomous Driving](https://mlsanity.com/paper/1707.02342/) (2017)
- [The Safety Filter: A Unified View of Safety-Critical Control in Autonomous Systems](https://mlsanity.com/paper/2309.05837/) (2023)
- [Algorithms for Overcoming the Curse of Dimensionality for Certain Hamilton-Jacobi Equations Arising in Control Theory and Elsewhere](https://mlsanity.com/paper/1605.01799/) (2016)
- [Solving Stabilize-Avoid Optimal Control via Epigraph Form and Deep Reinforcement Learning](https://mlsanity.com/paper/2305.14154/) (2023)
- [Verification of Neural Reachable Tubes via Scenario Optimization and Conformal Prediction](https://mlsanity.com/paper/2312.08604/) (2023)
- [Solving Minimum-Cost Reach Avoid using Reinforcement Learning](https://mlsanity.com/paper/2410.22600/) (2024)
- [On Safety and Liveness Filtering Using Hamilton-Jacobi Reachability Analysis](https://mlsanity.com/paper/2312.15347/) (2023)
- [Learning a Formally Verified Control Barrier Function in Stochastic Environment](https://mlsanity.com/paper/2403.19332/) (2024)
- [Exact Imposition of Safety Boundary Conditions in Neural Reachable Tubes](https://mlsanity.com/paper/2404.00814/) (2024)
- [Semi-Supervised Safe Visuomotor Policy Synthesis using Barrier Certificates](https://mlsanity.com/paper/2409.12616/) (2024)
- [Safe Approximate Dynamic Programming Via Kernelized Lipschitz Estimation](https://mlsanity.com/paper/1907.02151/) (2019)
- [Multi-Agent Path Integral Control for Interaction-Aware Motion Planning in Urban Canals](https://mlsanity.com/paper/2302.06547/) (2023)
- [Cooptimizing Safety and Performance with a Control-Constrained Formulation](https://mlsanity.com/paper/2409.06696/) (2024)
- [Collision Cone Control Barrier Functions: Experimental Validation on UGVs for Kinematic Obstacle Avoidance](https://mlsanity.com/paper/2310.10839/) (2023)

## Built on by (most cited first, 12 of 12)

- [V-OCBF: Learning Safety Filters from Offline Data via Value-Guided Offline Control Barrier Functions](https://mlsanity.com/paper/2512.10822/) (2025)
- [CP-NCBF: A Conformal Prediction-based Approach to Synthesize Verified Neural Control Barrier Functions](https://mlsanity.com/paper/2503.17395/) (2025)
- [Safe Flow Q-Learning: Offline Safe Reinforcement Learning with Reachability-Based Flow Policies](https://mlsanity.com/paper/2603.15136/) (2026)
- [Certified Approximate Reachability (CARe): Formal Error Bounds on Deep Learning of Reachable Sets](https://mlsanity.com/paper/2503.23912/) (2025)
- [Safety Certification in the Latent space using Control Barrier Functions and World Models](https://mlsanity.com/paper/2507.13871/) (2025)
- [Epigraph-Guided Flow Matching for Safe and Performant Offline Reinforcement Learning](https://mlsanity.com/paper/2602.08054/) (2026)
- [Neural Control Barrier Functions from Physics Informed Neural Networks](https://mlsanity.com/paper/2504.11045/) (2025)
- [Data-augmented Learning of Geodesic Distances in Irregular Domains through Soner Boundary Conditions](https://mlsanity.com/paper/2503.04579/) (2025)
- [RISE: Robust Imitation through Stochastic Encoding](https://mlsanity.com/paper/2503.12243/) (2025)
- [MAD-PINN: A Decentralized Physics-Informed Machine Learning Framework for Safe and Optimal Multi-Agent Control](https://mlsanity.com/paper/2509.23960/) (2025)
- [Safe-SDL:Establishing Safety Boundaries and Control Mechanisms for AI-Driven Self-Driving Laboratories](https://mlsanity.com/paper/2602.15061/) (2026)
- [ShieldVLA: Feasibility-Aware Safety Alignment for Vision-Language-Action Models](https://mlsanity.com/paper/2609.13231/) (2026)

## Abstract (arXiv)

As autonomous systems become more ubiquitous in daily life, ensuring high performance with guaranteed safety is crucial. However, safety and performance could be competing objectives, which makes their co-optimization difficult. Learning-based methods, such as Constrained Reinforcement Learning (CRL), achieve strong performance but lack formal safety guarantees due to safety being enforced as soft constraints, limiting their use in safety-critical settings. Conversely, formal methods such as Hamilton-Jacobi (HJ) Reachability Analysis and Control Barrier Functions (CBFs) provide rigorous safety assurances but often neglect performance, resulting in overly conservative controllers. To bridge this gap, we formulate the co-optimization of safety and performance as a state-constrained optimal control problem, where performance objectives are encoded via a cost function and safety requirements are imposed as state constraints. We demonstrate that the resultant value function satisfies a Hamilton-Jacobi-Bellman (HJB) equation, which we approximate efficiently using a novel physics-informed machine learning framework. In addition, we introduce a conformal prediction-based verification strategy to quantify the learning errors, recovering a high-confidence safety value function, along with a probabilistic error bound on performance degradation. Through several case studies, we demonstrate the efficacy of the proposed framework in enabling scalable learning of safe and performant controllers for complex, high-dimensional autonomous systems.