Hamilton-Jacobi Reachability

Least-restrictive safety filters based on Hamilton-Jacobi reachability provide strong safety guarantees by overriding a nominal controller only when the system reaches the boundary of the set of unsafe states defined as a Backward Reachable Tube (BRT). These guarantees, however, rely on the continuous-time nature of the underlying formulation. In practice, robotic systems apply control at discrete sampling intervals, which creates a mismatch where the system may jump into the unsafe BRT between updates, allowing failures that are theoretically avoidable. This work introduces a principled solution based on a one-step expanded BRT that predicts all states capable of reaching the true BRT within a single timestep. By using this expanded boundary as the activation condition for the safety filter, safety interventions occur early enough to ensure correctness under discrete-time execution. We formulate this expanded set as a modified reachability problem and compute it using standard continuous-time solvers.

Hybrid dynamical systems provide a powerful modeling framework for robotic systems, particularly in contact-rich environments. However, ensuring safety and performance in such systems remains challenging due to the intricate coupling between continuous dynamics and discrete mode transitions. In this work, we extend classical Hamilton-Jacobi (HJ) reachability analysis, a formal verification method for continuous-time nonlinear systems, to hybrid dynamical systems. Our framework characterizes safe sets for hybrid systems through a generalized value function defined over both discrete and continuous states while accounting for control constraints and model uncertainty. We additionally provide a numerical algorithm to compute this value function. Building on these safe sets, we propose two different mechanisms to integrate performance objectives. First, we introduce a hybrid least-restrictive safety filter that intervenes on both the discrete and continuous components of a nominal controller only when necessary to avoid unsafe states, thereby preserving nominal behavior whenever possible. Second, we formulate and compute hybrid backward reach-avoid tubes, enabling the simultaneous enforcement of safety and goal-reaching behavior, an extension not previously addressed within hybrid HJ reachability. This enables the synthesis of continuous and discrete control policies that guarantee both safety and task completion. We validate our framework through simulation studies and real-world experiments on a quadrupedal robot, demonstrating its effectiveness in hybrid mode planning and safety-critical applications.

Hamilton-Jacobi (HJ) reachability provides a mathematically rigorous framework for safe control of dynamical systems, but its practical application is bottlenecked by the computational complexity of solving Hamilton-Jacobi-Isaacs variational inequality PDEs in high dimensions. Physics-informed neural networks (PINNs) have recently emerged as a promising alternative to classical mesh-based solvers, yet their performance is highly sensitive to the choice of collocation sampling. In order to learn accurate safety value functions, existing PINNs-based HJ reachability solvers must rely on complex training pipelines and auxiliary supervision. In this work, we propose STEER2REACH (S2R), a PINNs-based HJ reachability solver that requires minimal modification on top of standard PINNs training. S2R's key contribution is a lightweight, low-overhead adaptive collocation sampling distribution constructed by steering forward trajectories using a combination of the optimal control and disturbance signals induced by the current value function, with injected stochastic exploration noise. We demonstrate that despite its simplicity, S2R achieves competitive--and in some cases improved--performance on safety metrics while reducing relative L2 error across a range of reachability benchmarks compared with SoTA MPC-guided HJ reachability solvers, all without requiring multi-stage training or MPC-based supervision.

Robot policies are becoming increasingly general, with vision-language-action (VLA) models enabling a single policy to execute diverse tasks specified in natural language. Safe deployment, however, requires adapting not only to new tasks but also to varying safety requirements across users, environments, and applications. Existing safety filters remain largely constraint-specific and thus must be redesigned or relearned when safety requirements change. In this paper, we investigate language-conditioned safety filtering, in which a Hamilton-Jacobi safety actor and critic are conditioned on language-specified constraints. We evaluate this formulation across pick-and-place, table-wiping, and block-stacking tasks in the vision-based setting, examining its ability to enforce language-specified constraints and transfer to unseen constraint instances within the evaluated constraint families. Our experiments provide evidence that language-conditioned safety filters reduce constraint violations and exhibit partial transfer to unseen constraint instances.

We introduce trajectory-regularized stochastic optimal control (TRSOC), which augments standard stochastic optimal control (SOC) with a Kullback--Leibler (KL) divergence between controlled and reference trajectory distributions. Using Girsanov's theorem, the trajectory KL reduces to a quadratic drift mismatch penalty, yielding a modified running cost that preserves the dynamic programming (DP) structure. We derive the corresponding Hamilton--Jacobi--Bellman (HJB) equation and characterize the optimal policy. In the linear-quadratic (LQ) setting, the formulation admits a closed-form solution with an augmented control cost. Experiments show that the regularization parameter induces a trade-off between performance-driven and reference-preserving behavior, including cases with reference dynamics learned from offline data.

Reachability analysis is central to safety-critical control, robotics, and neural network verification, but classical computational methods, such as Hamilton--Jacobi reachability and set propagation, scale poorly with state dimension. Sampling-based methods have emerged as a promising alternative, often providing finite-sample guarantees that bound the probability-mass left uncovered. However, an explicit account of how the geometry of the initial set, the dynamics, and the sampling law affect the accuracy of the estimator is not fully available in the literature. We study this by casting sampling-based reachable-set recovery as geometric support estimation over a family of problems specified by an initial set, its dynamics, and a sampling law. First, we identify two regularity properties, positive reach of the initial set's complement and Lipschitz continuity of the dynamics, that together make recovery well-posed: a probability-mass coverage guarantee can be upgraded to accuracy $r$ in Hausdorff distance. Second, we bound the resulting sample complexity: recovery is achievable with $\tilde{\mathcal{O}}\big((e^{3LT}/r)^n\big)$ samples, exponential in both the state dimension and the time horizon. Third, we show that neither can be removed: an minimax lower bound of $Ω\big((e^{LT}/r)^n\big)$ holds for every estimator, so the exponential dependence on dimension and the degradation over the horizon are both intrinsic, not artifacts of a particular method. Experiments on nonlinear systems confirm that adversarial sampling improves constants but not the scaling.

Diffusion models offer a powerful framework for sampling from complex probability densities by learning to reverse a noising process. A common approach involves solving for the time-reversed stochastic differential equation (SDE), which requires the score function of the evolving sample distribution. The logarithm of this distribution's density is governed by a Hamilton-Jacobi-Bellman (HJB) type partial differential equation (PDE). However, current methods for solving this PDE, such as PINNs or trajectory-based techniques, often suffer from long training times and significant sensitivity to hyperparameter tuning. In this work, we introduce a novel and efficient solver for the underlying HJB equation based on the functional tensor train (FTT) format. The FTT representation leverages latent low-rank structures to efficiently approximate high-dimensional functions, enabling both model compression and rapid computation. By integrating this efficient representation with a backward-in-time iterative scheme derived from backward stochastic differential equations (BSDEs), we develop a fast, robust and accurate sampling method. Our approach overcomes primary bottlenecks of existing techniques, enabling high-fidelity sampling from challenging target distributions with improved efficiency.

Safe multi-arm motion planning is a challenging problem in robotics due to its high dimensionality, coupled configuration space, and complex collision constraints. Centralized planners are capable of coordinating all arms but often face scalability limitations, restricting applicability in real-time settings. On the other hand, decentralized methods are scalable and recent deep learning-based approaches have shown promising results. However, these depend on accurate behavior prediction or coordination protocols and may fail when other arms act unpredictably. To address these challenges, we introduce a neural Hamilton-Jacobi Reachability (HJR) learning-based approach to approximate a safety value function that captures worst-case inter-arm safety constraints. We further develop a decentralized trajectory optimization framework that uses the learned HJR representation for real-time planning. The proposed method is scalable and data-efficient, generalizes across multi-manipulator systems, and outperforms state-of-the-art baselines on challenging multi-arm motion planning tasks.

Robots deployed in safety-critical environments must execute motions that are simultaneously robust to disturbances and provably safe from collisions. Dynamic Movement Primitives (DMPs) offer inherent stability, temporal flexibility, and efficient trajectory generalization from single demonstrations, but they lack formal safety certificates. Conversely, Hamilton-Jacobi (HJ) Reachability analysis provides a principled framework for computing worst-case safety margins and forward-invariant safe sets, but classical grid-based methods suffer from the curse of dimensionality and are impractical for real-time control. This paper introduces HJ-SafeDMP, a framework that integrates DMPs with learned HJ Reachability-based safety value functions to achieve provably safe, robust, and computationally efficient robot motion. We learn a Control Barrier Value Function (CBVF) from offline demonstration data using a model-free, finite-difference HJ recursion and deploy it as a real-time safety filter via a closed-form control law that modulates the DMP output. Unlike optimization-based CBF-QP approaches, our method achieves safety filtering without online quadratic program solves, preserving the computational efficiency of DMPs. We further incorporate an expectile-based offline learning objective that avoids querying out-of-distribution actions, and a conformal prediction calibration step that provides finite-sample probabilistic safety coverage. Experimental evaluation on a 7-DOF robot manipulator demonstrates that HJ-SafeDMP achieves formal safety guarantees with orders-of-magnitude faster execution than optimization-based baselines, while maintaining the robustness and adaptability of DMPs for human-robot interaction.

Stackelberg differential games (SDGs) provide a powerful framework for hierarchical decision-making in stochastic and continuous-time environments, yet their solution remains computationally challenging due to the complexity of traditional dynamic programming and Hamilton-Jacobi-Bellman-Isaacs (HJBI) methods, especially in high-dimensional systems. This paper proposes an entropy-regularized reinforcement learning (ERRL) approach for linear-quadratic SDGs (LQ-SDGs) within a continuous-time diffusion framework governed by Markovian regime switching. The key innovation lies in deriving exploratory weakly-coupled HJBI equations with entropy regularization, which promotes stochastic policies that actively avoid suboptimal equilibria -- a limitation of classical SDG methods. Neural networks are integrated to approximate regime-dependent value functions and solve high-dimensional partial differential equations (PDEs) efficiently, while a novel sampling technique enhances computational tractability. Numerical results demonstrate the effectiveness of the framework compared to conventional approaches, particularly in escaping suboptimal traps through exploratory policies. The study highlights the critical role of entropy regularization and neural network approximations in achieving robust solutions for hierarchical decision-making problems under abrupt environmental shifts.

A data-driven method is developed for approximating value functions in deterministic optimal control problems with nonlinear control-affine dynamics. The Pontryagin Maximum Principle optimality system is solved from multiple initial conditions to generate training data consisting of values, gradients, and Hessians of the value function, where Hessian information is obtained from a matrix Riccati equation along optimal trajectories. These quantities augment a weighted least-squares regression over sparse polynomial bases on hyperbolic cross index sets, with gradients and Hessians contributing additional linear equations per sample and substantially reducing sample complexity compared to value-only regression. Feedback laws are recovered analytically from the learned value function. In high dimensions, a partial Hessian strategy controls the cost of data generation. The approach is validated on problems of increasing state dimension, where second-order data augmentation is shown to improve approximation accuracy and closed-loop performance, with up to an order-of-magnitude reduction in the number of training samples required relative to lower-order methods.

We introduce $λ$-Reachability, a scalable approach to Hamilton--Jacobi safety analysis for high-dimensional robotic systems. Unlike prior discounted formulations that rely on fixed one-step Bellman updates, $λ$-Reachability employs a stochastic multi-step estimator of the safety value, using a geometrically distributed rollout horizon together with a randomly absorbed terminal. Conceptually analogous to TD($λ$), $λ$-Reachability interpolates between local self-consistency updates and long-horizon max-over-trajectory safety targets via an interpretable horizon-control parameter. Unlike TD($λ$), where the terminal value is always incorporated in learning targets, the terminal safety value in $λ$-Reachability is only used at a probability controlled by parameter $δ$. We formally show that for $δ<1$, the update induces a contraction mapping that allows temporal-difference learning; as $λ\to 1$, the estimator recovers the undiscounted reachability objective. We apply $λ$-Reachability to high-dimensional safety learning problems with both simulated and real humanoid robots under balance and collision avoidance constraints. Experimental results demonstrate that $λ$-Reachability significantly improves both safe-set boundary classification and safety margin estimation compared to single-step temporal-difference baselines.

Emergency collision avoidance under extreme driving conditions demands safety-critical control that accounts for both obstacle proximity and vehicle dynamic stability over a future time horizon, yet existing methods often rely on instantaneous or local safety evaluations. This paper proposes a safe reinforcement learning framework guided by a Hamilton-Jacobi (HJ) reachability based motion safety set that provides forward-looking safety supervision for constrained policy optimization. Specifically, a unified signed safety function is formulated by combining geometric collision margins and chassis stability limits, and is then extended through reachability analysis into a finite-horizon motion safety set that characterizes whether safety can be maintained under future vehicle state evolution. To enable practical computation, the motion safety set is approximated from offline extreme driving data, mitigating the computational burden of grid-based HJ solvers. The learned motion safety set is then embedded as a continuous safety cost into a constrained Markov decision process, and a PID-Lagrangian policy optimization scheme is employed to adaptively regulate the Lagrange multiplier for safety constraint enforcement. Simulation and real-vehicle experiments on low-adhesion obstacle-avoidance scenarios demonstrate that the proposed method achieves higher goal-reaching rates, produces smoother avoidance maneuvers, and maintains larger unified safety margins than baseline methods.

We propose a five-step diagnostic protocol for residual-trained neural HJB-PIDE solvers with control-dependent Lévy jumps, targeting a general failure mode of neural PDE methods: a learned solution can match headline scalar diagnostics while miscomputing an operator inside its training loss. The protocol pairs each neural solve with at least one from-scratch independent reference, decomposes the Hamiltonian into drift, diffusion, compensator, and nonlocal-integral components across a u-grid, and compares the value function and its low-order derivatives over a (t,x) grid before any argmax comparison. Applied to a standard CRRA-Merton-Variance-Gamma benchmark, it isolates a missing 1/2-mixture factor in the neural method's importance-proposal density that scaled the nonlocal integral by exactly half - a textbook signature of a constant proposal scale error, invisible to longer training, grid refinement, and truncation sweeps. With the bug corrected, four references - two finite-difference solvers with disjoint discretizations, the neural solver, and a semi-analytic scalar baseline obtained from CRRA homogeneity - agree on the optimal control to within ~2%. The constant-coefficient CRRA benchmark collapses by homogeneity to a scalar maximization, so the scalar baseline is the efficient method here; the contribution is the protocol, applicable in principle to non-homogeneous and higher-dimensional settings where neural HJB-PIDE solvers are genuinely needed. The episode is a concrete instance of a broader neural-PDE verification failure: pointwise agreement of a learned value or control can coexist with a systematically wrong nonlocal operator, so per-component and surface-level checks are needed before trusting the argmax policy.

This article presents a Hamilton--Jacobi (HJ) reachability framework for a two--satellite collision avoidance problem operating in the same circular orbit, where relative motion is modeled in the radial--tangential--normal (RTN) frame using planar Hill--Clohessy--Wiltshire (HCW) dynamics. We define the target state space as unsafe relative configurations in the orbit plane corresponding to minimum separation requirements consistent with Federal Communications Commission (FCC) orbital standards. The interaction between spacecraft is formulated as a zero--sum differential game, where Player 1 is the controlled satellite and Player 2 is modeled as a bounded adversarial disturbance with unknown intent. We present the HJ formulation and compute backward reachable sets that characterize relative states from which collision cannot be avoided under worst-case disturbances, while states outside this set admit provably collision-free trajectories. These reachable sets are integrated with supervisory hybrid control logic to determine when evasive maneuvers must be initiated, enabling mathematically grounded safety guarantees for scalability.

This work reformulates language generation as a stochastic optimal control problem, providing a unified theoretical perspective to analyze autoregressive and diffusion models and explain their limitations (Efficiency-Fidelity Paradox, Irreversibility Error Propagation, Optimization Tractability and Fidelity) in terms of combination of trajectory singularity, adjoint state vanishing, and gradient absence. To address these issues, we approximate the solution to the Hamilton-Jacobi-Bellman (HJB) equation, yielding an optimal policy that acts as a closed-loop controller. To bypass the intractability of directly solving the HJB PDE, we employ Flow Matching as the optimal trajectory solver within the rectified latent control space. This allows our Manta-LM with Global Integral Operator to approximate the global vector field, effectively realizing a model that simultaneously achieves high-fidelity text generation and efficient, low-cost parallel sampling. Empirically, our method achieves strong performance on language modeling and conditional generation tasks, while exhibiting improved stability, efficiency, and controllability.

Autonomous spacecraft docking requires control policies that simultaneously ensure collision avoidance and target reachability under coupled, high-dimensional translational-rotational dynamics. Hamilton-Jacobi (HJ) reachability provides formal reach-avoid guarantees, but classical solvers are limited to low-dimensional systems. Learning-based approaches have begun to scale HJ analysis, yet they struggle in reach-avoid settings, especially where goal and failure sets are tightly coupled, as in docking. We propose a learning-based Backward Reach-Avoid Tube (BRAT) framework that addresses this challenge by tightly integrating HJ structure with MPC-based supervision. In the offline phase, we train a neural approximation of the HJ value function using PDE-based losses augmented with curriculum-driven MPC supervision, which provides informative value targets and stabilizes training in regions where purely PDE-based methods fail. In the online phase, the learned value function is deployed through two real-time controllers: (i) a value gradient-driven controller, and (ii) a value-function-augmented terminal MPC that explicitly enforces reachability at the horizon. We evaluate the proposed method on a 6D planar docking problem against grid-based ground truth and then scale to the full 13D system. Across both settings, our approach outperforms existing methods in success rate and computational efficiency.

This paper investigates the continuous-time counterpart of the Q-function for entropy-regularized mean-field control (MFC) with controlled common noise, coined as q-function by Jia and Zhou (2023) in the single agent's model. We first show that, under discretely sampled actions, the value function in the exploratory formulation converges to the one in the relaxed control formulation as the time grid refines. Leveraging the relaxed control formulation, we derive the exploratory Hamilton-Jacobi-Bellman (HJB) equation, in which the controlled common noise gives rise to an additional nonlinear functional of policy, rendering the policy iteration intricate. Under certain concavity condition, we establish the existence and uniqueness of the optimal one-step policy iteration via a first-order condition using the partial linear functional derivative with respect to policy. The policy improvement at each iteration is verified by relating to an entropy-regularized optimization problem over the space of policies. In the mean-field setting, we introduce the integrated q-function (Iq-function) defined on the state distribution and the policy, and it is shown that an optimal policy is identified as a two-layer fixed point to the argmax operator of the Iq-function. Finally, we provide the explicit characterization of an optimal policy as a Gaussian distribution in the general linear-quadratic (LQ) setting.

Safety filters have been shown to be effective tools to ensure the safety of control systems with unsafe nominal policies. To address scalability challenges in traditional synthesis methods, learning-based approaches have been proposed for designing safety filters for systems with high-dimensional state and control spaces. However, the inevitable errors in the decisions of these models raise concerns about their reliability and the safety guarantees they offer. This paper presents Adaptive Conformal Filtering (ACoFi), a method that combines learned Hamilton-Jacobi reachability-based safety filters with adaptive conformal inference. Under ACoFi, the filter dynamically adjusts its switching criteria based on the observed errors in its predictions of the safety of actions. The range of possible safety values of the nominal policy's output is used to quantify uncertainty in safety assessment. The filter switches from the nominal policy to the learned safe one when that range suggests it might be unsafe. We show that ACoFi guarantees that the rate of incorrectly quantifying uncertainty in the predicted safety of the nominal policy is asymptotically upper bounded by a user-defined parameter. This gives a soft safety guarantee rather than a hard safety guarantee. We evaluate ACoFi in a Dubins car simulation and a Safety Gymnasium environment, empirically demonstrating that it significantly outperforms the baseline method that uses a fixed switching threshold by achieving higher learned safety values and fewer safety violations, especially in out-of-distribution scenarios.

Autonomous navigation requires planning to reach a goal safely and efficiently in complex and potentially dynamic environments. Graph search-based algorithms are widely adopted due to their generality and theoretical guarantees when equipped with admissible heuristics. However, the computational complexity of graph search grows rapidly with the dimensionality of the search space, often making real-time planning in dynamic environments intractable. In this paper, we combine offline Hamilton-Jacobi (HJ) reachability with online graph search to leverage the complementary strengths of both. Precomputed HJ value functions, used as informative heuristics and proactive safety constraints, amortize online computation of the graph search process. At the same time, graph search enables reachability-based reasoning to be incorporated into online planning, overcoming the long-standing challenge of HJ reachability requiring full knowledge of the environment. Extensive simulation studies and real-world experiments demonstrate that the proposed approach consistently outperforms baseline methods in terms of planning efficiency and navigation safety, in environments with and without human presence.

This paper studies obstacle avoidance under translation invariant dynamics using an avoid-side travel cost Hamilton Jacobi formulation. For running costs that are zero outside an obstacle and strictly negative inside it, we prove that the value function is non-positive everywhere, equals zero exactly outside the avoid set, and is strictly negative exactly on it. Under translation invariance, this yields a reuse principle: the value of any translated obstacle is obtained by translating a single template value function. We show that the pointwise minimum of translated template values exactly characterizes the union of the translated single-obstacle avoid sets and provides a conservative inner certificate of unavoidable collision in clutter. To reduce conservatism, we introduce a blockwise composition framework in which subsets of obstacles are merged and solved jointly. This yields a hierarchy of conservative certificates from singleton reuse to the exact clutter value, together with monotonicity under block merging and an exactness criterion based on the existence of a common clutter avoiding control. The framework is illustrated on a Dubins car example in a repeated clutter field.

Safe navigation of autonomous robots remains one of the core challenges in the field, especially in dynamic and uncertain environments. One prevalent approach is safety filtering based on control barrier functions (CBFs), which are easy to deploy but difficult to design. Motivated by the shortcomings of existing learning- and model-based methods, we propose a simple yet effective neural CBF design method for safe robot navigation in dynamic environments. We employ the idea of a composite CBF, where multiple neural CBFs are combined into a single CBF. Individual CBFs are trained using data generated offline via the Hamilton-Jacobi reachability framework to approximate the optimal safe set for single moving obstacles. Additionally, we use a residual neural architecture, ensuring that the estimated safe set does not intersect with the corresponding failure set. The method is extensively evaluated in simulation experiments for a ground robot and a quadrotor, comparing it against several baseline methods. The proposed method improves success rates by up to 18% over the strongest baseline, while maintaining comparable or lower path lengths and motion times. The method is also demonstrated in hardware experiments for both types of robots.

Safe multi-agent motion planning (MAMP) under task-induced constraints is a critical challenge in robotics. Many real-world scenarios require robots to navigate dynamic environments while adhering to manifold constraints imposed by tasks. For example, service robots must carry cups upright while avoiding collisions with humans or other robots. Despite recent advances in decentralized MAMP for high-dimensional systems, incorporating manifold constraints remains difficult. To address this, we propose a manifold-constrained Hamilton-Jacobi reachability (HJR) learning framework for decentralized MAMP. Our method solves HJR problems under manifold constraints to capture task-aware safety conditions, which are then integrated into a decentralized trajectory optimization planner. This enables robots to generate motion plans that are both safe and task-feasible without requiring assumptions about other agents' policies. Our approach generalizes across diverse manifold-constrained tasks and scales effectively to high-dimensional multi-agent manipulation problems. Experiments show that our method outperforms existing constrained motion planners and operates at speeds suitable for real-world applications. Video demonstrations are available at https://youtu.be/RYcEHMnPTH8 .

Co-optimizing safety and performance in large-scale multi-agent systems remains a fundamental challenge. Existing approaches based on multi-agent reinforcement learning (MARL), safety filtering, or Model Predictive Control (MPC) either lack strict safety guarantees, suffer from conservatism, or fail to scale effectively. We propose MAD-PINN, a decentralized physics-informed machine learning framework for solving the multi-agent state-constrained optimal control problem (MASC-OCP). Our method leverages an epigraph-based reformulation of SC-OCP to simultaneously capture performance and safety, and approximates its solution via a physics-informed neural network. Scalability is achieved by training the SC-OCP value function on reduced-agent systems and deploying them in a decentralized fashion, where each agent relies only on local observations of its neighbours for decision-making. To further enhance safety and efficiency, we introduce an Hamilton-Jacobi (HJ) reachability-based neighbour selection strategy to prioritize safety-critical interactions, and a receding-horizon policy execution scheme that adapts to dynamic interactions while reducing computational burden. Experiments on multi-agent navigation tasks demonstrate that MAD-PINN achieves superior safety-performance trade-offs, maintains scalability as the number of agents grows, and consistently outperforms state-of-the-art baselines.

Safe multi-arm motion planning is a challenging problem in robotics due to its high dimensionality, coupled configuration space, and complex collision constraints. Centralized planners are capable of coordinating all arms but often face scalability limitations, restricting applicability in real-time settings. On the other hand, decentralized methods are scalable and recent deep learning-based approaches have shown promising results. However, these depend on accurate behavior prediction or coordination protocols and may fail when other arms act unpredictably. To address these challenges, we introduce a neural Hamilton-Jacobi Reachability (HJR) learning-based approach to approximate a safety value function that captures worst-case inter-arm safety constraints. We further develop a decentralized trajectory optimization framework that uses the learned HJR representation for real-time planning. The proposed method is scalable and data-efficient, generalizes across multi-manipulator systems, and outperforms state-of-the-art baselines on challenging multi-arm motion planning tasks.

In this paper, we propose a Gaussian Process (GP)-based policy iteration framework for addressing both forward and inverse problems in Hamilton--Jacobi--Bellman (HJB) equations and mean field games (MFGs). Policy iteration is formulated as an alternating procedure between evaluating the value function under a fixed control policy and improving the policy. In our approach, we model the unknown fields using GPs within a policy-iteration framework that converts the nonlinear system into a sequence of linear PDE subproblems. Then, leveraging the linear structure, the updates for the value function and, in the MFG setting, the population density admit explicit representer formulas under linear PDE collocation constraints. The policy is subsequently updated pointwise via a Legendre transform step, which involves a low-dimensional maximization over the control variable. This maximization is explicit for standard quadratic costs. For smooth, strictly convex costs, this pointwise maximization is solved through its first-order optimality condition, whereas in constrained or non-smooth cases, it becomes a low-dimensional constrained maximization problem. To improve convergence, we incorporate the additive Schwarz acceleration as a preconditioning step following each policy update. Numerical experiments demonstrate the effectiveness of the Schwarz acceleration in improving computational efficiency.

We analyze a neural semi-discrete method for high-dimensional first-order Hamilton-Jacobi-Bellman (HJB) equations with known or learned dynamics. Centered differences and an artificial viscosity $Nh=O(h)$ define a monotone operator evaluated through $2d+1$ shifted network queries; policy iteration solves the resulting Bellman equation without a tensor grid. At fixed $h$, the sharp componentwise condition $\max_i|f_i|\le2N$ turns every frozen-policy operator into a nearest-neighbor Markov-chain generator with a policy-independent total jump rate. Uniformization gives whole-space well-posedness for measurable feedbacks, an explicit Poisson-tail bound on the numerical domain of dependence, and boundary-free localization. The representation also yields a posteriori policy-evaluation bounds that account for residual and learned-model errors. A greedy-gap analysis controls inexact policy iteration at fixed $h$; a separate consistency estimate connects the semi-discrete equation to the continuous HJB equation. Experiments reproduce the extremal tail, show rates consistent with $O(\sqrt h)$ and nearly $h$-independent exact-policy-iteration decay, and assess empirical estimator effectivity. A nonsmooth example shows that the continuous residual can miss a non-viscosity solution, whereas the shifted residual detects the defect. Further tests provide a structured interval-verified certificate calibration, an early-budget benefit of policy freezing for bang-bang control, and learned-dynamics diagnostics. A structured nonlinear problem with active compact-control constraints is tested against a manufactured semi-discrete reference through $d=1024$.