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.
arXiv:2608. 11480v1 Announce Type: cross Abstract: 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.
The paper introduces a mesh‑free policy iteration framework that blends classical dynamic programming with physics‑informed neural networks (PINNs) to solve high‑dimensional, nonconvex Hamilton–Jacobi–Isaacs (HJI) equations. The method alternates between solving linear second‑order PDEs under fixed feedback policies and updating controls via pointwise minimax optimization using automatic differentiation. The authors prove local uniform convergence of the value function iterates to the unique viscosity solution under standard Lipschitz and uniform ellipticity assumptions, and demonstrate the approach’s accuracy and scalability in two‑, five‑, and ten‑dimensional stochastic games, outperforming direct PINN solvers.
arXiv:2509. 23960v2 Announce Type: replace-cross Abstract: Co-optimizing safety and performance in large-scale multi-agent systems remains a fundamental challenge.
arXiv:2508. 01718v2 Announce Type: replace Abstract: We develop a physics-informed policy-iteration method for stationary second-order Hamilton--Jacobi--Bellman equations arising in continuous-time stochastic control.
VertexCBF is a framework that learns neural control barrier functions (CBFs) by approximating the stationary Hamilton–Jacobi value function with a neural network trained through physics‑informed and sparsely supervised learning. It exploits control‑affine dynamics and a convex polytope control set to generate supervision points via GPU‑parallel vertex‑restricted tree search, ensuring the learned CBF never exceeds the specified constraint function. The method was evaluated on 15 systems, outperforming baselines by recovering larger safe sets, and demonstrated on a mobile robot that safely avoids pedestrians using a neural CBF trained with this approach.
arXiv:2608.22026v1 Announce Type: new Abstract: Accurate simulation of the long-time evolution of systems governed by partial differential equations (PDEs) is central to scientific computing. Among e...
arXiv:2507.06625v4 Announce Type: replace-cross Abstract: Model Predictive Control (MPC) enables reliable trajectory optimization under dynamics constraints, but often depends on accurate dynamics mo...
arXiv:2607. 16177v1 Announce Type: new Abstract: Reinforcement learning (RL) has recently emerged as a promising feedback control strategy for nonlinear and complex dynamical systems.
arXiv:2607. 14233v1 Announce Type: cross Abstract: Physics-informed neural networks (PINNs) have had a broad research impact in modeling domains governed by partial differential equations (PDE).
arXiv:2510. 21874v2 Announce Type: replace-cross Abstract: Unmanned aerial vehicles (UAVs) operating in dynamic wind fields must generate safe and energy-efficient trajectories under physical and environmental constraints.
The paper introduces the Physics-Informed Stochastic Configuration Machine (PI‑SCM), a backpropagation‑free neural network designed for solving nonlinear differential equations. By analytically evaluating local Jacobians, PI‑SCM linearizes the physical loss, enabling optimal weight determination through generalized linear least squares and avoiding iterative nonlinear optimization. The authors present a progressive algorithmic suite—PI‑SC‑I, PI‑SC‑II, and PI‑SC‑III—prove their universal approximation properties, and show through experiments that PI‑SCM achieves high‑fidelity predictions and parameter identification while accelerating training by orders of magnitude compared to standard PINNs.
arXiv:2609.36942v1 Announce Type: cross Abstract: Learning neural-network models of dynamical systems with safety guarantees is a fundamental requirement for their deployment in safety-critical setti...