arXiv:2605. 07116v2 Announce Type: replace Abstract: We analyze a neural semi-discrete method for high-dimensional first-order Hamilton-Jacobi-Bellman (HJB) equations with known or learned dynamics.
By Minseok Kim, Yeongjong Kim, Namkyeong Cho, Yeoneung Kim
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.
By Hee Jun Yang, Minjung Gim, Yeoneung Kim
arXiv:2606. 24999v1 Announce Type: new Abstract: High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging.
By Yanwei Jia, Du Ouyang, Huy\^en Pham, Xun Yu Zhou
arXiv:2607. 21644v1 Announce Type: new Abstract: We present a goal-agnostic control framework for partial differential equations (PDEs) built around a joint-embedding predictive architecture (JEPA).
By Jonathan Gallagher, Roberto Guglielmi
The paper presents a convergence framework for deep $V$‑learning over a finite horizon $H$, deriving explicit bounds on policy loss by decomposing the Bellman update error into six residuals. It shows how $L^s$ concentrability controls expected $L^1$ loss, quantifies the impact of shared sampling across horizon levels, and provides optimal and near‑optimal sample allocations for statistical error rates. The work also establishes sharp action‑gap bounds under a margin condition, transfers optimal‑gap results to frozen‑iterate gaps, and offers consistency guarantees for generative‑reset approximate‑ERM procedures with exact action scores.
By Yury Kolomeytsev
arXiv:2512. 19643v2 Announce Type: replace Abstract: Numerical simulation of time-dependent partial differential equations (PDEs) is central to scientific and engineering applications, but high-fidelity solvers are often prohibitively expensive for long-horizon or time-critical settings.
By Rajyasri Roy, Dibyajyoti Nayak, Somdatta Goswami